Hyperspectral Unmixing Overview: Geometrical, Statistical, and Sparse Regression-Based Approaches
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paper_title: Multiparameter Receiver Operating Characteristic Analysis for Signal Detection and Classification

paper_content:

Receiver operating characteristic (ROC) analysis is a widely used evaluation tool in signal processing and communications, and medical diagnosis for performance analysis. It utilizes 2-D curves plotted by detection rate (P D) against false alarm rate (P F) to assess effectiveness of a detector, sensor/device for detection. However, P D and P F are actually dependent parameters resulting from a more crucial but implicit parameter hidden in the ROC curves, threshold ? , which is determined by the cost of implementing a detector or sensor/device, except only the case that when the Bayes theory is used for detection, ? is completely determined by the Bayes cost. This paper extends the traditional ROC analysis for single-signal detection to detection and classification of multiple signals. It also explores relationships among the three parameters, P D, P F, and ? , and further develops a new concept of multiparameter ROC analysis, which uses 3-D ROC curves plotted by three parameters, P D, P F, and ?, to evaluate effectiveness of detection performance based on interrelationship among P D, P F, and ?, rather then only P D and P F used by 2-D ROC analysis. As a result of a 3-D ROC curve, three 2-D ROC curves can be also derived: the conventional 2-D ROC curve plotted by P D versus P F and two new 2-D ROC curves plotted based on P D versus ? and P F versus ?. In order to demonstrate the utility of 3-D ROC analysis, four applications are considered: hyperspectral target detection, medical diagnosis, chemical/biological agent detection, and biometric recognition.
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paper_title: Kernel Based Subspace Projection of Near Infrared Hyperspectral Images of Maize Kernels

paper_content:

In this paper we present an exploratory analysis of hyper-spectral 900-1700 nm images of maize kernels. The imaging device is a line scanning hyper spectral camera using a broadband NIR illumination. In order to explore the hyperspectral data we compare a series of subspace projection methods including principal component analysis and maximum autocorrelation factor analysis. The latter utilizes the fact that interesting phenomena in images exhibit spatial autocorrelation. However, linear projections often fail to grasp the underlying variability on the data. Therefore we propose to use so-called kernel version of the two afore-mentioned methods. The kernel methods implicitly transform the data to a higher dimensional space using non-linear transformations while retaining the computational complexity. Analysis on our data example illustrates that the proposed kernel maximum autocorrelation factor transform outperform the linear methods as well as kernel principal components in producing interesting projections of the data.
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paper_title: Earth system science related imaging spectroscopy - An assessment

paper_content:

The science of spectroscopy has existed for more than three centuries, and imaging spectroscopy for the Earth system for three decades. We first discuss the historical background of spectroscopy, followed by imaging spectroscopy, introducing a common definition for the latter. The relevance of imaging spectroscopy is then assessed using a comprehensive review of the cited literature. Instruments, technological advancements and (pre-)processing approaches are discussed to set the scene for application related advancements. We demonstrate these efforts using four examples that represent progress due to imaging spectroscopy, namely (i) bridging scaling gaps from molecules to ecosystems using coupled radiative transfer models (ii) assessing surface heterogeneity including clumping, (iii) physical based (inversion) modeling, and iv) assessing interaction of light with the Earth surface. Recent advances of imaging spectroscopy contributions to the Earth system sciences are discussed. We conclude by summarizing the achievements of thirty years of imaging spectroscopy and strongly recommend this community to increase its efforts to convince relevant stakeholders of the urgency to acquire the highest quality imaging spectrometer data for Earth observation from operational satellites capable of collecting consistent data for climatically-relevant periods of time.
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paper_title: Anomaly detection from hyperspectral imagery

paper_content:

We develop anomaly detectors, i.e., detectors that do not presuppose a signature model of one or more dimensions, for three clutter models: the local normal model, the global normal mixture model, and the global linear mixture model. The local normal model treats the neighborhood of a pixel as having a normal probability distribution. The normal mixture model considers the observation from each pixel as arising from one of several possible classes such that each class has a normal probability distribution. The linear mixture model considers each observation to be a linear combination of fixed spectra, known as endmembers, that are, or may be, associated with materials in the scene, and the coefficients, interpreted as fractional abundance, are constrained to be nonnegative and sum to one. We show how the generalized likelihood ratio test (GLRT) may be used to derive anomaly detectors for the local normal and global normal mixture models. The anomaly detector applied with the linear mixture approach proceeds by identifying target like endmembers based on properties of the histogram of the abundance estimates and employing a matched filter in the space of abundance estimates. To overcome the limitations of the individual models, we develop a joint decision logic, based on a maximum entropy probability model and the GLRT, that utilizes multiple decision statistics, and we apply this approach using the detection statistics derived from the three clutter models. Examples demonstrate that the joint decision logic can improve detection performance in comparison with the individual anomaly detectors. We also describe the application of linear prediction filters to repeated images of the same area to detect changes that occur within the scene over time.
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paper_title: Detection and Analysis of the Intestinal Ischemia Using Visible and Invisible Hyperspectral Imaging

paper_content:

Intestinal ischemia, or inadequate blood flow to the intestine, is caused by a variety of disorders and conditions. The quickness with which the problem is brought to medical attention for diagnosis and treatment has great effects on the outcome of ischemic injury. Recently, hyperspectral sensors have advanced and emerged as compact imaging tools that can be utilized in medical diagnostics. Hyperspectral imaging provides a powerful tool for noninvasive tissue analyses. In this paper, the hyperspectral camera, with visible and invisible wavelengths, has been evaluated for detection and analysis of intestinal ischemia during surgeries. This technique can help the surgeon to quickly find ischemic tissues. Two cameras, a visible-to-near-infrared camera (400-1000 nm) and an infrared camera (900-1700 nm) were used to capture the hyperspectral images. Vessels supplying an intestinal segment of a pig were clamped to simulate ischemic conditions. A key wavelength range that provides the best differentiation between normal and ischemic intestine was determined from all wavelengths that potentially reduces the amount of data collected in subsequent work. The data were classified using two filters that were designed to discriminate the ischemic intestinal regions.
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paper_title: Quantitative determination of mineral types and abundances from reflectance spectra using principal components analysis

paper_content:

A procedure was developed for analyzing remote reflectance spectra, including multispectral images, that quantifies parameters such as types of mineral mixtures, the abundances of mixed minerals, and particle sizes. Principal components analysis (PCA) reduced the spectral dimensionality and allowed testing the uniqueness and validity of spectral mixing models. By analyzing variations in the overall spectral reflectance curves we identified the type of spectral mixture, quantified mineral abundances, and identified the effects of particle size. The results demonstrate an advantage in classification accuracy over classical forms of analysis that ignore effects of particle-size or mineral-mixture systematics on spectra. The approach is applicable to remote sensing data of planetary surfaces for quantitative determinations of mineral abundances.
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paper_title: Pharmaceutical applications of vibrational chemical imaging and chemometrics: a review.

paper_content:

The emergence of chemical imaging (CI) has gifted spectroscopy an additional dimension. Chemical imaging systems complement chemical identification by acquiring spatially located spectra that enable visualization of chemical compound distributions. Such techniques are highly relevant to pharmaceutics in that the distribution of excipients and active pharmaceutical ingredient informs not only a product's behavior during manufacture but also its physical attributes (dissolution properties, stability, etc.). The rapid image acquisition made possible by the emergence of focal plane array detectors, combined with publication of the Food and Drug Administration guidelines for process analytical technology in 2001, has heightened interest in the pharmaceutical applications of CI, notably as a tool for enhancing drug quality and understanding process. Papers on the pharmaceutical applications of CI have been appearing in steadily increasing numbers since 2000. The aim of the present paper is to give an overview of infrared, near-infrared and Raman imaging in pharmaceutics. Sections 2 and 3 deal with the theory, device set-ups, mode of acquisition and processing techniques used to extract information of interest. Section 4 addresses the pharmaceutical applications.
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paper_title: HYPERSPECTRAL REFLECTANCE AND FLUORESCENCE IMAGING SYSTEM FOR FOOD QUALITY AND SAFETY

paper_content:

This article presents a laboratory-based hyperspectral imaging system designed and developed by the Instrumentation and Sensing Laboratory, U.S. Department of Agriculture, Beltsville, Maryland. The spectral range is from 430 to 930 nm with spectral resolution of approximately 10 nm (full width at half maximum) and spatial resolution better than 1 mm. Our system is capable of reflectance and fluorescence measurements with the use of dual illumination sources where fluorescence emissions are measured with ultraviolet (UV-A) excitation. We present the calibrations and image-correction procedures for the system artifacts and heterogeneous responses caused by the optics, sensor, and lighting conditions throughout the spectrum region for reflectance and fluorescence. The results of the fluorescence correction method showed that the system responses throughout the spectrum region were normalized to within 0.5% error. The versatility of the hyperspectral imaging system was demonstrated with sample fluorescence and reflectance images of a normal apple and an apple with fungal contamination and bruised spots. The primary use of the imaging system in our laboratory is to conduct food safety and quality research. However, we envision that this unique system can be used in a number of scientific applications.
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paper_title: Spectral unmixing

paper_content:

Spectral unmixing using hyperspectral data represents a significant step in the evolution of remote decompositional analysis that began with multispectral sensing. It is a consequence of collecting data in greater and greater quantities and the desire to extract more detailed information about the material composition of surfaces. Linear mixing is the key assumption that has permitted well-known algorithms to be adapted to the unmixing problem. In fact, the resemblance of the linear mixing model to system models in other areas has permitted a significant legacy of algorithms from a wide range of applications to be adapted to unmixing. However, it is still unclear whether the assumption of linearity is sufficient to model the mixing process in every application of interest. It is clear, however, that the applicability of models and techniques is highly dependent on the variety of circumstances and factors that give rise to mixed pixels. The outputs of spectral unmixing, endmember, and abundance estimates are important for identifying the material composition of mixtures.
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paper_title: Blind Decomposition of Transmission Light Microscopic Hyperspectral Cube Using Sparse Representation

paper_content:

In this paper, we address the problem of fully automated decomposition of hyperspectral images for transmission light microscopy. The hyperspectral images are decomposed into spectrally homogeneous compounds. The resulting compounds are described by their spectral characteristics and optical density. We present the multiplicative physical model of image formation in transmission light microscopy, justify reduction of a hyperspectral image decomposition problem to a blind source separation problem, and provide method for hyperspectral restoration of separated compounds. In our approach, dimensionality reduction using principal component analysis (PCA) is followed by a blind source separation (BSS) algorithm. The BSS method is based on sparsifying transformation of observed images and relative Newton optimization procedure. The presented method was verified on hyperspectral images of biological tissues. The method was compared to the existing approach based on nonnegative matrix factorization. Experiments showed that the presented method is faster and better separates the biological compounds from imaging artifacts. The results obtained in this work may be used for improving automatic microscope hardware calibration and computer-aided diagnostics.
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paper_title: Detection algorithms for hyperspectral imaging applications

paper_content:

We introduce key concepts and issues including the effects of atmospheric propagation upon the data, spectral variability, mixed pixels, and the distinction between classification and detection algorithms. Detection algorithms for full pixel targets are developed using the likelihood ratio approach. Subpixel target detection, which is more challenging due to background interference, is pursued using both statistical and subspace models for the description of spectral variability. Finally, we provide some results which illustrate the performance of some detection algorithms using real hyperspectral imaging (HSI) data. Furthermore, we illustrate the potential deviation of HSI data from normality and point to some distributions that may serve in the development of algorithms with better or more robust performance. We therefore focus on detection algorithms that assume multivariate normal distribution models for HSI data.
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paper_title: Classification of Sound and Stained Wheat Grains Using Visible and near Infrared Hyperspectral Image Analysis

paper_content:

Near infrared hyperspectral image analysis has been used to classify individual wheat grains representing 24 different Australian varieties as sound or as being discoloured by one of the commercially important blackpoint, field fungi or pink stains. The study used a training set of 188 grains and a test set of 665 grains. The spectra were smoothed and then standardised by dividing each spectrum by its mean, so that the analysis was based solely on spectral shape. Penalised discriminant analysis was first used for pixel classification and then a simple rule for grain classification was developed. Overall classification accuracies of 95% were achieved over the 420–2500 nm wavelength range, as well as reduced ranges of 420–1000 nm and 420–700 nm.
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paper_title: The airborne visible/infrared imaging spectrometer (AVIRIS)

paper_content:

Abstract The Airborne Visible/Infrared Imaging Spectrometer (AVIRIS) is a facility consisting of a flight system, a ground data system, a calibration facility, and a full-time operations team. The facility was developed by the Jet Propulsion Laboratory (JPL) under funding from the National Aeronautics and Space Administration (NASA). NASA also provides funding for operations and maintenance. The flight system is a whisk-broom imager that acquires data in 224 narrow, contiguous spectral bands covering the solar reflected portion of the electromagnetic spectrum. It is flown aboard the NASA high altitude ER-2 research aircraft. The ground data system is a facility dedicated to the processing and distribution of data acquired by AVIRIS. It operates year round at JPL. The calibration facility consists of a calibration laboratory at JPL and a suite of field instruments and procedures for performing inflight calibration of AVIRIS. A small team of engineers, technicians and scientists supports a yearly operations schedule that includes 6 months of flight operations, 6 months of routine ground maintenance of the flight system, and year-round data processing and distribution. Details of the AVIRIS system, its performance history, and future plans are described.
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paper_title: NIR spectrometry for counterfeit drug detection: A feasibility study

paper_content:

Express-methods for detection of counterfeit drugs are of vital necessity. Visual control, dissociating tests or simple color reaction tests reveal only very rough forgeries. The feasibility of information-rich NIR-measurements as an analytical method together with multivariate calibration for mathematical data processing for false drugs detection is demonstrated. Also, multivariate hyperspectral image analysis is applied providing additional diagnostic information. Hyperspectral imaging is becoming a useful diagnostic tool for identifying non-homogeneous spatial regions of drug formulation. Two types of drugs are used to demonstrate the applicability of these approaches.
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paper_title: Hyperspectral imaging – an emerging process analytical tool for food quality and safety control

paper_content:

Hyperspectral imaging (HSI) is an emerging platform technology that integrates conventional imaging and spectroscopy to attain both spatial and spectral information from an object. Although HSI was originally developed for remote sensing, it has recently emerged as a powerful process analytical tool for non-destructive food analysis. This paper provides an introduction to hyperspectral imaging: HSI equipment, image acquisition and processing are described; current limitations and likely future applications are discussed. In addition, recent advances in the application of HSI to food safety and quality assessment are reviewed, such as contaminant detection, defect identification, constituent analysis and quality evaluation.
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paper_title: Imaging Spectroscopy and the Airborne Visible/Infrared Imaging Spectrometer (AVIRIS)

paper_content:

Abstract Imaging spectroscopy is of growing interest as a new approach to Earth remote sensing. The Airborne Visible/Infrared Imaging Spectrometer (AVIRIS) was the first imaging sensor to measure the solar reflected spectrum from 400 nm to 2500 nm at 10 nm intervals. The calibration accuracy and signal-to-noise of AVIRIS remain unique. The AVIRIS system as well as the science research and applications have evolved significantly in recent years. The initial design and upgraded characteristics of the AVIRIS system are described in terms of the sensor, calibration, data system, and flight operation. This update on the characteristics of AVIRIS provides the context for the science research and applications that use AVIRIS data acquired in the past several years. Recent science research and applications are reviewed spanning investigations of atmospheric correction, ecology and vegetation, geology and soils, inland and coastal waters, the atmosphere, snow and ice hydrology, biomass burning, environmental hazards, satellite simulation and calibration, commercial applications, spectral algorithms, human infrastructure, as well as spectral modeling.
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paper_title: Signal processing for hyperspectral image exploitation

paper_content:

Electro-optical remote sensing involves the acquisition of information about an object or scene without coming into physical contact with it. This is achieved by exploiting the fact that the materials comprising the various objects in a scene reflect, absorb, and emit electromagnetic radiation in ways characteristic of their molecular composition and shape. If the radiation arriving at the sensor is measured at each wavelength over a sufficiently broad spectral band, the resulting spectral signature, or simply spectrum, can be used (in principle) to uniquely characterize and identify any given material. An important function of hyperspectral signal processing is to eliminate the redundancy in the spectral and spatial sample data while preserving the high-quality features needed for detection, discrimination, and classification. This dimensionality reduction is implemented in a scene-dependent (adaptive) manner and may be implemented as a distinct step in the processing or as an integral part of the overall algorithm. The most widely used algorithm for dimensionality reduction is principal component analysis (PCA) or, equivalently, Karhunen-Loeve transformation.
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paper_title: Recent Advances in Techniques for Hyperspectral Image Processing

paper_content:

Imaging spectroscopy, also known as hyperspectral imaging, has been transformed in less than thirty years from being a sparse research tool into a commodity product available to a broad user community. Currently, there is a need for standardized data processing techniques able to take into account the special properties of hyperspec- tral data. In this paper, we provide a seminal view on recent advances in techniques for hyperspectral image processing. Our main focus is on the design of techniques able to deal with the high-dimensional nature of the data, and to integrate the spa- tial and spectral information. Performance of the discussed techniques is evaluated in different analysis scenarios. To satisfy time-critical constraints in specific appli- cations, we also develop efficient parallel implementations of some of the discussed algorithms. Combined, these parts provide an excellent snapshot of the state-of- the-art in those areas, and offer a thoughtful perspective on future potentials and emerging challenges in the design of robust hyperspectral imaging algorithms.
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paper_title: Hyperspectral image data analysis

paper_content:

The fundamental basis for space-based remote sensing is that information is potentially available from the electromagnetic energy field arising from the Earth's surface and, in particular, from the spatial, spectral, and temporal variations in that field. Rather than focusing on the spatial variations, which imagery perhaps best conveys, why not move on to look at how the spectral variations might be used. The idea was to enlarge the size of a pixel until it includes an area that is characteristic from a spectral response standpoint for the surface cover to be discriminated. The article includes an example of an image space representation, using three bands to simulate a color IR photograph of an airborne hyperspectral data set over the Washington, DC, mall.
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paper_title: Fuzzy Spectral and Spatial Feature Integration for Classification of Nonferrous Materials in Hyperspectral Data

paper_content:

Hyperspectral data allows the construction of more elaborate models to sample the properties of the nonferrous materials than the standard RGB color representation. In this paper, the nonferrous waste materials are studied as they cannot be sorted by classical procedures due to their color, weight and shape similarities. The experimental results presented in this paper reveal that factors such as the various levels of oxidization of the waste materials and the slight differences in their chemical composition preclude the use of the spectral features in a simplistic manner for robust material classification. To address these problems, the proposed FUSSER (fuzzy spectral and spatial classifier) algorithm detailed in this paper merges the spectral and spatial features to obtain a combined feature vector that is able to better sample the properties of the nonferrous materials than the single pixel spectral features when applied to the construction of multivariate Gaussian distributions. This approach allows the implementation of statistical region merging techniques in order to increase the performance of the classification process. To achieve an efficient implementation, the dimensionality of the hyperspectral data is reduced by constructing bio-inspired spectral fuzzy sets that minimize the amount of redundant information contained in adjacent hyperspectral bands. The experimental results indicate that the proposed algorithm increased the overall classification rate from 44% using RGB data up to 98% when the spectral-spatial features are used for nonferrous material classification.
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paper_title: Spectral mixture modeling - A new analysis of rock and soil types at the Viking Lander 1 site. [on Mars]

paper_content:

A Viking Lander 1 image was modeled as mixtures of reflectance spectra of palagonite dust, gray andesitelike rock, and a coarse rocklike soil. The rocks are covered to varying degrees by dust but otherwise appear unweathered. Rocklike soil occurs as lag deposits in deflation zones around stones and on top of a drift and as a layer in a trench dug by the lander. This soil probably is derived from the rocks by wind abrasion and/or spallation. Dust is the major component of the soil and covers most of the surface. The dust is unrelated spectrally to the rock but is equivalent to the global-scale dust observed telescopically. A new method was developed to model a multispectral image as mixtures of end-member spectra and to compare image spectra directly with laboratory reference spectra. The method for the first time uses shade and secondary illumination effects as spectral end-members; thus the effects of topography and illumination on all scales can be isolated or removed. The image was calibrated absolutely from the laboratory spectra, in close agreement with direct calibrations. The method has broad applications to interpreting multispectral images, including satellite images.
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paper_title: Forensic analysis of bioagents by X-ray and TOF-SIMS hyperspectral imaging.

paper_content:

Hyperspectral imaging combined with multivariate statistics is an approach to microanalysis that makes the maximum use of the large amount of data potentially collected in forensics analysis. This study examines the efficacy of using hyperspectral imaging-enabled microscopies to identify chemical signatures in simulated bioagent materials. This approach allowed for the ready discrimination between all samples in the test. In particular, the hyperspectral imaging approach allowed for the identification of particles with trace elements that would have been missed with a more traditional approach to forensic microanalysis. The importance of combining signals from multiple length scales and analytical sensitivities is discussed.
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paper_title: Algorithm taxonomy for hyperspectral unmixing

paper_content:

In this paper, we introduce a set of taxonomies that hierarchically organize and specify algorithms associated with hyperspectral unmixing. Our motivation is to collectively organize and relate algorithms in order to assess the current state-of-the-art in the field and to facilitate objective comparisons between methods. The hyperspectral sensing community is populated by investigators with disparate scientific backgrounds and, speaking in their respective languages, efforts in spectral unmixing developed within disparate communities have inevitably led to duplication. We hope our analysis removes this ambiguity and redundancy by using a standard vocabulary, and that the presentation we provide clearly summarizes what has and has not been done. As we shall see, the framework for the taxonomies derives its organization from the fundamental, philosophical assumptions imposed on the problem, rather than the common calculations they perform, or the similar outputs they might yield.
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paper_title: Optimal linear spectral unmixing

paper_content:

The optimal estimate of ground cover components of a linearly mixed spectral pixel in remote-sensing imagery is investigated. The problem is formulated as two consecutive constrained least-squares (LS) problems: the first problem concerns the estimation of the end-member spectra (EMS), and the second concerns the estimate, within each mixed pixel, of ground cover class proportions (CCPs) given the estimated EMS. For the EMS estimation problem, the authors propose a total least-squares (TLS) solution as an alternative to the conventional LS approach. The authors pose the CCP estimation problem as a constrained LS optimization problem. Then, they solve for exact solution using a quadratic programming (QP) method, as opposed to the Lagrange multiplier (LM)-based approximated solution proposed by Settle and Drake (1993). Preliminary computer experiments indicated that the TLS-estimated EMS always leads to better estimates of CCP than that of the LS-estimated EMS.
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paper_title: Survey of geometric and statistical unmixing algorithms for hyperspectral images

paper_content:

Spectral mixture analysis (also called spectral unmixing) has been an alluring exploitation goal since the earliest days of imaging spectroscopy. No matter the spatial resolution, the spectral signatures collected in natural environments are invariably a mixture of the signatures of the various materials found within the spatial extent of the ground instantaneous field view of the imaging instrument. In this paper, we give a comprehensive enumeration of the unmixing methods used in practice, because of their implementation in widely used software packages, and those published in the literature. We have structured the review according to the basic computational approach followed by the algorithms, with particular attention to those based on the computational geometry formulation, and statistical approaches with a probabilistic foundation. The quantitative assessment of some available techniques in both categories provides an opportunity to review recent advances and to anticipate future developments.
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paper_title: Reflectance spectroscopy: Quantitative analysis techniques for remote sensing applications

paper_content:

Several methods for the analysis of remotely sensed reflectance data are compared, including empirical methods and scattering theories, both of which are important for solving remote sensing problems. The concept of the photon mean optical path length and the implications for use in modeling reflectance spectra are presented. It is shown that the mean optical path length in a particulate surface is in rough inverse proportion to the square root of the absorption coefficient. Thus, the stronger absorber a material is, the less photons will penetrate into the surface. The concept of apparent absorbance (-In reflectance) is presented, and it is shown that absorption bands, which are Gaussian in shape when plotted as absorption coefficient (true absorbance) versus photon energy, are also Gaussians in apparent absorbance. However, the Gaussians in apparent absorbance have a smaller intensity and a width which is a factor of √2 larger. An apparent continuum in a reflectance spectrum is modeled as a mathematical function used to isolate a particular absorption feature for analysis. It is shown that a continuum should be removed by dividing it into the reflectance spectrum or subtracting it from the apparent absorbance and that the fitting of Gaussians to absorption features should be done using apparent absorbance versus photon energy. Kubelka-Munk theory is only valid for materials with small total absorption and for bihemispherical reflectance, which are rarely encountered in geologic remote sensing. It is shown that the recently advocated bidirectional reflectance theories have the potential for use in deriving mineral abundance from a reflectance spectrum.
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paper_title: Hyperspectral image classification and dimensionality reduction: an orthogonal subspace projection approach

paper_content:

Most applications of hyperspectral imagery require processing techniques which achieve two fundamental goals: 1) detect and classify the constituent materials for each pixel in the scene; 2) reduce the data volume/dimensionality, without loss of critical information, so that it can be processed efficiently and assimilated by a human analyst. The authors describe a technique which simultaneously reduces the data dimensionality, suppresses undesired or interfering spectral signatures, and detects the presence of a spectral signature of interest. The basic concept is to project each pixel vector onto a subspace which is orthogonal to the undesired signatures. This operation is an optimal interference suppression process in the least squares sense. Once the interfering signatures have been nulled, projecting the residual onto the signature of interest maximizes the signal-to-noise ratio and results in a single component image that represents a classification for the signature of interest. The orthogonal subspace projection (OSP) operator can be extended to k-signatures of interest, thus reducing the dimensionality of k and classifying the hyperspectral image simultaneously. The approach is applicable to both spectrally pure as well as mixed pixels. >
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paper_title: Least squares subspace projection approach to mixed pixel classification for hyperspectral images

paper_content:

An orthogonal subspace projection (OSP) method using linear mixture modeling was recently explored in hyperspectral image classification and has shown promise in signature detection, discrimination, and classification. In this paper, the OSP is revisited and extended by three unconstrained least squares subspace projection approaches, called signature space OSP, target signature space OSP, and oblique subspace projection, where the abundances of spectral signatures are not known a priori but need to be estimated, a situation to which the OSP cannot be directly applied. The proposed three subspace projection methods can be used not only to estimate signature abundance, but also to classify a target signature at subpixel scale so as to achieve subpixel detection. As a result, they can be viewed as a posteriori OSP as opposed to OSP, which can be thought of as a priori OSP. In order to evaluate these three approaches, their associated least squares estimation errors are cast as a signal detection problem ill the framework of the Neyman-Pearson detection theory so that the effectiveness of their generated classifiers can be measured by receiver operating characteristics (ROC) analysis. All results are demonstrated by computer simulations and Airborne Visible/Infrared Imaging Spectrometer (AVIRIS) data.
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paper_title: A quantitative and comparative analysis of endmember extraction algorithms from hyperspectral data

paper_content:

Linear spectral unmixing is a commonly accepted approach to mixed-pixel classification in hyperspectral imagery. This approach involves two steps. First, to find spectrally unique signatures of pure ground components, usually known as endmembers, and, second, to express mixed pixels as linear combinations of endmember materials. Over the past years, several algorithms have been developed for autonomous and supervised endmember extraction from hyperspectral data. Due to a lack of commonly accepted data and quantitative approaches to substantiate new algorithms, available methods have not been rigorously compared by using a unified scheme. In this paper, we present a comparative study of standard endmember extraction algorithms using a custom-designed quantitative and comparative framework that involves both the spectral and spatial information. The algorithms considered in this study represent substantially different design choices. A database formed by simulated and real hyperspectral data collected by the Airborne Visible and Infrared Imaging Spectrometer (AVIRIS) is used to investigate the impact of noise, mixture complexity, and use of radiance/reflectance data on algorithm performance. The results obtained indicate that endmember selection and subsequent mixed-pixel interpretation by a linear mixture model are more successful when methods combining spatial and spectral information are applied.
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paper_title: Spectral unmixing

paper_content:

Spectral unmixing using hyperspectral data represents a significant step in the evolution of remote decompositional analysis that began with multispectral sensing. It is a consequence of collecting data in greater and greater quantities and the desire to extract more detailed information about the material composition of surfaces. Linear mixing is the key assumption that has permitted well-known algorithms to be adapted to the unmixing problem. In fact, the resemblance of the linear mixing model to system models in other areas has permitted a significant legacy of algorithms from a wide range of applications to be adapted to unmixing. However, it is still unclear whether the assumption of linearity is sufficient to model the mixing process in every application of interest. It is clear, however, that the applicability of models and techniques is highly dependent on the variety of circumstances and factors that give rise to mixed pixels. The outputs of spectral unmixing, endmember, and abundance estimates are important for identifying the material composition of mixtures.
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paper_title: Mixed pixels classification

paper_content:

There are two major approaches in spectral unmixing: linear and non-linear ones. They are appropriate for different types of mixture, namely checkerboard mixtures and intimate mixtures. The two approaches are briefly reviewed. Then in a carefully controlled laboratory experiment, the limitations and applicability of two of the methods (a linear and a non- linear one) are compared, in the context of unmixing an intimate mixture.© (1998) COPYRIGHT SPIE--The International Society for Optical Engineering. Downloading of the abstract is permitted for personal use only.
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paper_title: Hyperspectral unmixing: geometrical, statistical, and sparse regression-based approaches

paper_content:

Hyperspectral instruments acquire electromagnetic energy scattered within their ground instantaneous field view ::: in hundreds of spectral channels with high spectral resolution. Very often, however, owing to low spatial resolution ::: of the scanner or to the presence of intimate mixtures (mixing of the materials at a very small scale) in the scene, ::: the spectral vectors (collection of signals acquired at different spectral bands from a given pixel) acquired by the ::: hyperspectral scanners are actually mixtures of the spectral signatures of the materials present in the scene. ::: Given a set of mixed spectral vectors, spectral mixture analysis (or spectral unmixing) aims at estimating the ::: number of reference materials, also called endmembers, their spectral signatures, and their fractional abundances. ::: Spectral unmixing is, thus, a source separation problem where, under a linear mixing model, the sources are the ::: fractional abundances and the endmember spectral signatures are the columns of the mixing matrix. As such, ::: the independent component analysis (ICA) framework came naturally to mind to unmix spectral data. However, ::: the ICA crux assumption of source statistical independence is not satisfied in spectral applications, since the ::: sources are fractions and, thus, non-negative and sum to one. As a consequence, ICA-based algorithms have ::: severe limitations in the area of spectral unmixing, and this has fostered new unmixing research directions taking ::: into account geometric and statistical characteristics of hyperspectral sources. ::: This paper presents an overview of the principal research directions in hyperspectral unmixing. The presentations ::: is organized into four main topics: i) mixing models, ii) signal subspace identification, iii) geometrical-based ::: spectral unmixing, (iv) statistical-based spectral unmixing, and (v) sparse regression-based unmixing. In each ::: topic, we describe what physical or mathematical problems are involved and summarize state-of-the-art algorithms ::: to address these problems.
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paper_title: Image processing software for imaging spectrometry data analysis

paper_content:

Abstract The advent of a new generation of remote sensing instruments, called imaging spectrometers, promises to provide scientists a greatly enhanced capability for detailed observations of the earth's surface. These instruments collect image data in literally hundreds of spectral channels simultaneously from the near ultraviolet through the short wavelength infrared, and are capable in many cases of providing direct surface materials identification in a manner similar to that used in laboratory reflectance spectroscopy. The volume and complexity of data produced by these instruments offers a significant challenge to traditional multispectral image analysis methods, and in fact requires the development of new approaches to efficiently manage and analyze these data sets. This paper describes a software system specifically designed to provide the science user with a powerful set of tools for carrying out exploratory analysis of imaging spectrometer data utilizing only modest computational resources.
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paper_title: Theory of Reflectance and Emittance Spectroscopy

paper_content:

Acknowledgements 1. Introduction 2. Electromagnetic wave propagation 3. The absorption of light 4. Specular reflection 5. Single particle scattering: perfect spheres 6. Single particle scattering: irregular particles 7. Propagation in a nonuniform medium: the equation of radiative transfer 8. The bidirectional reflectance of a semi-infinite medium 9. The opposition effect 10. A miscellany of bidirectional reflectances and related quantities 11. Integrated reflectances and planetary photometry 12. Photometric effects of large scale roughness 13. Polarization 14. Reflectance spectroscopy 15. Thermal emission and emittance spectroscopy 16. Simultaneous transport of energy by radiation and conduction Appendix A. A brief review of vector calculus Appendix B. Functions of a complex variable Appendix C. The wave equation in spherical coordinates Appendix D. Fraunhoffer diffraction by a circular hole Appendix E. Table of symbols Bibliography Index.
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paper_title: On the relationship between spectral unmixing and subspace projection

paper_content:

A linear transformation was recently recommended for application on hyperspectral imagery. This note shows that the method is completely equivalent to extracting proportional ground cover by standard means, but is less efficient than the more usual methods of spectral unmixing.
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paper_title: Nonlinear spectral mixing models for vegetative and soil surfaces

paper_content:

Abstract In this article we apply an analytical solution of the radiosity equation to compute vegetation indices, reflectance spectra, and the spectral bidirectional reflectance distribution function for simple canopy geometries. We show that nonlinear spectral mixing occurs due to multiple reflection and transmission from surfaces. We compare radiosity-derived spectra with single scattering or linear mixing models. We also develop a simple model to predict the reflectance spectrum of binary and ternary mineral mixtures of faceted surfaces. The two facet model is validated by measurements of the reflectance.
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paper_title: Confidence in linear spectral unmixing of single pixels

paper_content:

The authors propose a method that estimates the membership of a mixed pixel in various possible mixture proportions. The method relies on the creation of model mixtures and the estimation of their local density in the vicinity of the pixel under consideration.
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paper_title: Spectral Mixture Analysis of Hyperspectral Scenes Using Intelligently Selected Training Samples

paper_content:

In this letter, we address the use of artificial neural networks for spectral mixture analysis of hyperspectral scenes. We specifically focus on the issue of how to effectively train neural network architectures in the context of spectral mixture analysis applications. To address this issue, a multilayer perceptron neural architecture is combined with techniques for intelligent selection and labeling of training samples directly obtained from the input data, thus maximizing the information that can be obtained from those samples while reducing the need for a priori information about the scene. The proposed approach is compared to unconstrained and fully constrained linear mixture models using hyperspectral data sets acquired (in the laboratory) from artificial forest scenes, using the compact airborne spectrographic imaging system. The Spreading of Photons for Radiation INTerception (SPRINT) canopy model, which assumes detailed knowledge about object geometry, was employed to evaluate the results obtained by the different methods. Our results show that the proposed approach, when trained with both pure and mixed training samples (generated automatically without prior information) can provide similar results to those provided by SPRINT, using very few labeled training samples. An application to real airborne data using a set of hyperspectral images collected at different altitudes by the digital airborne imaging spectrometer 7915 and the reflective optics system imaging spectrometer, operating simultaneously at multiple spatial resolutions, is also presented and discussed.
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paper_title: The Kubelka-Munk Diffuse Reflectance Formula Revisited

paper_content:

Abstract We use an integral equation approach to finding the Kubelka-Munk (KM) diffuse reflectance formula and extend the result by finding the apparent path length and total intensity distribution inside an infinite, homogeneous, diffusely reflecting medium with isotropic scattering. We then expand the approach to three dimensions to show that the KM formula is correct for total diffuse reflectance when scattering, excitation, and detection are all isotropic. We obtain simple and exact results for the angular distribution of diffuse reflection and for the total diffuse reflectance when the incident light has an isotropic angular distribution, when it strikes at a single angle of elevation, and when it has the steady-state angular distribution. This work includes some results that employ Chandrasekhar's H function, so we also provide a program for the rapid evaluation of H. “[Supplementary material is available for this article. Go to the publisher's online edition of Applied Spectroscopy Reviews for the ...
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paper_title: N-FINDR: an algorithm for fast autonomous spectral end-member determination in hyperspectral data

paper_content:

The analysis of hyperspectral data sets requires the determination of certain basis spectra called 'end-members.' Once these spectra are found, the image cube can be 'unmixed' into the fractional abundance of each material in each pixel. There exist several techniques for accomplishing the determination of the end-members, most of which involve the intervention of a trained geologist. Often these-end-members are assumed to be present in the image, in the form of pure, or unmixed, pixels. In this paper a method based upon the geometry of convex sets is proposed to find a unique set of purest pixels in an image. The technique is based on the fact that in N spectral dimensions, the N-volume contained by a simplex formed of the purest pixels is larger than any other volume formed from any other combination of pixels. The algorithm works by 'inflating' a simplex inside the data, beginning with a random set of pixels. For each pixel and each end-member, the end-member is replaced with the spectrum of the pixel and the volume is recalculated. If it increases, the spectrum of the new pixel replaces that end-member. This procedure is repeated until no more replacements are done. This algorithm successfully derives end-members in a synthetic data set, and appears robust with less than perfect data. Spectral end-members have been extracted for the AVIRIS Cuprite data set which closely match reference spectra, and resulting abundance maps match published mineral maps.
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paper_title: Linear Versus Nonlinear PCA for the Classification of Hyperspectral Data Based on the Extended Morphological Profiles

paper_content:

Morphological profiles (MPs) have been proposed in recent literature as aiding tools to achieve better results for classification of remotely sensed data. MPs are in general built using features containing most of the information content of the data, such as the components derived from principal component analysis (PCA). Recently, nonlinear PCA (NLPCA), performed by autoassociative neural network, has emerged as a good unsupervised technique to fit the information content of hyperspectral data into few components. The aim of this letter is to investigate the classification accuracies obtained using extended MPs built from the features of NPCA. A comparison of the two approaches has been validated on two different data sets having different spatial and spectral resolutions/coverages, over the same ground truth, and also using two different classification algorithms. The results show that NLPCA permits one to obtain better classification accuracies than using linear PCA.
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paper_title: A comparison of kernel functions for intimate mixture models

paper_content:

In previous work, kernel methods were introduced as a way to generalize the linear mixing model. This work led to a new set of algorithms that performed the unmixing of hyperspectral imagery in a reproducing kernel Hilbert space. By processing the imagery in this space different types of unmixing could be introduced - including an approximation of intimate mixtures. Whereas previous research focused on developing the mathematical foundation for kernel unmixing, this paper focuses on the selection of the kernel function. Experiments are conducted on real-world hyperspectral data using a linear, a radial-basis function, a polynomial, and a proposed physicsbased kernel. Results show which kernels provide the best ability to perform intimate unmixing.
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paper_title: Mapping intimate mixtures using an adaptive kernel-based technique

paper_content:

In previous work, kernel methods were introduced as a way to generalize the linear mixing model for hyperspectral data. This work led to a new adaptive kernel unmixing method that both identified and unmixed linearly and intimately mixed pixels. However, the results from this previous research was limited to lab-based data where the endmembers were known a-priori and atmospheric effects were absent. This paper documents the results of the adaptive kernel-based unmixing techniques on real-world hyperspectral data collected over Smith Island, Virginia, USA. The results show that the adaptive kernel unmixing method can readily identify where nonlinear mixtures exist in the image even when perfect knowledge of the endmembers and the reflectance cannot be known.
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paper_title: On the use of small training sets for neural network-based characterization of mixed pixels in remotely sensed hyperspectral images

paper_content:

In this work, neural network-based models involved in hyperspectral image spectra separation are considered. Focus is on how to select the most highly informative samples for effectively training the neural architecture. This issue is addressed here by several new algorithms for intelligent selection of training samples: (1) a border-training algorithm (BTA) which selects training samples located in the vicinity of the hyperplanes that can optimally separate the classes; (2) a mixed-signature algorithm (MSA) which selects the most spectrally mixed pixels in the hyperspectral data as training samples; and (3) a morphological-erosion algorithm (MEA) which incorporates spatial information (via mathematical morphology concepts) to select spectrally mixed training samples located in spatially homogeneous regions. These algorithms, along with other standard techniques based on orthogonal projections and a simple Maximin-distance algorithm, are used to train a multi-layer perceptron (MLP), selected in this work as a representative neural architecture for spectral mixture analysis. Experimental results are provided using both a database of nonlinear mixed spectra with absolute ground truth and a set of real hyperspectral images, collected at different altitudes by the digital airborne imaging spectrometer (DAIS 7915) and reflective optics system imaging spectrometer (ROSIS) operating simultaneously at multiple spatial resolutions.
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paper_title: Pixel Unmixing in Hyperspectral Data by Means of Neural Networks

paper_content:

Neural networks (NNs) are recognized as very effective techniques when facing complex retrieval tasks in remote sensing. In this paper, the potential of NNs has been applied in solving the unmixing problem in hyperspectral data. In its complete form, the processing scheme uses an NN architecture consisting of two stages: the first stage reduces the dimension of the input vector, while the second stage performs the mapping from the reduced input vector to the abundance percentages. The dimensionality reduction is performed by the so-called autoassociative NNs, which yield a nonlinear principal component analysis of the data. The evaluation of the whole performance is carried out for different sets of experimental data. The first one is provided by the Airborne Hyperspectral Scanner. The second set consists of images from the Compact High-Resolution Imaging Spectrometer on board the Project for On-Board Autonomy satellite, and it includes multiangle and multitemporal acquisitions. The third set is represented by Airborne Visible/InfraRed Imaging Spectrometer measurements. A quantitative performance analysis has been carried out in terms of effectiveness in the dimensionality reduction phase and in terms of the accuracy in the final estimation. The results obtained, when compared with those produced by appropriate benchmark techniques, show the advantages of this approach.
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paper_title: A generalized kernel for areal and intimate mixtures

paper_content:

In previous work, kernel methods were introduced as a way to generalize the linear mixing model for hyperspectral data. This work led to a new physics-based kernel that allowed accurate unmixing of intimate mixtures. Unfortunately, the new physics-based kernel did not perform well on linear mixtures; thus, different kernels had to be used for different mixtures. Ideally, a single unified kernel that can perform both unmixing of areal and intimate mixtures would be desirable. This paper presents such a kernel that can automatically identify the underlying mixture type from the data and perform the correct unmixing method. Results on real-world, ground-truthed intimate and linear mixtures demonstrate the ability of this new data-driven kernel to perform generalized unmixing of hyperspectral data.
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paper_title: Bilinear models for nonlinear unmixing of hyperspectral images

paper_content:

This paper compares several nonlinear models recently introduced for hyperspectral image unmixing. All these models consist of bilinear models that have shown interesting properties for hyperspectral images subjected to multipath effects. The first part of this paper presents different algorithms allowing the parameters of these models to be estimated. The relevance and flexibility of these models for spectral unmixing are then investigated by comparing the reconstruction errors and spectral angle mappers computed from synthetic and real dataset. This kind of study is important to determine which mixture model should be used in practical applications for hyper-spectral image unmixing.
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paper_title: A Model of Spectral Albedo of Particulate Surfaces: Implications for Optical Properties of the Moon

paper_content:

Abstract A simple one-dimensional geometrical-optics model for spectral albedo of powdered surfaces, in particular of lunar regolith, is presented. As distinct from, e.g., the Kubelka–Munk formula, which deals with two effective parameters of a medium, the suggested model uses spectra of optical constants of the medium materials. Besides, our model is invertible, i.e., allows estimations of spectral absorption using albedo spectrum, ifa prioridata on the real part of refractive index and surface porosity are known. The model has been applied to interpret optical properties of the Moon. In particular, it has been shown that: (1) both color indices and depth of absorption bands for regolith-like surfaces depend on particle size, which should be taken into account when correlations between these optical characteristics and abundance of Fe and Ti in the lunar regolith are studied; (2) fine-grained reduced iron occurring in regolith particles affects band minima positions in reflectance spectra of lunar pyroxenes and, consequently, affects the result of determination of pyroxene types and Fe abundance by Adams' method.
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paper_title: A quantitative and comparative analysis of linear and nonlinear spectral mixture models using radial basis function neural networks

paper_content:

A radial basis function neural network (RBFNN) is developed to examine two mixing models, linear and nonlinear spectral mixtures, which describe the spectra collected by both airborne and laboratory-based spectrometers. The authors examine the possibility that there may be naturally occurring situations where the typically used linear model may not provide the most accurate resultant spectral description. Under such a circumstance, a nonlinear model may better describe the mixing mechanism.
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paper_title: Spectral unmixing

paper_content:

Spectral unmixing using hyperspectral data represents a significant step in the evolution of remote decompositional analysis that began with multispectral sensing. It is a consequence of collecting data in greater and greater quantities and the desire to extract more detailed information about the material composition of surfaces. Linear mixing is the key assumption that has permitted well-known algorithms to be adapted to the unmixing problem. In fact, the resemblance of the linear mixing model to system models in other areas has permitted a significant legacy of algorithms from a wide range of applications to be adapted to unmixing. However, it is still unclear whether the assumption of linearity is sufficient to model the mixing process in every application of interest. It is clear, however, that the applicability of models and techniques is highly dependent on the variety of circumstances and factors that give rise to mixed pixels. The outputs of spectral unmixing, endmember, and abundance estimates are important for identifying the material composition of mixtures.
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paper_title: Calculation of Geodesic Distances in Nonlinear Mixing Models: Application to the Generalized Bilinear Model

paper_content:

Recently, several nonlinear techniques have been proposed in hyperspectral image processing for classification and unmixing applications. A popular data-driven approach for treating nonlinear problems employs the geodesic distances on the data manifold as property of interest. These geodesic distances are approximated by the shortest path distances in a nearest neighbor graph constructed in the data cloud. Although this approach often works well in practical applications, the graph-based approximation of these geodesic distances often fails to capture correctly the true nonlinear structure of the manifold, causing deviations in the subsequent algorithms. On the other hand, several model-based nonlinear techniques have been introduced as well and have the advantage that one can, in theory, calculate the geodesic distances analytically. In this letter, we demonstrate how one can calculate the true geodesics, and their lengths, on any manifold induced by a nonlinear hyperspectral mixing model. We introduce the required techniques from differential geometry, show how the constraints on the abundances can be integrated in these techniques, and present a numerical method for finding a solution of the geodesic equations. We demonstrate this technique on the recently developed generalized bilinear model, which is a flexible model for the nonlinearities introduced by secondary reflections. As an application of the technique, we demonstrate that multidimensional scaling applied to these geodesic distances can be used as a preprocessing step to linear unmixing, yielding better unmixing results on nonlinear data when compared to principal component analysis and outperforming ISOMAP.
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paper_title: Nonlinear unmixing of hyperspectral images using radial basis functions and orthogonal least squares

paper_content:

This paper studies a linear radial basis function network (RBFN) for unmixing hyperspectral images. The proposed RBFN assumes that the observed pixel reflectances are nonlinear mixtures of known end-members (extracted from a spectral library or estimated with an end-member extraction algorithm), with unknown proportions (usually referred to as abundances). We propose to estimate the model abundances using a linear combination of radial basis functions whose weights are estimated using training samples. The main contribution of this paper is to study an orthogonal least squares algorithm which allows the number of RBFN centers involved in the abundance estimation to be significantly reduced. The resulting abundance estimator is combined with a fully constrained estimation procedure ensuring positivity and sum-to-one constraints for the abundances. The performance of the nonlinear unmixing strategy is evaluated with simulations conducted on synthetic and real data.
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paper_title: Kernel fully constrained least squares abundance estimates

paper_content:

A critical step for fitting a linear mixing model to hyperspectral imagery is the estimation of the abundances. The abundances are the percentage of each end member within a given pixel; therefore, they should be non-negative and sum to one. With the advent of kernel based algorithms for hyperspectral imagery, kernel based abundance estimates have become necessary. This paper presents such an algorithm that estimates the abundances in the kernel feature space while maintaining the non-negativity and sum-to-one constraints. The usefulness of the algorithm is shown using the AVIRIS Cuprite, Nevada image.
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paper_title: Quantitative abundance estimates from bidirectional reflectance measurements

paper_content:

A simplified approach to the problem of determining the relative proportion of minerals in a mixture from a reflectance spectrum of the mixture is presented. Fundamental to this approach is a priori information concerning reflectance spectra of the minerals in the mixture and some estimate of the particle sizes of the mixture components. Reflectance spectra of intimate mixtures are a systematic but nonlinear combination of the spectra of the minerals in the mixtures. Equations for bidirectional reflectance are used to linearize the systematics of spectral mixing for intimate mixtures. The equations are simplified by assuming that particulate media scatter light isotropically at phase angles between 15 o and 40 o. This method for linearizing the mixing systematics is used to determine mineral relative geometric cross sections (proportional to mass fraction/density x particle diameter) from reflectance spectra of mixtures of igneous rock-forming minerals (oilvine, magnetite, enstatite, and anorthite) and to determine endmember relative geometric cross sections from reflectance spectra of mixtures of terrestrial desert soils. Since particle diameters are known, the mass fractions of the mixture components are also calculated. For materials without strongly absorbing components, the accuracy of abundance determinations is better than 5%. The results indicate that the method presented can be used to accurately determine the relative proportions of components (minerals or complex endmembers) in a mixture from a reflectance spectrum of the mixture given information of the endmembers in the mixture, reflectance spectra of the endmembers, and an estimate of particle sizes of the respective mixture components.
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paper_title: Nonlinear Hyperspectral Mixture Analysis for tree cover estimates in orchards

paper_content:

Abstract Accurate monitoring of spatial and temporal variation in tree cover provides essential information for steering management practices in orchards. In this light, the present study investigates the potential of Hyperspectral Mixture Analysis. Specific focus lies on a thorough study of non-linear mixing effects caused by multiple photon scattering. In a series of experiments the importance of multiple scattering is demonstrated while a novel conceptual Nonlinear Spectral Mixture Analysis approach is presented and successfully tested on in situ measured mixed pixels in Citrus sinensis L. orchards. The rationale behind the approach is the redistribution of nonlinear fractions (i.e., virtual fractions) among the actual physical ground cover entities (e.g., tree, soil). These ‘virtual’ fractions, which account for the extent and nature of multiple photon scattering only have a physical meaning at the spectral level but cannot be interpreted as an actual physical part of the ground cover. Results illustrate that the effect of multiple scattering on Spectral Mixture Analysis is significant as the linear approach provides a mean relative root mean square error (RMSE) for tree cover fraction estimates of 27%. While traditional nonlinear approaches only slightly reduce this error (RMSE = 23%), important improvements are obtained for the novel Nonlinear Spectral Mixture Analysis approach (RMSE = 12%).
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paper_title: Nonlinear unmixing of hyperspectral images using a generalized bilinear model

paper_content:

This paper studies a generalized bilinear model and a hierarchical Bayesian algorithm for unmixing hyperspectral images. The proposed model is a generalization of the accepted linear mixing model but also of a bilinear model recently introduced in the literature. Appropriate priors are chosen for its parameters in particular to satisfy the positivity and sum-to-one constraints for the abundances. The joint posterior distribution of the unknown parameter vector is then derived. A Metropolis-within-Gibbs algorithm is proposed which allows samples distributed according to the posterior of interest to be generated and to estimate the unknown model parameters. The performance of the resulting unmixing strategy is evaluated via simulations conducted on synthetic and real data.
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paper_title: Supervised Nonlinear Spectral Unmixing Using a Postnonlinear Mixing Model for Hyperspectral Imagery

paper_content:

This paper presents a nonlinear mixing model for hyperspectral image unmixing. The proposed model assumes that the pixel reflectances are nonlinear functions of pure spectral components contaminated by an additive white Gaussian noise. These nonlinear functions are approximated using polynomial functions leading to a polynomial postnonlinear mixing model. A Bayesian algorithm and optimization methods are proposed to estimate the parameters involved in the model. The performance of the unmixing strategies is evaluated by simulations conducted on synthetic and real data.
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paper_title: Nonlinear Spectral Mixture Analysis for Hyperspectral Imagery in an Unknown Environment

paper_content:

Nonlinear spectral mixture analysis for hyperspectral imagery is investigated without prior information about the image scene. A simple but effective nonlinear mixture model is adopted, where the multiplication of each pair of endmembers results in a virtual endmember representing multiple scattering effect during pixel construction process. The analysis is followed by linear unmixing for abundance estimation. Due to a large number of nonlinear terms being added in an unknown environment, the following abundance estimation may contain some errors if most of the endmembers do not really participate in the mixture of a pixel. We take advantage of the developed endmember variable linear mixture model (EVLMM) to search the actual endmember set for each pixel, which yields more accurate abundance estimation in terms of smaller pixel reconstruction error, smaller residual counts, and more pixel abundances satisfying sum-to-one and nonnegativity constraints.
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paper_title: Comparative study between a new nonlinear model and common linear model for analysing laboratory simulated-forest hyperspectral data

paper_content:

The spectral unmixing of mixed pixels is a key factor in remote sensing images, especially for hyperspectral imagery. A commonly used approach to spectral unmixing has been linear unmixing. However, the question of whether linear or nonlinear processes dominate spectral signatures of mixed pixels is still an unresolved matter. In this study, we put forward a new nonlinear model for inferring end-member fractions within hyperspectral scenes. This study focuses on comparing the nonlinear model with a linear model. A detail comparative analysis of the fractions 'sunlit crown', 'sunlit background' and 'shadow' between the two methods was carried out through visualization, and comparing with supervised classification using a database of laboratory simulated-forest scenes. Our results show that the nonlinear model of spectral unmixing outperforms the linear model, especially in the scenes with translucent crown on a white background. A nonlinear mixture model is needed to account for the multiple scattering between tree crowns and background.
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paper_title: Modal mineralogy of planetary surfaces from visible and near-infrared spectral data

paper_content:

Real planetary surfaces are composed of several to many different minerals and ices. Deconvolving a reflectance spectrum to material abundance in an unambiguous way is difficult, because the spectra are complex nonlinear functions of grain size, abundance, and material opacity. Multiple scattering models can provide approximate solutions to the radiative transfer in a particulate medium. The paper examines the different approaches which deal with the theory of radiative transfer on atmosphereless bodies. We present the relative merits of two scattering theories based on the equivalent slab model: the extensively used Hapke theory [1] and the Shkuratov theory [2]. The performances of the two models for determining mineral abundance in multicomponent mixtures are also evaluated using laboratory data. Finally, one application on real planetary surfaces will be shown.
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paper_title: Nonlinear mixture model for hyperspectral unmixing

paper_content:

This paper addresses the problem of unmixing hyperspectral images, when the light suffers multiple interactions ::: among distinct endmembers. In these scenarios, linear unmixing has poor accuracy since the multiple light ::: scattering effects are not accounted for by the linear mixture model. ::: Herein, a nonlinear scenario composed by a single layer of vegetation above the soil is considered. For this ::: class of scene, the adopted mixing model, takes into account the second-order scattering interactions. Higher ::: order interactions are assumed negligible. A semi-supervised unmixing method is proposed and evaluated with ::: simulated and real hyperspectral data sets.
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paper_title: Robust uncertainty principles: exact signal reconstruction from highly incomplete frequency information

paper_content:

This paper considers the model problem of reconstructing an object from incomplete frequency samples. Consider a discrete-time signal f/spl isin/C/sup N/ and a randomly chosen set of frequencies /spl Omega/. Is it possible to reconstruct f from the partial knowledge of its Fourier coefficients on the set /spl Omega/? A typical result of this paper is as follows. Suppose that f is a superposition of |T| spikes f(t)=/spl sigma//sub /spl tau//spl isin/T/f(/spl tau/)/spl delta/(t-/spl tau/) obeying |T|/spl les/C/sub M//spl middot/(log N)/sup -1/ /spl middot/ |/spl Omega/| for some constant C/sub M/>0. We do not know the locations of the spikes nor their amplitudes. Then with probability at least 1-O(N/sup -M/), f can be reconstructed exactly as the solution to the /spl lscr//sub 1/ minimization problem. In short, exact recovery may be obtained by solving a convex optimization problem. We give numerical values for C/sub M/ which depend on the desired probability of success. Our result may be interpreted as a novel kind of nonlinear sampling theorem. In effect, it says that any signal made out of |T| spikes may be recovered by convex programming from almost every set of frequencies of size O(|T|/spl middot/logN). Moreover, this is nearly optimal in the sense that any method succeeding with probability 1-O(N/sup -M/) would in general require a number of frequency samples at least proportional to |T|/spl middot/logN. The methodology extends to a variety of other situations and higher dimensions. For example, we show how one can reconstruct a piecewise constant (one- or two-dimensional) object from incomplete frequency samples - provided that the number of jumps (discontinuities) obeys the condition above - by minimizing other convex functionals such as the total variation of f.
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paper_title: Compressed sensing

paper_content:

Suppose x is an unknown vector in Ropfm (a digital image or signal); we plan to measure n general linear functionals of x and then reconstruct. If x is known to be compressible by transform coding with a known transform, and we reconstruct via the nonlinear procedure defined here, the number of measurements n can be dramatically smaller than the size m. Thus, certain natural classes of images with m pixels need only n=O(m1/4log5/2(m)) nonadaptive nonpixel samples for faithful recovery, as opposed to the usual m pixel samples. More specifically, suppose x has a sparse representation in some orthonormal basis (e.g., wavelet, Fourier) or tight frame (e.g., curvelet, Gabor)-so the coefficients belong to an lscrp ball for 0<ples1. The N most important coefficients in that expansion allow reconstruction with lscr2 error O(N1/2-1p/). It is possible to design n=O(Nlog(m)) nonadaptive measurements allowing reconstruction with accuracy comparable to that attainable with direct knowledge of the N most important coefficients. Moreover, a good approximation to those N important coefficients is extracted from the n measurements by solving a linear program-Basis Pursuit in signal processing. The nonadaptive measurements have the character of "random" linear combinations of basis/frame elements. Our results use the notions of optimal recovery, of n-widths, and information-based complexity. We estimate the Gel'fand n-widths of lscrp balls in high-dimensional Euclidean space in the case 0<ples1, and give a criterion identifying near- optimal subspaces for Gel'fand n-widths. We show that "most" subspaces are near-optimal, and show that convex optimization (Basis Pursuit) is a near-optimal way to extract information derived from these near-optimal subspaces
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paper_title: Emergence of simple-cell receptive field properties by learning a sparse code for natural images

paper_content:

THE receptive fields of simple cells in mammalian primary visual cortex can be characterized as being spatially localized, oriented1–4 and bandpass (selective to structure at different spatial scales), comparable to the basis functions of wavelet transforms5,6. One approach to understanding such response properties of visual neurons has been to consider their relationship to the statistical structure of natural images in terms of efficient coding7–12. Along these lines, a number of studies have attempted to train unsupervised learning algorithms on natural images in the hope of developing receptive fields with similar properties13–18, but none has succeeded in producing a full set that spans the image space and contains all three of the above properties. Here we investigate the proposal8,12 that a coding strategy that maximizes sparseness is sufficient to account for these properties. We show that a learning algorithm that attempts to find sparse linear codes for natural scenes will develop a complete family of localized, oriented, bandpass receptive fields, similar to those found in the primary visual cortex. The resulting sparse image code provides a more efficient representation for later stages of processing because it possesses a higher degree of statistical independence among its outputs.
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paper_title: Algorithm taxonomy for hyperspectral unmixing

paper_content:

In this paper, we introduce a set of taxonomies that hierarchically organize and specify algorithms associated with hyperspectral unmixing. Our motivation is to collectively organize and relate algorithms in order to assess the current state-of-the-art in the field and to facilitate objective comparisons between methods. The hyperspectral sensing community is populated by investigators with disparate scientific backgrounds and, speaking in their respective languages, efforts in spectral unmixing developed within disparate communities have inevitably led to duplication. We hope our analysis removes this ambiguity and redundancy by using a standard vocabulary, and that the presentation we provide clearly summarizes what has and has not been done. As we shall see, the framework for the taxonomies derives its organization from the fundamental, philosophical assumptions imposed on the problem, rather than the common calculations they perform, or the similar outputs they might yield.
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paper_title: Survey of geometric and statistical unmixing algorithms for hyperspectral images

paper_content:

Spectral mixture analysis (also called spectral unmixing) has been an alluring exploitation goal since the earliest days of imaging spectroscopy. No matter the spatial resolution, the spectral signatures collected in natural environments are invariably a mixture of the signatures of the various materials found within the spatial extent of the ground instantaneous field view of the imaging instrument. In this paper, we give a comprehensive enumeration of the unmixing methods used in practice, because of their implementation in widely used software packages, and those published in the literature. We have structured the review according to the basic computational approach followed by the algorithms, with particular attention to those based on the computational geometry formulation, and statistical approaches with a probabilistic foundation. The quantitative assessment of some available techniques in both categories provides an opportunity to review recent advances and to anticipate future developments.
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paper_title: Spectral unmixing

paper_content:

Spectral unmixing using hyperspectral data represents a significant step in the evolution of remote decompositional analysis that began with multispectral sensing. It is a consequence of collecting data in greater and greater quantities and the desire to extract more detailed information about the material composition of surfaces. Linear mixing is the key assumption that has permitted well-known algorithms to be adapted to the unmixing problem. In fact, the resemblance of the linear mixing model to system models in other areas has permitted a significant legacy of algorithms from a wide range of applications to be adapted to unmixing. However, it is still unclear whether the assumption of linearity is sufficient to model the mixing process in every application of interest. It is clear, however, that the applicability of models and techniques is highly dependent on the variety of circumstances and factors that give rise to mixed pixels. The outputs of spectral unmixing, endmember, and abundance estimates are important for identifying the material composition of mixtures.
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paper_title: Mixed pixels classification

paper_content:

There are two major approaches in spectral unmixing: linear and non-linear ones. They are appropriate for different types of mixture, namely checkerboard mixtures and intimate mixtures. The two approaches are briefly reviewed. Then in a carefully controlled laboratory experiment, the limitations and applicability of two of the methods (a linear and a non- linear one) are compared, in the context of unmixing an intimate mixture.© (1998) COPYRIGHT SPIE--The International Society for Optical Engineering. Downloading of the abstract is permitted for personal use only.
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paper_title: Minimum-volume transforms for remotely sensed data

paper_content:

Scatter diagrams for multispectral remote sensing data tend to be triangular, in the two-band case, pyramidal for three bands, and so on. They radiate away from the so-called darkpoint, which represents the scanner's response to an un-illuminated target. A minimum-volume transform may be described (provisionally) as a nonorthogonal linear transformation of the multivariate data to new axes passing through the dark point, with directions chosen such that they (for two bands), or the new coordinate planes (for three bands, etc.) embrace the data cloud as tightly as possible. The reason for the observed shapes of scatter diagrams is to be found in the theory of linear mixing at the subfootprint scale. Thus, suitably defined, minimum-volume transforms can often be used to unmix images into new spatial variables showing the proportions of the different cover types present, a type of enhancement that is not only intense, but physically meaningful. The present paper furnishes details for constructing computer programs to effect this operation. It will serve as a convenient technical source that may be referenced in subsequent, more profusely illustrated publications that address the intended application, the mapping of surface mineralogy. >
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paper_title: Hyperspectral unmixing: geometrical, statistical, and sparse regression-based approaches

paper_content:

Hyperspectral instruments acquire electromagnetic energy scattered within their ground instantaneous field view ::: in hundreds of spectral channels with high spectral resolution. Very often, however, owing to low spatial resolution ::: of the scanner or to the presence of intimate mixtures (mixing of the materials at a very small scale) in the scene, ::: the spectral vectors (collection of signals acquired at different spectral bands from a given pixel) acquired by the ::: hyperspectral scanners are actually mixtures of the spectral signatures of the materials present in the scene. ::: Given a set of mixed spectral vectors, spectral mixture analysis (or spectral unmixing) aims at estimating the ::: number of reference materials, also called endmembers, their spectral signatures, and their fractional abundances. ::: Spectral unmixing is, thus, a source separation problem where, under a linear mixing model, the sources are the ::: fractional abundances and the endmember spectral signatures are the columns of the mixing matrix. As such, ::: the independent component analysis (ICA) framework came naturally to mind to unmix spectral data. However, ::: the ICA crux assumption of source statistical independence is not satisfied in spectral applications, since the ::: sources are fractions and, thus, non-negative and sum to one. As a consequence, ICA-based algorithms have ::: severe limitations in the area of spectral unmixing, and this has fostered new unmixing research directions taking ::: into account geometric and statistical characteristics of hyperspectral sources. ::: This paper presents an overview of the principal research directions in hyperspectral unmixing. The presentations ::: is organized into four main topics: i) mixing models, ii) signal subspace identification, iii) geometrical-based ::: spectral unmixing, (iv) statistical-based spectral unmixing, and (v) sparse regression-based unmixing. In each ::: topic, we describe what physical or mathematical problems are involved and summarize state-of-the-art algorithms ::: to address these problems.
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paper_title: Spectral mixture modeling - A new analysis of rock and soil types at the Viking Lander 1 site. [on Mars]

paper_content:

A Viking Lander 1 image was modeled as mixtures of reflectance spectra of palagonite dust, gray andesitelike rock, and a coarse rocklike soil. The rocks are covered to varying degrees by dust but otherwise appear unweathered. Rocklike soil occurs as lag deposits in deflation zones around stones and on top of a drift and as a layer in a trench dug by the lander. This soil probably is derived from the rocks by wind abrasion and/or spallation. Dust is the major component of the soil and covers most of the surface. The dust is unrelated spectrally to the rock but is equivalent to the global-scale dust observed telescopically. A new method was developed to model a multispectral image as mixtures of end-member spectra and to compare image spectra directly with laboratory reference spectra. The method for the first time uses shade and secondary illumination effects as spectral end-members; thus the effects of topography and illumination on all scales can be isolated or removed. The image was calibrated absolutely from the laboratory spectra, in close agreement with direct calibrations. The method has broad applications to interpreting multispectral images, including satellite images.
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paper_title: Hyperspectral Subspace Identification

paper_content:

Signal subspace identification is a crucial first step in many hyperspectral processing algorithms such as target detection, change detection, classification, and unmixing. The identification of this subspace enables a correct dimensionality reduction, yielding gains in algorithm performance and complexity and in data storage. This paper introduces a new minimum mean square error-based approach to infer the signal subspace in hyperspectral imagery. The method, which is termed hyperspectral signal identification by minimum error, is eigen decomposition based, unsupervised, and fully automatic (i.e., it does not depend on any tuning parameters). It first estimates the signal and noise correlation matrices and then selects the subset of eigenvalues that best represents the signal subspace in the least squared error sense. State-of-the-art performance of the proposed method is illustrated by using simulated and real hyperspectral images.
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paper_title: Improved Manifold Coordinate Representations of Large-Scale Hyperspectral Scenes

paper_content:

In recent publications, we have presented a data-driven approach to representing the nonlinear structure of hyperspectral imagery using manifold coordinates. The approach relies on graph methods to derive geodesic distances on the high-dimensional hyperspectral data manifold. From these distances, a set of intrinsic manifold coordinates that parameterizes the data manifold is derived. Scaling the solution relied on divide-conquer-and-merge strategies for the. manifold coordinates because of the computational and memory scaling of the geodesic coordinate calculations. In this paper, we improve the scaling performance of isometric mapping (ISOMAP) and achieve full-scene global manifold coordinates while removing artifacts generated by the original methods. The CPU time of the enhanced ISOMAP approach scales as O(·N log 2 (N)), where N is the number of samples, while the memory requirement is bounded by O(N log(N)). Full hyperspectral scenes of O(10 6 ) samples or greater are obtained via a reconstruction algorithm, which allows insertion of large numbers of samples into a representative "backbone" manifold obtained for a smaller but representative set of O(10 5 ) samples. We provide a classification example using a coastal hyperspectral scene to illustrate the approach.
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paper_title: A new look at the statistical model identification

paper_content:

The history of the development of statistical hypothesis testing in time series analysis is reviewed briefly and it is pointed out that the hypothesis testing procedure is not adequately defined as the procedure for statistical model identification. The classical maximum likelihood estimation procedure is reviewed and a new estimate minimum information theoretical criterion (AIC) estimate (MAICE) which is designed for the purpose of statistical identification is introduced. When there are several competing models the MAICE is defined by the model and the maximum likelihood estimates of the parameters which give the minimum of AIC defined by AIC = (-2)log-(maximum likelihood) + 2(number of independently adjusted parameters within the model). MAICE provides a versatile procedure for statistical model identification which is free from the ambiguities inherent in the application of conventional hypothesis testing procedure. The practical utility of MAICE in time series analysis is demonstrated with some numerical examples.
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paper_title: An information theoretic comparison of projection pursuit and principal component features for classification of Landsat TM imagery of central Colorado

paper_content:

Projection pursuit (PP) and principal component analysis (PCA) projections derived from Landsat Thematic Mapper (TM) imagery of central Colorado were compared. While PCA is a simple subset of the general class of PP algorithms, it cannot distinguish Gaussian from non-Gaussian distributions, since it maximizes projected variance. PP algorithms, which maximize higher-order statistics, can be used to find skew or multi-modal projections in order to reveal underlying class structure. These data projections have greater fidelity to underlying land-cover distributions. On sequestered test data, PP projections improved separation of individual categories from a few percent to as much as 24%. PP performance exceeded that of PCA for all but one of the 14 land-cover categories.
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paper_title: Statistical Signal Processing Detection Estimation And Time Series Analysis

paper_content:

Thank you for reading statistical signal processing detection estimation and time series analysis. Maybe you have knowledge that, people have look hundreds times for their chosen novels like this statistical signal processing detection estimation and time series analysis, but end up in harmful downloads. Rather than reading a good book with a cup of tea in the afternoon, instead they juggled with some malicious bugs inside their laptop.
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paper_title: Hyperspectral Subspace Identification

paper_content:

Signal subspace identification is a crucial first step in many hyperspectral processing algorithms such as target detection, change detection, classification, and unmixing. The identification of this subspace enables a correct dimensionality reduction, yielding gains in algorithm performance and complexity and in data storage. This paper introduces a new minimum mean square error-based approach to infer the signal subspace in hyperspectral imagery. The method, which is termed hyperspectral signal identification by minimum error, is eigen decomposition based, unsupervised, and fully automatic (i.e., it does not depend on any tuning parameters). It first estimates the signal and noise correlation matrices and then selects the subset of eigenvalues that best represents the signal subspace in the least squared error sense. State-of-the-art performance of the proposed method is illustrated by using simulated and real hyperspectral images.
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paper_title: Automatic reduction of hyperspectral imagery using wavelet spectral analysis

paper_content:

Hyperspectral imagery provides richer information about materials than multispectral imagery. The new larger data volumes from hyperspectral sensors present a challenge for traditional processing techniques. For example, the identification of each ground surface pixel by its corresponding spectral signature is still difficult because of the immense volume of data. Conventional classification methods may not be used without dimension reduction preprocessing. This is due to the curse of dimensionality, which refers to the fact that the sample size needed to estimate a function of several variables to a given degree of accuracy grows exponentially with the number of variables. Principal component analysis (PCA) has been the technique of choice for dimension reduction. However, PCA is computationally expensive and does not eliminate anomalies that can be seen at one arbitrary band. Spectral data reduction using automatic wavelet decomposition could be useful. This is because it preserves the distinctions among spectral signatures. It is also computed in automatic fashion and can filter data anomalies. This is due to the intrinsic properties of wavelet transforms that preserves high- and low-frequency features, therefore preserving peaks and valleys found in typical spectra. Compared to PCA, for the same level of data reduction, we show that automatic wavelet reduction yields better or comparable classification accuracy for hyperspectral data, while achieving substantial computational savings.
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paper_title: A transformation for ordering multispectral data in terms of image quality with implications for noise removal

paper_content:

A transformation known as the maximum noise fraction (MNF) transformation, which always produces new components ordered by image quality, is presented. It can be shown that this transformation is equivalent to principal components transformations when the noise variance is the same in all bands and that it reduces to a multiple linear regression when noise is in one band only. Noise can be effectively removed from multispectral data by transforming to the MNF space, smoothing or rejecting the most noisy components, and then retransforming to the original space. In this way, more intense smoothing can be applied to the MNF components with high noise and low signal content than could be applied to each band of the original data. The MNF transformation requires knowledge of both the signal and noise covariance matrices. Except when the noise is in one band only, the noise covariance matrix needs to be estimated. One procedure for doing this is discussed and examples of cleaned images are presented. >
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paper_title: Real-time analysis of hyperspectral data sets using NRL's ORASIS algorithm

paper_content:

The covered lantern project was initiated by the central MASINT Technology Coordination Office to demonstrate the tactical use of hyperspectral imagery with real time processing capability. We report on the design and use of the HYCORDER system developed for Covered Lantern that was tested in June 1995. The HYCORDER system consisted of an imaging spectrometer flying in a Pioneer Uncrewed Aeronautical Vehicle and a ground based real time analysis and visualization system. The camera was intensified allowing dawn to dusk operation. The spectral information was downlinked to the analysis system as standard analog video. The analysis system was constructed from 17 Texas Instrument C44 DSPs controlled by a 200 MHz Pentium Pro PC. A real time, parallel version of NRL's optical real-time adaptive spectral identification system algorithm was developed for this system. The system was capable of running continuously, allowing for broad area coverage. The algorithm was adaptive, accommodating changing lighting conditions and terrain. The general architecture of the algorithm will be discussed as well as results from the test.© (1997) COPYRIGHT SPIE--The International Society for Optical Engineering. Downloading of the abstract is permitted for personal use only.
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paper_title: Manifold learning techniques for the analysis of hyperspectral ocean data

paper_content:

A useful technique in hyperspectral data analysis is dimensionality reduction, which replaces the original high dimensional data with low dimensional representations. Usually this is done with linear techniques such as linear mixing or principal components (PCA). While often useful, there is no a priori reason for believing that the data is actually linear. ::: ::: Lately there has been renewed interest in modeling high dimensional data using nonlinear techniques such as manifold learning (ML). In ML, the data is assumed to lie on a low dimensional, possibly curved surface (or manifold). The goal is to discover this manifold and therefore find the best low dimensional representation of the data. ::: ::: Recently, researchers at the Naval Research Lab have begun to model hyperspectral data using ML. We continue this work by applying ML techniques to hyperspectral ocean water data. We focus on water since there are underlying physical reasons for believing that the data lies on a certain type of nonlinear manifold. In particular, ocean data is influenced by three factors: the water parameters, the bottom type, and the depth. For fixed water and bottom types, the spectra that arise by varying the depth will lie on a nonlinear, one dimensional manifold (i.e. a curve). Generally, water scenes will contain a number of different water and bottom types, each combination of which leads to a distinct curve. In this way, the scene may be modeled as a union of one dimensional curves. In this paper, we investigate the use of manifold learning techniques to separate the various curves, thus partitioning the scene into homogeneous areas. We also discuss ways in which these techniques may be able to derive various scene characteristics such as bathymetry.
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paper_title: Constrained band selection for hyperspectral imagery

paper_content:

Constrained energy minimization (CEM) has shown effective in hyperspectral target detection. It linearly constrains a desired target signature while minimizing interfering effects caused by other unknown signatures. This paper explores this idea for band selection and develops a new approach to band selection, referred to as constrained band selection (CBS) for hyperspectral imagery. It interprets a band image as a desired target signature vector while considering other band images as unknown signature vectors. As a result, the proposed CBS using the concept of the CEM to linearly constrain a band image, while also minimizing band correlation or dependence provided by other band images, is referred to as CEM-CBS. Four different criteria referred to as Band Correlation Minimization (BCM), Band Correlation Constraint (BCC), Band Dependence Constraint (BDC), and Band Dependence Minimization (BDM) are derived for CEM-CBS.. Since dimensionality resulting from conversion of a band image to a vector may be huge, the CEM-CBS is further reinterpreted as linearly constrained minimum variance (LCMV)-based CBS by constraining a band image as a matrix where the same four criteria, BCM, BCC, BDC, and BDM, can be also used for LCMV-CBS. In order to determine the number of bands required to select p, a recently developed concept, called virtual dimensionality, is used to estimate the p. Once the p is determined, a set of p desired bands can be selected by the CEM/LCMV-CBS. Finally, experiments are conducted to substantiate the proposed CEM/LCMV-CBS four criteria, BCM, BCC, BDC, and BDM, in comparison with variance-based band selection, information divergence-based band selection, and uniform band selection.
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paper_title: Spatially Coherent Nonlinear Dimensionality Reduction and Segmentation of Hyperspectral Images

paper_content:

The nonlinear dimensionality reduction and its effects on vector classification and segmentation of hyperspectral images are investigated in this letter. In particular, the way dimensionality reduction influences and helps classification and segmentation is studied. The proposed framework takes into account the nonlinear nature of high-dimensional hyperspectral images and projects onto a lower dimensional space via a novel spatially coherent locally linear embedding technique. The spatial coherence is introduced by comparing pixels based on their local surrounding structure in the image domain and not just on their individual values as classically done. This spatial coherence in the image domain across the multiple bands defines the high-dimensional local neighborhoods used for the dimensionality reduction. This spatial coherence concept is also extended to the segmentation and classification stages that follow the dimensionality reduction, introducing a modified vector angle distance. We present the underlying concepts of the proposed framework and experimental results showing the significant classification improvements
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paper_title: Unsupervised hyperspectral image analysis with projection pursuit

paper_content:

Principal components analysis (PCA) is effective at compressing information in multivariate data sets by computing orthogonal projections that maximize the amount of data variance. Unfortunately, information content in hyperspectral images does not always coincide with such projections. The authors propose an application of projection pursuit (PP), which seeks to find a set of projections that are "interesting," in the sense that they deviate from the Gaussian distribution assumption. Once these projections are obtained, they can be used for image compression, segmentation, or enhancement for visual analysis. To find these projections, a two-step iterative process is followed where they first search for a projection that maximizes a projection index based on the information divergence of the projection's estimated probability distribution from the Gaussian distribution and then reduce the rank by projecting the data onto the subspace orthogonal to the previous projections. To calculate each projection, they use a simplified approach to maximizing the projection index, which does not require an optimization algorithm. It searches for a solution by obtaining a set of candidate projections from the data and choosing the one with the highest projection index. The effectiveness of this method is demonstrated through simulated examples as well as data from the hyperspectral digital imagery collection experiment (HYDICE) and the spatially enhanced broadband array spectrograph system (SEBASS).
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paper_title: Estimation of number of spectrally distinct signal sources in hyperspectral imagery

paper_content:

With very high spectral resolution, hyperspectral sensors can now uncover many unknown signal sources which cannot be identified by visual inspection or a priori. In order to account for such unknown signal sources, we introduce a new definition, referred to as virtual dimensionality (VD) in this paper. It is defined as the minimum number of spectrally distinct signal sources that characterize the hyperspectral data from the perspective view of target detection and classification. It is different from the commonly used intrinsic dimensionality (ID) in the sense that the signal sources are determined by the proposed VD based only on their distinct spectral properties. These signal sources may include unknown interfering sources, which cannot be identified by prior knowledge. With this new definition, three Neyman-Pearson detection theory-based thresholding methods are developed to determine the VD of hyperspectral imagery, where eigenvalues are used to measure signal energies in a detection model. In order to evaluate the performance of the proposed methods, two information criteria, an information criterion (AIC) and minimum description length (MDL), and the factor analysis-based method proposed by Malinowski, are considered for comparative analysis. As demonstrated in computer simulations, all the methods and criteria studied in this paper may work effectively when noise is independent identically distributed. This is, unfortunately, not true when some of them are applied to real image data. Experiments show that all the three eigenthresholding based methods (i.e., the Harsanyi-Farrand-Chang (HFC), the noise-whitened HFC (NWHFC), and the noise subspace projection (NSP) methods) produce more reliable estimates of VD compared to the AIC, MDL, and Malinowski's empirical indicator function, which generally overestimate VD significantly. In summary, three contributions are made in this paper, 1) an introduction of the new definition of VD, 2) three Neyman-Pearson detection theory-based thresholding methods, HFC, NWHFC, and NSP derived for VD estimation, and 3) experiments that show the AIC and MDL commonly used in passive array processing and the second-order statistic-based Malinowski's method are not effective measures in VD estimation.
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paper_title: Modeling by shortest data description

paper_content:

The number of digits it takes to write down an observed sequence x"1, ..., x"N of a time series depends on the model with its parameters that one assumes to have generated the observed data. Accordingly, by finding the model which minimizes the description length one obtains estimates of both the integer-valued structure parameters and the real-valued system parameters.
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paper_title: Curvilinear Component Analysis: a Self-Organizing Neural Network for Nonlinear Mapping of Data Sets

paper_content:

We present a new strategy called "curvilinear component analysis" (CCA) for dimensionality reduction and representation of multidimensional data sets. The principle of CCA is a self-organized neural network performing two tasks: vector quantization (VQ) of the submanifold in the data set (input space); and nonlinear projection (P) of these quantizing vectors toward an output space, providing a revealing unfolding of the submanifold. After learning, the network has the ability to continuously map any new point from one space into another: forward mapping of new points in the input space, or backward mapping of an arbitrary position in the output space.
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paper_title: Principal Component Analysis

paper_content:

When large multivariate datasets are analyzed, it is often desirable to reduce their dimensionality. Principal component analysis is one technique for doing this. It replaces the p original variables by a smaller number, q, of derived variables, the principal components, which are linear combinations of the original variables. Often, it is possible to retain most of the variability in the original variables with q very much smaller than p. Despite its apparent simplicity, principal component analysis has a number of subtleties, and it has many uses and extensions. A number of choices associated with the technique are briefly discussed, namely, covariance or correlation, how many components, and different normalization constraints, as well as confusion with factor analysis. Various uses and extensions are outlined. ::: ::: ::: Keywords: ::: ::: dimension reduction; ::: factor analysis; ::: multivariate analysis; ::: variance maximization
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paper_title: Detection of signals by information theoretic criteria

paper_content:

A new approach is presented to the problem of detecting the number of signals in a multichannel time-series, based on the application of the information theoretic criteria for model selection introduced by Akaike (AIC) and by Schwartz and Rissanen (MDL). Unlike the conventional hypothesis testing based approach, the new approach does not requite any subjective threshold settings; the number of signals is obtained merely by minimizing the AIC or the MDL criteria. Simulation results that illustrate the performance of the new method for the detection of the number of signals received by a sensor array are presented.
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paper_title: Enhancement of high spectral resolution remote-sensing data by a noise-adjusted principal components transform

paper_content:

High-spectral-resolution remote-sensing data are first transformed so that the noise covariance matrix becomes the identity matrix. Then the principal components transform is applied. This transform is equivalent to the maximum noise fraction transform and is optimal in the sense that it maximizes the signal-to-noise ratio (SNR) in each successive transform component, just as the principal component transform maximizes the data variance in successive components. Application of this transform requires knowledge or an estimate of the noise covariance matrix of the data. The effectiveness of this transform for noise removal is demonstrated in both the spatial and spectral domains. Results that demonstrate the enhancement of geological mapping and detection of alteration mineralogy in data from the Pilbara region of Western Australia, including mapping of the occurrence of pyrophyllite over an extended area, are presented. >
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paper_title: Exploiting manifold geometry in hyperspectral imagery

paper_content:

A new algorithm for exploiting the nonlinear structure of hyperspectral imagery is developed and compared against the de facto standard of linear mixing. This new approach seeks a manifold coordinate system that preserves geodesic distances in the high-dimensional hyperspectral data space. Algorithms for deriving manifold coordinates, such as isometric mapping (ISOMAP), have been developed for other applications. ISOMAP guarantees a globally optimal solution, but is computationally practical only for small datasets because of computational and memory requirements. Here, we develop a hybrid technique to circumvent ISOMAP's computational cost. We divide the scene into a set of smaller tiles. The manifolds derived from the individual tiles are then aligned and stitched together to recomplete the scene. Several alignment methods are discussed. This hybrid approach exploits the fact that ISOMAP guarantees a globally optimal solution for each tile and the presumed similarity of the manifold structures derived from different tiles. Using land-cover classification of hyperspectral imagery in the Virginia Coast Reserve as a test case, we show that the new manifold representation provides better separation of spectrally similar classes than one of the standard linear mixing models. Additionally, we demonstrate that this technique provides a natural data compression scheme, which dramatically reduces the number of components needed to model hyperspectral data when compared with traditional methods such as the minimum noise fraction transform.
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paper_title: Applying nonlinear manifold learning to hyperspectral data for land cover classification

paper_content:

The shortest path k-nearest neighbor classifier (SkNN), that utilizes nonlinear manifold learning, is proposed for analysis of hyperspectral data. In contrast to classifiers that deal with the high dimensional feature space directly, this approach uses the pairwise distance matrix over a nonlinear manifold to classify novel observations. Because manifold learning preserves the local pairwise distances and updates distances of a sample to samples beyond the user-defined neighborhood along the shortest path on the manifold, similar samples are moved into closer proximity. High classification accuracies are achieved by using the simple k-nearest neighbor (kNN) classifier. SkNN was applied to hyperspectral data collected by the Hyperion sensor on the EO1 satellite over the Okavango Delta of Botswana. Classification accuracies and generalization capability are compared to those achieved by the best basis binary hierarchical classifier, the hierarchical support vector machine classifier, and the k-nearest neighbor classifier on both the original data and a subset of its principal components.
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paper_title: Information-theory-based band selection and utility evaluation for reflective spectral systems

paper_content:

We have developed a methodology for wavelength band selection. This methodology can be used in system design studies to provide an optimal sensor cost, data reduction, and data utility trade-off relative to a specific application. The methodology combines an information theory- based criterion for band selection with a genetic algorithm to search for a near-optimal solution. We have applied this methodology to 612 material spectra from a combined database to determine the band locations for 6, 9, 15, 30, and 60- band sets in the 0.42 to 2.5 microns spectral region that permit the best material separation. These optimal bands sets were then evaluated in terms of their utility related to anomaly ddetection and material identification using multi-band data cubes generated from two HYDICE cubes. The optimal band locations and their corresponding entropies are given in this paper. Our optimal band locations for the 6, 9, and 15-band sets are compared to the bands of existing multi-band systems such as Landsat 7, Multispectral Thermal Imager, Advanced Land Imager, Daedalus, and M7. Also presented are the anomaly detection and material identification results obtained from our generalted multi- band data cubes. Comparisons are made between these exploitation results with those obtained from the original 210-band HYDICE data cubes.
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paper_title: Hyperspectral image processing using locally linear embedding

paper_content:

We describe a method of processing hyperspectral images of natural scenes that uses a combination of k-means clustering and locally linear embedding (LLE). The primary goal is to assist anomaly detection by preserving spectral uniqueness among the pixels. In order to reduce redundancy among the pixels, adjacent pixels which are spectrally similar are grouped using the k-means clustering algorithm. Representative pixels from each cluster are chosen and passed to the LLE algorithm, where the high dimensional spectral vectors are encoded by a low dimensional mapping. Finally, monochromatic and tri-chromatic images are constructed from the k-means cluster assignments and LLE vector mappings. The method generates images where differences in the original spectra are reflected in differences in the output vector assignments. An additional benefit of mapping to a lower dimensional space is reduced data size. When spectral irregularities are added to a patch of the hyperspectral images, again the method successfully generated color assignments that detected the changes in the spectra.
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paper_title: Noise reduction of hyperspectral imagery using hybrid spatial-spectral derivative-domain wavelet shrinkage

paper_content:

In this paper, a new noise reduction algorithm is introduced and applied to the problem of denoising hyperspectral imagery. This algorithm resorts to the spectral derivative domain, where the noise level is elevated, and benefits from the dissimilarity of the signal regularity in the spatial and the spectral dimensions of hyperspectral images. The performance of the new algorithm is tested on two different hyperspectral datacubes: an Airborne Visible/Infrared Imaging Spectrometer (AVIRIS) datacube that is acquired in a vegetation-dominated site and a simulated AVIRIS datacube that simulates a geological site. The new algorithm provides signal-to-noise-ratio improvement up to 84.44% and 98.35% in the first and the second datacubes, respectively.
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paper_title: Hyperspectral Subspace Identification

paper_content:

Signal subspace identification is a crucial first step in many hyperspectral processing algorithms such as target detection, change detection, classification, and unmixing. The identification of this subspace enables a correct dimensionality reduction, yielding gains in algorithm performance and complexity and in data storage. This paper introduces a new minimum mean square error-based approach to infer the signal subspace in hyperspectral imagery. The method, which is termed hyperspectral signal identification by minimum error, is eigen decomposition based, unsupervised, and fully automatic (i.e., it does not depend on any tuning parameters). It first estimates the signal and noise correlation matrices and then selects the subset of eigenvalues that best represents the signal subspace in the least squared error sense. State-of-the-art performance of the proposed method is illustrated by using simulated and real hyperspectral images.
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paper_title: Image Denoising by Sparse 3-D Transform-Domain Collaborative Filtering

paper_content:

We propose a novel image denoising strategy based on an enhanced sparse representation in transform domain. The enhancement of the sparsity is achieved by grouping similar 2D image fragments (e.g., blocks) into 3D data arrays which we call "groups." Collaborative Altering is a special procedure developed to deal with these 3D groups. We realize it using the three successive steps: 3D transformation of a group, shrinkage of the transform spectrum, and inverse 3D transformation. The result is a 3D estimate that consists of the jointly filtered grouped image blocks. By attenuating the noise, the collaborative filtering reveals even the finest details shared by grouped blocks and, at the same time, it preserves the essential unique features of each individual block. The filtered blocks are then returned to their original positions. Because these blocks are overlapping, for each pixel, we obtain many different estimates which need to be combined. Aggregation is a particular averaging procedure which is exploited to take advantage of this redundancy. A significant improvement is obtained by a specially developed collaborative Wiener filtering. An algorithm based on this novel denoising strategy and its efficient implementation are presented in full detail; an extension to color-image denoising is also developed. The experimental results demonstrate that this computationally scalable algorithm achieves state-of-the-art denoising performance in terms of both peak signal-to-noise ratio and subjective visual quality.
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paper_title: A Convex Analysis-Based Minimum-Volume Enclosing Simplex Algorithm for Hyperspectral Unmixing

paper_content:

Hyperspectral unmixing aims at identifying the hidden spectral signatures (or endmembers) and their corresponding proportions (or abundances) from an observed hyperspectral scene. Many existing approaches to hyperspectral unmixing rely on the pure-pixel assumption, which may be violated for highly mixed data. A heuristic unmixing criterion without requiring the pure-pixel assumption has been reported by Craig: The endmember estimates are determined by the vertices of a minimum-volume simplex enclosing all the observed pixels. In this paper, using convex analysis, we show that the hyperspectral unmixing by Craig's criterion can be formulated as an optimization problem of finding a minimum-volume enclosing simplex (MVES). An algorithm that cyclically solves the MVES problem via linear programs (LPs) is also proposed. Some Monte Carlo simulations are provided to demonstrate the efficacy of the proposed MVES algorithm.
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paper_title: Minimum-volume transforms for remotely sensed data

paper_content:

Scatter diagrams for multispectral remote sensing data tend to be triangular, in the two-band case, pyramidal for three bands, and so on. They radiate away from the so-called darkpoint, which represents the scanner's response to an un-illuminated target. A minimum-volume transform may be described (provisionally) as a nonorthogonal linear transformation of the multivariate data to new axes passing through the dark point, with directions chosen such that they (for two bands), or the new coordinate planes (for three bands, etc.) embrace the data cloud as tightly as possible. The reason for the observed shapes of scatter diagrams is to be found in the theory of linear mixing at the subfootprint scale. Thus, suitably defined, minimum-volume transforms can often be used to unmix images into new spatial variables showing the proportions of the different cover types present, a type of enhancement that is not only intense, but physically meaningful. The present paper furnishes details for constructing computer programs to effect this operation. It will serve as a convenient technical source that may be referenced in subsequent, more profusely illustrated publications that address the intended application, the mapping of surface mineralogy. >
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paper_title: Spectral mixture analysis for subpixel vegetation fractions in the urban environment: How to incorporate endmember variability?

paper_content:

In the urban environment both quality of life and surface biophysical processes are closely related to the presence of vegetation. Spectral mixture analysis (SMA) has been frequently used to derive subpixel vegetation information from remotely sensed imagery in urban areas, where the underlying landscapes are assumed to be composed of a few fundamental components, called endmembers. A critical step in SMA is to identify the endmembers and their corresponding spectral signatures. A common practice in SMA assumes a constant spectral signature for each endmember. In fact, the spectral signatures of endmembers may vary from pixel to pixel due to changes in biophysical (e.g. leaves, stems and bark) and biochemical (e.g. chlorophyll content) composition. This study developed a Bayesian Spectral Mixture Analysis (BSMA) model to understand the impact of endmember variability on the derivation of subpixel vegetation fractions in an urban environment. BSMA incorporates endmember spectral variability in the unmixing process based on Bayes Theorem. In traditional SMA, each endmember is represented by a constant signature, while BSMA uses the endmember signature probability distribution in the analysis. BSMA has the advantage of maximally capturing the spectral variability of an image with the least number of endmembers. In this study, the BSMA model is first applied to simulated images, and then to Ikonos and Landsat ETM+ images. BSMA leads to an improved estimate of subpixel vegetation fractions, and provides uncertainty information for the estimates. The study also found that the traditional SMA using the statistical means of the signature distributions as endmember signatures produces subpixel endmember fractions with almost the same and sometimes even better accuracy than those from BSMA except without uncertainty information for the estimates. However, using the modes of signature distributions as endmembers may result in serious bias in subpixel endmember fractions derived from traditional SMA.
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paper_title: Endmember bundles: a new approach to incorporating endmember variability into spectral mixture analysis

paper_content:

Accuracy of vegetation cover fractions, computed with spectral mixture analysis, may be compromised by variation in canopy structure and biochemistry when a single endmember represents top-of-canopy reflectance. In this article, endmember variability is incorporated into mixture analysis by representing each endmember by a set or bundle of spectra, each of which could reasonably be the reflectance of an instance of the endmember. Endmember bundles are constructed from the data itself by an extension to a previously described method of manually deriving endmembers from remotely sensed data. Applied to remotely sensed images, bundle unmixing produces maximum and minimum fraction images bounding the correct cover fractions and specifying error due to endmember variability. In this article, endmember bundles and bounding fraction images were created for an airborne visible/infrared imaging spectrometer (AVIRIS) subscene simulated with a canopy radiative transfer/geometric-optical model. Variation in endmember reflectance was achieved using ranges of parameter values including leaf area index (LAI) and tissue optical properties observed In a North Texas savanna. The subscene's spatial pattern was based on a 1992 Landsat Thematic Mapper image of the study region. Bounding fraction images bracketed the cover fractions of the simulated data for 98% of the pixels for soil, 97% for senescent grass and 93% for trees. Averages of bounding images estimated fractional coverage used in the simulation with an average error of /spl les/0.05, a significant improvement over previous methods with important implications for regional-scale research on vegetation extent and dynamics.
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paper_title: Sequential N-FINDR algorithms

paper_content:

Abstract N-finder algorithm (N-FINDR) is probably one of most popular and widely used algorithms used for endmember extraction. Three major obstacles need to be overcome in its practical implementation. One is that the number of endmembers must be known a priori. A second one is the use of random initial endmembers to initialize N-FINDR, which results in inconsistent final results of extracted endm embers. A third one is its very expensive computational cost caused by an exhaustive search. While the first two issues can be resolved by a recently developed concept, virtual dimensionality (VD) and custom-designed initialization algorithms respectively, the third issue seems to remain challenging. This paper addresses the latter issue by re-designing N-FINDR which can generate one endmember at a time sequentially in a successive fashion to ease computational complexity. Such resulting algorithm is called SeQuential N-FINDR (SQ N-FINDR) as opposed to the orig inal N-FINDR referred to as SiMultaneous N-FINDR (SM N-FINDR) which generates all endmembers simultaneously at once. Two variants of SQ N-FINDR can be further derived to reduce computational complexity. Interestingly, e xperimental results show that SQ N-FINDR can perform as well as SM-N-FINDR if initial endmem bers are appropriately selected. Key words: Endmember extraction. Mixed pixels. Subpixel targets. Target Embeddedness (TE). Target Implantation (TI).
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paper_title: Autonomous single-pass endmember approximation using lattice auto-associative memories

paper_content:

We propose a novel method for the autonomous determination of endmembers that employs recent results from the theory of lattice based auto-associative memories. In contrast to several other existing methods, the endmembers determined by the proposed method are physically linked to the data set spectra. Numerical examples are provided to illustrate lattice theoretical concepts and a hyperspectral image subcube, from the Cuprite site in Nevada, is used to find all endmember candidates in a single pass.
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paper_title: N-FINDR: an algorithm for fast autonomous spectral end-member determination in hyperspectral data

paper_content:

The analysis of hyperspectral data sets requires the determination of certain basis spectra called 'end-members.' Once these spectra are found, the image cube can be 'unmixed' into the fractional abundance of each material in each pixel. There exist several techniques for accomplishing the determination of the end-members, most of which involve the intervention of a trained geologist. Often these-end-members are assumed to be present in the image, in the form of pure, or unmixed, pixels. In this paper a method based upon the geometry of convex sets is proposed to find a unique set of purest pixels in an image. The technique is based on the fact that in N spectral dimensions, the N-volume contained by a simplex formed of the purest pixels is larger than any other volume formed from any other combination of pixels. The algorithm works by 'inflating' a simplex inside the data, beginning with a random set of pixels. For each pixel and each end-member, the end-member is replaced with the spectrum of the pixel and the volume is recalculated. If it increases, the spectrum of the new pixel replaces that end-member. This procedure is repeated until no more replacements are done. This algorithm successfully derives end-members in a synthetic data set, and appears robust with less than perfect data. Spectral end-members have been extracted for the AVIRIS Cuprite data set which closely match reference spectra, and resulting abundance maps match published mineral maps.
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paper_title: The sequential maximum angle convex cone (SMACC) endmember model

paper_content:

A new endmember extraction method has been developed that is based on a convex cone model for representing vector data. The endmembers are selected directly from the data set. The algorithm for finding the endmembers is sequential: the convex cone model starts with a single endmember and increases incrementally in dimension. Abundance maps are simultaneously generated and updated at each step. A new endmember is identified based on the angle it makes with the existing cone. The data vector making the maximum angle with the existing cone is chosen as the next endmember to add to enlarge the endmember set. The algorithm updates the abundances of previous endmembers and ensures that the abundances of previous and current endmembers remain positive or zero. The algorithm terminates when all of the data vectors are within the convex cone, to some tolerance. The method offers advantages for hyperspectral data sets where high correlation among channels and pixels can impair un-mixing by standard techniques. The method can also be applied as a band-selection tool, finding end-images that are unique and forming a convex cone for modeling the remaining hyperspectral channels. The method is described and applied to hyperspectral data sets.
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paper_title: A New Growing Method for Simplex-Based Endmember Extraction Algorithm

paper_content:

A new growing method for simplex-based endmember extraction algorithms (EEAs), called simplex growing algorithm (SGA), is presented in this paper. It is a sequential algorithm to find a simplex with the maximum volume every time a new vertex is added. In order to terminate this algorithm a recently developed concept, virtual dimensionality (VD), is implemented as a stopping rule to determine the number of vertices required for the algorithm to generate. The SGA improves one commonly used EEA, the N-finder algorithm (N-FINDR) developed by Winter, by including a process of growing simplexes one vertex at a time until it reaches a desired number of vertices estimated by the VD, which results in a tremendous reduction of computational complexity. Additionally, it also judiciously selects an appropriate initial vector to avoid a dilemma caused by the use of random vectors as its initial condition in the N-FINDR where the N-FINDR generally produces different sets of final endmembers if different sets of randomly generated initial endmembers are used. In order to demonstrate the performance of the proposed SGA, the N-FINDR and two other EEAs, pixel purity index, and vertex component analysis are used for comparison.
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paper_title: Two lattice computing approaches for the unsupervised segmentation of hyperspectral images

paper_content:

Endmembers for the spectral unmixing analysis of hyperspectral images are sets of affinely independent vectors, which define a convex polytope covering the data points that represent the pixel image spectra. Strong lattice independence (SLI) is a property defined in the context of lattice associative memories convergence analysis. Recent results show that SLI implies affine independence, confirming the value of lattice associative memories for the study of endmember induction algorithms. In fact, SLI vector sets can be easily deduced from the vectors composing the lattice auto-associative memories (LAM). However, the number of candidate endmembers found by this algorithm is very large, so that some selection algorithm is needed to obtain the full benefits of the approach. In this paper we explore the unsupervised segmentation of hyperspectral images based on the abundance images computed, first, by an endmember selection algorithm and, second, by a previously proposed heuristically defined algorithm. We find their results comparable on a qualitative basis.
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paper_title: Vertex component analysis: a fast algorithm to unmix hyperspectral data

paper_content:

Given a set of mixed spectral (multispectral or hyperspectral) vectors, linear spectral mixture analysis, or linear unmixing, aims at estimating the number of reference substances, also called endmembers, their spectral signatures, and their abundance fractions. This paper presents a new method for unsupervised endmember extraction from hyperspectral data, termed vertex component analysis (VCA). The algorithm exploits two facts: (1) the endmembers are the vertices of a simplex and (2) the affine transformation of a simplex is also a simplex. In a series of experiments using simulated and real data, the VCA algorithm competes with state-of-the-art methods, with a computational complexity between one and two orders of magnitude lower than the best available method.
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paper_title: Mapping target signatures via partial unmixing of AVIRIS data

paper_content:

A complete spectral unmixing of a complicated AVIRIS scene may not always be possible or even desired. High quality data of spectrally complex areas are very high dimensional and are consequently difficult to fully unravel. Partial unmixing provides a method of solving only that fraction of the data inversion problem that directly relates to the specific goals of the investigation. Many applications of imaging spectrometry can be cast in the form of the following question: 'Are my target signatures present in the scene, and if so, how much of each target material is present in each pixel?' This is a partial unmixing problem. The number of unmixing endmembers is one greater than the number of spectrally defined target materials. The one additional endmember can be thought of as the composite of all the other scene materials, or 'everything else'. Several workers have proposed partial unmixing schemes for imaging spectrometry data, but each has significant limitations for operational application. The low probability detection methods described by Farrand and Harsanyi and the foreground-background method of Smith et al are both examples of such partial unmixing strategies. The new method presented here builds on these innovative analysis concepts, combining their different positive attributes while attempting to circumvent their limitations. This new method partially unmixes AVIRIS data, mapping apparent target abundances, in the presence of an arbitrary and unknown spectrally mixed background. It permits the target materials to be present in abundances that drive significant portions of the scene covariance. Furthermore it does not require a priori knowledge of the background material spectral signatures. The challenge is to find the proper projection of the data that hides the background variance while simultaneously maximizing the variance amongst the targets.
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paper_title: A Simplex Volume Maximization Framework for Hyperspectral Endmember Extraction

paper_content:

In the late 1990s, Winter proposed an endmember extraction belief that has much impact on endmember extraction techniques in hyperspectral remote sensing. The idea is to find a maximum-volume simplex whose vertices are drawn from the pixel vectors. Winter's belief has stimulated much interest, resulting in many different variations of pixel search algorithms, widely known as N-FINDR, being proposed. In this paper, we take a continuous optimization perspective to revisit Winter's belief, where the aim is to provide an alternative framework of formulating and understanding Winter's belief in a systematic manner. We first prove that, fundamentally, the existence of pure pixels is not only sufficient for the Winter problem to perfectly identify the ground-truth endmembers but also necessary. Then, under the umbrella of the Winter problem, we derive two methods using two different optimization strategies. One is by alternating optimization. The resulting algorithm turns out to be an N-FINDR variant, but, with the proposed formulation, we can pin down some of its convergence characteristics. Another is by successive optimization; interestingly, the resulting algorithm is found to exhibit some similarity to vertex component analysis. Hence, the framework provides linkage and alternative interpretations to these existing algorithms. Furthermore, we propose a robust worst case generalization of the Winter problem for accounting for perturbed pixel effects in the noisy scenario. An algorithm combining alternating optimization and projected subgradients is devised to deal with the problem. We use both simulations and real data experiments to demonstrate the viability and merits of the proposed algorithms.
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paper_title: A lattice matrix method for hyperspectral image unmixing

paper_content:

In this manuscript we propose a method for the autonomous determination of endmembers in hyperspectral imagery based on recent theoretical advancements on lattice auto-associative memories. Given a hyperspectral image, the lattice algebra approach finds in a single-pass all possible candidate endmembers from which various affinely independent sets of final endmembers may be derived. In contrast to other endmember detection methods, the endmembers found using two dual canonical lattice matrices are geometrically linked to the data set spectra. The mathematical foundation of the proposed method is first described in some detail followed by application examples that illustrate the key steps of the proposed lattice based method.
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paper_title: Unmixing of Hyperspectral Images using Bayesian Non-negative Matrix Factorization with Volume Prior

paper_content:

Hyperspectral imaging can be used in assessing the quality of foods by decomposing the image into constituents such as protein, starch, and water. Observed data can be considered a mixture of underlying characteristic spectra (endmembers), and estimating the constituents and their abundances requires efficient algorithms for spectral unmixing. We present a Bayesian spectral unmixing algorithm employing a volume constraint and propose an inference procedure based on Gibbs sampling. We evaluate the method on synthetic and real hyperspectral data of wheat kernels. Results show that our method perform as good or better than existing volume constrained methods. Further, our method gives credible intervals for the endmembers and abundances, which allows us to asses the confidence of the results.
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paper_title: A Convex Analysis-Based Minimum-Volume Enclosing Simplex Algorithm for Hyperspectral Unmixing

paper_content:

Hyperspectral unmixing aims at identifying the hidden spectral signatures (or endmembers) and their corresponding proportions (or abundances) from an observed hyperspectral scene. Many existing approaches to hyperspectral unmixing rely on the pure-pixel assumption, which may be violated for highly mixed data. A heuristic unmixing criterion without requiring the pure-pixel assumption has been reported by Craig: The endmember estimates are determined by the vertices of a minimum-volume simplex enclosing all the observed pixels. In this paper, using convex analysis, we show that the hyperspectral unmixing by Craig's criterion can be formulated as an optimization problem of finding a minimum-volume enclosing simplex (MVES). An algorithm that cyclically solves the MVES problem via linear programs (LPs) is also proposed. Some Monte Carlo simulations are provided to demonstrate the efficacy of the proposed MVES algorithm.
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paper_title: Piece-wise convex spatial-spectral unmixing of hyperspectral imagery using possibilistic and fuzzy clustering

paper_content:

Imaging spectroscopy refers to methods for identifying materials in a scene using cameras that digitize light into hundreds of spectral bands. Each pixel in these images consists of vectors representing the amount of light reflected in the different spectral bands from the physical location corresponding to the pixel. Images of this type are called hyperspectral images. Hyperspectral image analysis differs from traditional image analysis in that, in addition to the spatial information inherent in an image, there is abundant spectral information at the pixel or sub-pixel level that can be used to identify materials in the scene. Spectral unmixing techniques attempt to identify the material spectra in a scene down to the sub-pixel level. In this paper, a piece-wise convex hyperspectral unmixing algorithm using both spatial and spectral image information is presented. The proposed method incorporates possibilistic and fuzzy clustering methods. The typicality and membership estimates from those methods can be combined with traditional material proportion estimates to produce more meaningful proportion estimates than obtained with previous spectral unmixing algorithms. An analysis of the utility of using all three estimates produce a better estimate is given using real hyperspectral imagery.
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paper_title: Collaborative nonnegative matrix factorization for remotely sensed hyperspectral unmixing

paper_content:

In this paper, we develop a new algorithm for hyperspectral unmixing which can provide suitable endmembers (and their corresponding abundances) in a single step. Hence, the algorithm does not require a previous subspace identification step to estimate the number of endmembers as it can cope with the two most likely scenarios in practice (i.e., the number of endmembers is correctly determined or overestimated a priori). The proposed approach, termed collaborative NMF (CoNMF), uses a collaborative regularization prior which forces the abundances corresponding to the overestimated endmembers to zero, such that it is guaranteed that only the true endmembers have fractional abundance contributions and the estimation of the number of endmembers is not required in advance. The obtained experimental results demonstrate that the proposed method exhibits very good performance in case the number of endmember is not available a priori.
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paper_title: Spatially-smooth piece-wise convex endmember detection

paper_content:

An endmember detection and spectral unmixing algorithm that uses both spatial and spectral information is presented. This method, Spatial Piece-wise Convex Multiple Model Endmember Detection (Spatial P-COMMEND), autonomously estimates multiple sets of endmembers and performs spectral unmixing for input hyperspectral data. Spatial P-COMMEND does not restrict the estimated endmembers to define a single convex region during spectral unmixing. Instead, a piece-wise convex representation is used that can effectively represent non-convex hyperspectral data. Spatial P-COMMEND drives neighboring pixels to be unmixed by the same set of endmembers encouraging spatially-smooth unmixing results.
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paper_title: Minimum Volume Simplex Analysis: A Fast Algorithm to Unmix Hyperspectral Data

paper_content:

This paper presents a new method of minimum volume class for hyperspectral unmixing, termed minimum volume simplex analysis (MVSA). The underlying mixing model is linear; i.e., the mixed hyperspectral vectors are modeled by a linear mixture of the end-member signatures weighted by the correspondent abundance fractions. MVSA approaches hyperspectral unmixing by fitting a minimum volume simplex to the hyperspectral data, constraining the abundance fractions to belong to the probability simplex. The resulting optimization problem is solved by implementing a sequence of quadratically constrained subproblems. In a final step, the hard constraint on the abundance fractions is replaced with a hinge type loss function to account for outliers and noise. We illustrate the state-of-the-art performance of the MVSA algorithm in unmixing simulated data sets. We are mainly concerned with the realistic scenario in which the pure pixel assumption (i.e., there exists at least one pure pixel per end member) is not fulfilled. In these conditions, the MVSA yields much better performance than the pure pixel based algorithms.
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paper_title: Bayesian nonnegative Matrix Factorization with volume prior for unmixing of hyperspectral images

paper_content:

In hyperspectral image analysis the objective is to unmix a set of acquired pixels into pure spectral signatures (end-members) and corresponding fractional abundances. The Non-negative Matrix Factorization (NMF) methods have received a lot of attention for this unmixing process. Many of these NMF based unmixing algorithms are based on sparsity regularization encouraging pure spectral endmembers, but this is not optimal for certain applications, such as foods, where abundances are not sparse. The pixels will theoretically lie on a simplex and hence the endmembers can be estimated as the vertices of the smallest enclosing simplex. In this context we present a Bayesian framework employing a volume constraint for the NMF algorithm, where the posterior distribution is numerically sampled from using a Gibbs sampling procedure. We evaluate the method on synthetical and real hyperspectral data of wheat kernels.
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paper_title: A comparison of deterministic and probabilistic approaches to endmember representation

paper_content:

The piece-wise convex multiple model endmember detection algorithm (P-COMMEND) and the Piece-wise Convex End-member detection (PCE) algorithm autonomously estimate many sets of endmembers to represent a hyperspectral image. A piece-wise convex model with several sets of endmembers is more effective for representing non-convex hyperspectral imagery over the standard convex geometry model (or linear mixing model). The terms of the objective function in P-COMMEND are based on geometric properties of the input data and the endmember estimates. In this paper, the P-COMMEND algorithm is extended to autonomously determine the number of sets of endmembers needed. The number of sets of endmembers, or convex regions, is determined by incorporating the competitive agglomeration algorithm into P-COMMEND. Results are shown comparing the Competitive Agglomeration P-COMMEND (CAP) algorithm to results found using the statistical PCE endmember detection method.
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paper_title: Modelling cognitive representations

paper_content:

This thesis analyzes the modelling of visual cognitive representations based on extracting cognitive components from the MNIST dataset of handwritten digits using simple unsupervised linear and non-linear matrix factorizations both non-negative and unconstrained based on gradient descent learning. We introduce two different classes of generative models for modelling the cognitive data: Mixture Models and Deep Network Models. Mixture models based on K-Means, Guassian and factor analyzer kernel functions are presented as simple generative models in a general framework. From simulations we analyze the generative properties of these models and show how they render insufficient to proper model the complex distribution of the visual cognitive data. Motivated by the introduction of deep belief nets by Hinton et al. [12] we propose a simpler generative deep network model based on cognitive components. A theoretical framework is presented as individual modules for building a generative hierarchical network model. We analyze the performance in terms of classification and generation of MNIST digits and show how our simplifications compared to Hinton et al. [12] leads to degraded performance. In this respect we outline the differences and conjecture obvious improvements.
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paper_title: A Variable Splitting Augmented Lagrangian Approach to Linear Spectral Unmixing

paper_content:

This paper presents a new linear hyperspectral unmixing method of the minimum volume class, termed simplex identification via split augmented Lagrangian (SISAL). Following Craig's seminal ideas, hyperspectral linear unmixing amounts to finding the minimum vol­ ume simplex containing the hyperspectral vectors. This is a noncon­ vex optimization problem with convex constraints. In the proposed approach, the positivity constraints, forcing the spectral vectors to belong to the convex hull of the endmember signatures, are replaced by soft constraints. The obtained problem is solved by a sequence of augmented Lagrangian optimizations. The resulting algorithm is very fast and able so solve problems far beyond the reach of the current state-of-the art algorithms. The effectiveness of SISAL is illustrated with simulated data.
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paper_title: Fully Constrained Least Squares Spectral Unmixing by Simplex Projection

paper_content:

We present a new algorithm for linear spectral mixture analysis, which is capable of supervised unmixing of hyperspectral data while respecting the constraints on the abundance coefficients. This simplex-projection unmixing algorithm is based upon the equivalence of the fully constrained least squares problem and the problem of projecting a point onto a simplex. We introduce several geometrical properties of high-dimensional simplices and combine them to yield a recursive algorithm for solving the simplex-projection problem. A concrete implementation of the algorithm for large data sets is provided, and the algorithm is benchmarked against well-known fully constrained least squares unmixing (FCLSU) techniques, on both artificial data sets and real hyperspectral data collected over the Cuprite mining region. Unlike previous algorithms for FCLSU, the presented algorithm possesses no optimization steps and is completely analytical, severely reducing the required processing power.
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paper_title: Minimum-volume transforms for remotely sensed data

paper_content:

Scatter diagrams for multispectral remote sensing data tend to be triangular, in the two-band case, pyramidal for three bands, and so on. They radiate away from the so-called darkpoint, which represents the scanner's response to an un-illuminated target. A minimum-volume transform may be described (provisionally) as a nonorthogonal linear transformation of the multivariate data to new axes passing through the dark point, with directions chosen such that they (for two bands), or the new coordinate planes (for three bands, etc.) embrace the data cloud as tightly as possible. The reason for the observed shapes of scatter diagrams is to be found in the theory of linear mixing at the subfootprint scale. Thus, suitably defined, minimum-volume transforms can often be used to unmix images into new spatial variables showing the proportions of the different cover types present, a type of enhancement that is not only intense, but physically meaningful. The present paper furnishes details for constructing computer programs to effect this operation. It will serve as a convenient technical source that may be referenced in subsequent, more profusely illustrated publications that address the intended application, the mapping of surface mineralogy. >
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paper_title: ICE: a statistical approach to identifying endmembers in hyperspectral images

paper_content:

Several of the more important endmember-finding algorithms for hyperspectral data are discussed and some of their shortcomings highlighted. A new algorithm - iterated constrained endmembers (ICE) - which attempts to address these shortcomings is introduced. An example of its use is given. There is also a discussion of the advantages and disadvantages of normalizing spectra before the application of ICE or other endmember-finding algorithms.
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paper_title: Endmember Extraction From Highly Mixed Data Using Minimum Volume Constrained Nonnegative Matrix Factorization

paper_content:

Endmember extraction is a process to identify the hidden pure source signals from the mixture. In the past decade, numerous algorithms have been proposed to perform this estimation. One commonly used assumption is the presence of pure pixels in the given image scene, which are detected to serve as endmembers. When such pixels are absent, the image is referred to as the highly mixed data, for which these algorithms at best can only return certain data points that are close to the real endmembers. To overcome this problem, we present a novel method without the pure-pixel assumption, referred to as the minimum volume constrained nonnegative matrix factorization (MVC-NMF), for unsupervised endmember extraction from highly mixed image data. Two important facts are exploited: First, the spectral data are nonnegative; second, the simplex volume determined by the endmembers is the minimum among all possible simplexes that circumscribe the data scatter space. The proposed method takes advantage of the fast convergence of NMF schemes, and at the same time eliminates the pure-pixel assumption. The experimental results based on a set of synthetic mixtures and a real image scene demonstrate that the proposed method outperforms several other advanced endmember detection approaches
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paper_title: Multispectral and hyperspectral image analysis with convex cones

paper_content:

A new approach to multispectral and hyperspectral image analysis is presented. This method, called convex cone analysis (CCA), is based on the bet that some physical quantities such as radiance are nonnegative. The vectors formed by discrete radiance spectra are linear combinations of nonnegative components, and they lie inside a nonnegative, convex region. The object of CCA is to find the boundary points of this region, which can be used as endmember spectra for unmixing or as target vectors for classification. To implement this concept, the authors find the eigenvectors of the sample spectral correlation matrix of the image. Given the number of endmembers or classes, they select as many eigenvectors corresponding to the largest eigenvalues. These eigenvectors are used as a basis to form linear combinations that have only nonnegative elements, and thus they lie inside a convex cone. The vertices of the convex cone will be those points whose spectral vector contains as many zero elements as the number of eigenvectors minus one. Accordingly, a mixed pixel can be decomposed by identifying the vertices that were used to form its spectrum. An algorithm for finding the convex cone boundaries is presented, and applications to unsupervised unmixing and classification are demonstrated with simulated data as well as experimental data from the hyperspectral digital imagery collection experiment (HYDICE).
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paper_title: Multiple model endmember detection based on spectral and spatial information

paper_content:

We introduce a new spectral mixture analysis approach. Unlike most available approaches that only use the spectral information, this approach uses the spectral and spatial information available in the hyperspectral data. Moreover, it does not assume a global convex geometry model that encompasses all the data but rather multiple local convex models. Both the multiple model boundaries and the model's endmembers and abundances are fuzzy. This allows points to belong to multiple groups with different membership degrees. Our approach is based on minimizing a joint objective function to simultaneously learn the underling fuzzy multiple convex geometry models and find a robust estimate of the model's endmembers and abundances.
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paper_title: Algorithms for Non-negative Matrix Factorization

paper_content:

Non-negative matrix factorization (NMF) has previously been shown to be a useful decomposition for multivariate data. Two different multiplicative algorithms for NMF are analyzed. They differ only slightly in the multiplicative factor used in the update rules. One algorithm can be shown to minimize the conventional least squares error while the other minimizes the generalized Kullback-Leibler divergence. The monotonic convergence of both algorithms can be proven using an auxiliary function analogous to that used for proving convergence of the Expectation-Maximization algorithm. The algorithms can also be interpreted as diagonally rescaled gradient descent, where the rescaling factor is optimally chosen to ensure convergence.
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paper_title: Fully Constrained Linear Spectral Unmixing: Analytic Solution Using Fuzzy Sets

paper_content:

The linear mixture model is a convenient way to describe image pixels as a linear combination of pure spectra - termed end-members. The fractional contribution from each end-member is calculated through inversion of the linear model. Despite the simplicity of the model, a nonnegativity constraint that is imposed on the fractions leads to an unmixing problem for which it is hard to find a closed analytical solution. Current solutions to this problem involve iterative algorithms, which are computationally intensive and not appropriate for unmixing large number of pixels. This paper presents an algorithm to build fuzzy membership functions that are equivalent to the least square solution of the fully constrained linear spectral unmixing problem. The efficiency and effectiveness of the proposed solution is demonstrated using both simulated and real data.
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paper_title: Chance-Constrained Robust Minimum-Volume Enclosing Simplex Algorithm for Hyperspectral Unmixing

paper_content:

Effective unmixing of hyperspectral data cube under a noisy scenario has been a challenging research problem in remote sensing arena. A branch of existing hyperspectral unmixing algorithms is based on Craig's criterion, which states that the vertices of the minimum-volume simplex enclosing the hyperspectral data should yield high fidelity estimates of the endmember signatures associated with the data cloud. Recently, we have developed a minimum-volume enclosing simplex (MVES) algorithm based on Craig's criterion and validated that the MVES algorithm is very useful to unmix highly mixed hyperspectral data. However, the presence of noise in the observations expands the actual data cloud, and as a consequence, the endmember estimates obtained by applying Craig-criterion-based algorithms to the noisy data may no longer be in close proximity to the true endmember signatures. In this paper, we propose a robust MVES (RMVES) algorithm that accounts for the noise effects in the observations by employing chance constraints. These chance constraints in turn control the volume of the resulting simplex. Under the Gaussian noise assumption, the chance-constrained MVES problem can be formulated into a deterministic nonlinear program. The problem can then be conveniently handled by alternating optimization, in which each subproblem involved is handled by using sequential quadratic programming solvers. The proposed RMVES is compared with several existing benchmark algorithms, including its predecessor, the MVES algorithm. Monte Carlo simulations and real hyperspectral data experiments are presented to demonstrate the efficacy of the proposed RMVES algorithm.
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paper_title: Robust Endmember detection using L1 norm factorization

paper_content:

The results from L 1 -Endmembers display the algorithm's stability and accuracy with increasing levels of noise. The algorithm was extremely stable in the number of endmembers when compared to the SPICE algorithm and the Virtual Dimensionality methods for estimating the number of endmembers. Furthermore, the results shown for this algorithm were generated with the same parameter set for all of the data sets, from two-dimensional data to 51-dimensional real hyperspectral data. This indicates L 1 -Endmembers may lack of sensitivity to parameter value settings.
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paper_title: Hyperspectral Unmixing via $L_{1/2}$ Sparsity-Constrained Nonnegative Matrix Factorization

paper_content:

Hyperspectral unmixing is a crucial preprocessing step for material classification and recognition. In the last decade, nonnegative matrix factorization (NMF) and its extensions have been intensively studied to unmix hyperspectral imagery and recover the material end-members. As an important constraint for NMF, sparsity has been modeled making use of the L1 regularizer. Unfortunately, the L1 regularizer cannot enforce further sparsity when the full additivity constraint of material abundances is used, hence limiting the practical efficacy of NMF methods in hyperspectral unmixing. In this paper, we extend the NMF method by incorporating the L1/2 sparsity constraint, which we name L1/2 -NMF. The L1/2 regularizer not only induces sparsity but is also a better choice among Lq(0 <; q <; 1) regularizers. We propose an iterative estimation algorithm for L1/2-NMF, which provides sparser and more accurate results than those delivered using the L1 norm. We illustrate the utility of our method on synthetic and real hyperspectral data and compare our results to those yielded by other state-of-the-art methods.
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paper_title: PCE: Piecewise Convex Endmember Detection

paper_content:

A new hyperspectral endmember detection method that represents endmembers as distributions, autonomously partitions the input data set into several convex regions, and simultaneously determines endmember distributions (EDs) and proportion values for each convex region is presented. Spectral unmixing methods that treat endmembers as distributions or hyperspectral images as piecewise convex data sets have not been previously developed. Piecewise convex endmember (PCE) detection can be viewed in two parts. The first part, the ED detection algorithm, estimates a distribution for each endmember rather than estimating a single spectrum. By using EDs, PCE can incorporate an endmember's inherent spectral variation and the variation due to changing environmental conditions. ED uses a new sparsity-promoting polynomial prior while estimating abundance values. The second part of PCE partitions the input hyperspectral data set into convex regions and estimates EDs and proportions for each of these regions. The number of convex regions is determined autonomously using the Dirichlet process. PCE is effective at handling highly mixed hyperspectral images where all of the pixels in the scene contain mixtures of multiple endmembers. Furthermore, each convex region found by PCE conforms to the convex geometry model for hyperspectral imagery. This model requires that the proportions associated with a pixel be nonnegative and sum to one. Algorithm results on hyperspectral data indicate that PCE produces endmembers that represent the true ground-truth classes of the input data set. The algorithm can also effectively represent endmembers as distributions, thus incorporating an endmember's spectral variability.
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paper_title: Unmixing of Hyperspectral Images using Bayesian Non-negative Matrix Factorization with Volume Prior

paper_content:

Hyperspectral imaging can be used in assessing the quality of foods by decomposing the image into constituents such as protein, starch, and water. Observed data can be considered a mixture of underlying characteristic spectra (endmembers), and estimating the constituents and their abundances requires efficient algorithms for spectral unmixing. We present a Bayesian spectral unmixing algorithm employing a volume constraint and propose an inference procedure based on Gibbs sampling. We evaluate the method on synthetic and real hyperspectral data of wheat kernels. Results show that our method perform as good or better than existing volume constrained methods. Further, our method gives credible intervals for the endmembers and abundances, which allows us to asses the confidence of the results.
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paper_title: Algorithm taxonomy for hyperspectral unmixing

paper_content:

In this paper, we introduce a set of taxonomies that hierarchically organize and specify algorithms associated with hyperspectral unmixing. Our motivation is to collectively organize and relate algorithms in order to assess the current state-of-the-art in the field and to facilitate objective comparisons between methods. The hyperspectral sensing community is populated by investigators with disparate scientific backgrounds and, speaking in their respective languages, efforts in spectral unmixing developed within disparate communities have inevitably led to duplication. We hope our analysis removes this ambiguity and redundancy by using a standard vocabulary, and that the presentation we provide clearly summarizes what has and has not been done. As we shall see, the framework for the taxonomies derives its organization from the fundamental, philosophical assumptions imposed on the problem, rather than the common calculations they perform, or the similar outputs they might yield.
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paper_title: Joint Bayesian endmember extraction and linear unmixing for hyperspectral imagery

paper_content:

This paper studies a fully Bayesian algorithm for endmember extraction and abundance estimation for hyperspectral imagery. Each pixel of the hyperspectral image is decomposed as a linear combination of pure endmember spectra following the linear mixing model. The estimation of the unknown endmember spectra is conducted in a unified manner by generating the posterior distribution of abundances and endmember parameters under a hierarchical Bayesian model. This model assumes conjugate prior distributions for these parameters, accounts for non-negativity and full-additivity constraints, and exploits the fact that the endmember proportions lie on a lower dimensional simplex. A Gibbs sampler is proposed to overcome the complexity of evaluating the resulting posterior distribution. This sampler generates samples distributed according to the posterior distribution and estimates the unknown parameters using these generated samples. The accuracy of the joint Bayesian estimator is illustrated by simulations conducted on synthetic and real AVIRIS images.
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paper_title: Implementation strategies for hyperspectral unmixing using Bayesian source separation

paper_content:

Bayesian positive source separation (BPSS) is a useful unsupervised approach for hyperspectral data unmixing, where numerical nonnegativity of spectra and abundances has to be ensured, such as in remote sensing. Moreover, it is sensible to impose a sum-to-one (full additivity) constraint to the estimated source abundances in each pixel. Even though nonnegativity and full additivity are two necessary properties to get physically interpretable results, the use of BPSS algorithms has so far been limited by high computation time and large memory requirements due to the Markov chain Monte Carlo calculations. An implementation strategy that allows one to apply these algorithms on a full hyperspectral image, as it is typical in earth and planetary science, is introduced. The effects of pixel selection and the impact of such sampling on the relevance of the estimated component spectra and abundance maps, as well as on the computation times, are discussed. For that purpose, two different data sets have been used: a synthetic one and a real hyperspectral image from Mars.
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paper_title: Hyperspectral Unmixing Based on Mixtures of Dirichlet Components

paper_content:

This paper introduces a new unsupervised hyperspectral unmixing method conceived to linear but highly mixed hyperspectral data sets, in which the simplex of minimum volume, usually estimated by the purely geometrically based algorithms, is far way from the true simplex associated with the endmembers. The proposed method, an extension of our previous studies, resorts to the statistical framework. The abundance fraction prior is a mixture of Dirichlet densities, thus automatically enforcing the constraints on the abundance fractions imposed by the acquisition process, namely, nonnegativity and sum-to-one. A cyclic minimization algorithm is developed where the following are observed: 1) The number of Dirichlet modes is inferred based on the minimum description length principle; 2) a generalized expectation maximization algorithm is derived to infer the model parameters; and 3) a sequence of augmented Lagrangian-based optimizations is used to compute the signatures of the endmembers. Experiments on simulated and real data are presented to show the effectiveness of the proposed algorithm in unmixing problems beyond the reach of the geometrically based state-of-the-art competitors.
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paper_title: Bayesian separation of spectral sources under non-negativity and full additivity constraints

paper_content:

This paper addresses the problem of separating spectral sources which are linearly mixed with unknown proportions. The main difficulty of the problem is to ensure the full additivity (sum-to-one) of the mixing coefficients and non-negativity of sources and mixing coefficients. A Bayesian estimation approach based on Gamma priors was recently proposed to handle the non-negativity constraints in a linear mixture model. However, incorporating the full additivity constraint requires further developments. This paper studies a new hierarchical Bayesian model appropriate to the non-negativity and sum-to-one constraints associated to the regressors and regression coefficients of linear mixtures. The estimation of the unknown parameters of this model is performed using samples generated using an appropriate Gibbs sampler. The performance of the proposed algorithm is evaluated through simulation results conducted on synthetic mixture models. The proposed approach is also applied to the processing of multicomponent chemical mixtures resulting from Raman spectroscopy.
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paper_title: Unmixing Hyperspectral Data

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In hyperspectral imagery one pixel typically consists of a mixture of the reflectance spectra of several materials, where the mixture coefficients correspond to the abundances of the constituting materials. We assume linear combinations of reflectance spectra with some additive normal sensor noise and derive a probabilistic MAP framework for analyzing hyperspectral data. As the material reflectance characteristics are not know a priori, we face the problem of unsupervised linear unmixing. The incorporation of different prior information (e.g. positivity and normalization of the abundances) naturally leads to a family of interesting algorithms, for example in the noise-free case yielding an algorithm that can be understood as constrained independent component analysis (ICA). Simulations underline the usefulness of our theory.
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paper_title: Does independent component analysis play a role in unmixing hyperspectral data

paper_content:

Independent component analysis (ICA) has recently been proposed as a tool to unmix hyperspectral data. ICA is founded on two assumptions: 1) the observed spectrum vector is a linear mixture of the constituent spectra (endmember spectra) weighted by the correspondent abundance fractions (sources); 2)sources are statistically independent. Independent factor analysis (IFA) extends ICA to linear mixtures of independent sources immersed in noise. Concerning hyperspectral data, the first assumption is valid whenever the multiple scattering among the distinct constituent substances (endmembers) is negligible, and the surface is partitioned according to the fractional abundances. The second assumption, however, is violated, since the sum of abundance fractions associated to each pixel is constant due to physical constraints in the data acquisition process. Thus, sources cannot be statistically independent, this compromising the performance of ICA/IFA algorithms in hyperspectral unmixing. This paper studies the impact of hyperspectral source statistical dependence on ICA and IFA performances. We conclude that the accuracy of these methods tends to improve with the increase of the signature variability, of the number of endmembers, and of the signal-to-noise ratio. In any case, there are always endmembers incorrectly unmixed. We arrive to this conclusion by minimizing the mutual information of simulated and real hyperspectral mixtures. The computation of mutual information is based on fitting mixtures of Gaussians to the observed data. A method to sort ICA and IFA estimates in terms of the likelihood of being correctly unmixed is proposed.
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paper_title: Blind Decomposition of Transmission Light Microscopic Hyperspectral Cube Using Sparse Representation

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In this paper, we address the problem of fully automated decomposition of hyperspectral images for transmission light microscopy. The hyperspectral images are decomposed into spectrally homogeneous compounds. The resulting compounds are described by their spectral characteristics and optical density. We present the multiplicative physical model of image formation in transmission light microscopy, justify reduction of a hyperspectral image decomposition problem to a blind source separation problem, and provide method for hyperspectral restoration of separated compounds. In our approach, dimensionality reduction using principal component analysis (PCA) is followed by a blind source separation (BSS) algorithm. The BSS method is based on sparsifying transformation of observed images and relative Newton optimization procedure. The presented method was verified on hyperspectral images of biological tissues. The method was compared to the existing approach based on nonnegative matrix factorization. Experiments showed that the presented method is faster and better separates the biological compounds from imaging artifacts. The results obtained in this work may be used for improving automatic microscope hardware calibration and computer-aided diagnostics.
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paper_title: Analyzing hyperspectral data with independent component analysis

paper_content:

Hyperspectral image sensors provide images with a large number of contiguous spectral channels per pixel and enable information about different materials within a pixel to be obtained. The problem of spectrally unmixing materials may be viewed as a specific case of the blind source separation problem where data consists of mixed signals and the goal is to determine the contribution of each mineral to the mix without prior knowledge of the minerals in the mix. The technique of independent component analysis (ICA) assumes that the spectral components are close to statistically independent and provides an unsupervised method for blind source separation. We introduce contextual ICA in the context of hyperspectral data analysis and apply the method to mineral data from synthetically mixed minerals and real image signatures.© (1998) COPYRIGHT SPIE--The International Society for Optical Engineering. Downloading of the abstract is permitted for personal use only.
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paper_title: Endmember Extraction From Highly Mixed Data Using Minimum Volume Constrained Nonnegative Matrix Factorization

paper_content:

Endmember extraction is a process to identify the hidden pure source signals from the mixture. In the past decade, numerous algorithms have been proposed to perform this estimation. One commonly used assumption is the presence of pure pixels in the given image scene, which are detected to serve as endmembers. When such pixels are absent, the image is referred to as the highly mixed data, for which these algorithms at best can only return certain data points that are close to the real endmembers. To overcome this problem, we present a novel method without the pure-pixel assumption, referred to as the minimum volume constrained nonnegative matrix factorization (MVC-NMF), for unsupervised endmember extraction from highly mixed image data. Two important facts are exploited: First, the spectral data are nonnegative; second, the simplex volume determined by the endmembers is the minimum among all possible simplexes that circumscribe the data scatter space. The proposed method takes advantage of the fast convergence of NMF schemes, and at the same time eliminates the pure-pixel assumption. The experimental results based on a set of synthetic mixtures and a real image scene demonstrate that the proposed method outperforms several other advanced endmember detection approaches
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paper_title: Hyperspectral unmixing algorithm via dependent component analysis

paper_content:

This paper introduces a new method to blindly unmix hyperspectral data, termed dependent component analysis (DECA). This method decomposes a hyperspectral images into a collection of reflectance (or radiance) spectra of the materials present in the scene (end member signatures) and the corresponding abundance fractions at each pixel. DECA assumes that each pixel is a linear mixture of the end-members signatures weighted by the correspondent abundance fractions. These abundances are modeled as mixtures of Dirichlet densities, thus enforcing the constraints on abundance fractions imposed by the acquisition process, namely non-negativity and constant sum. The mixing matrix is inferred by a generalized expectation-maximization (GEM) type algorithm. This method overcomes the limitations of unmixing methods based on independent component analysis (ICA) and on geometrical based approaches. The effectiveness of the proposed method is illustrated using simulated data based on U.S.G.S. laboratory spectra and real hyperspectral data collected by the AVIRIS sensor over Cuprite, Nevada.
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paper_title: On the decomposition of Mars hyperspectral data by ICA and Bayesian positive source separation

paper_content:

The surface of Mars is currently being imaged with an unprecedented combination of spectral and spatial resolution. This high resolution, and its spectral range, gives the ability to pinpoint chemical species on the surface and the atmosphere of Mars more accurately than before. The subject of this paper is to present a method to extract informations on these chemicals from hyperspectral images. A first approach, based on independent component analysis (ICA) [P. Comon, Independent component analysis, a new concept? Signal Process. 36 (3) (1994) 287-314], is able to extract artifacts and locations of CO"2 and H"2O ices. However, the main independence assumption and some basic properties (like the positivity of images and spectra) being unverified, the reliability of all the independent components (ICs) is weak. For improving the component extraction and consequently the endmember classification, a combination of spatial ICA with spectral Bayesian positive source separation (BPSS) [S. Moussaoui, D. Brie, A. Mohammad-Djafari, C. Carteret, Separation of non-negative mixture of non-negative sources using a Bayesian approach and MCMC sampling, IEEE Trans. Signal Process. 54 (11) (2006) 4133-4145] is proposed. To reduce the computational burden, the basic idea is to use spatial ICA yielding a rough classification of pixels, which allows selection of small, but relevant, number of pixels. Then, BPSS is applied for the estimation of the source spectra using the spectral mixtures provided by this reduced set of pixels. Finally, the abundances of the components are assessed on the whole pixels of the images. Results of this approach are shown and evaluated by comparison with available reference spectra.
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paper_title: Separation of Non-Negative Mixture of Non-Negative Sources Using a Bayesian Approach and MCMC Sampling

paper_content:

This paper addresses blind-source separation in the case where both the source signals and the mixing coefficients are non-negative. The problem is referred to as non-negative source separation and the main application concerns the analysis of spectrometric data sets. The separation is performed in a Bayesian framework by encoding non-negativity through the assignment of Gamma priors on the distributions of both the source signals and the mixing coefficients. A Markov chain Monte Carlo (MCMC) sampling procedure is proposed to simulate the resulting joint posterior density from which marginal posterior mean estimates of the source signals and mixing coefficients are obtained. Results obtained with synthetic and experimental spectra are used to discuss the problem of non-negative source separation and to illustrate the effectiveness of the proposed method
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paper_title: Imaging Spectroscopy and the Airborne Visible/Infrared Imaging Spectrometer (AVIRIS)

paper_content:

Abstract Imaging spectroscopy is of growing interest as a new approach to Earth remote sensing. The Airborne Visible/Infrared Imaging Spectrometer (AVIRIS) was the first imaging sensor to measure the solar reflected spectrum from 400 nm to 2500 nm at 10 nm intervals. The calibration accuracy and signal-to-noise of AVIRIS remain unique. The AVIRIS system as well as the science research and applications have evolved significantly in recent years. The initial design and upgraded characteristics of the AVIRIS system are described in terms of the sensor, calibration, data system, and flight operation. This update on the characteristics of AVIRIS provides the context for the science research and applications that use AVIRIS data acquired in the past several years. Recent science research and applications are reviewed spanning investigations of atmospheric correction, ecology and vegetation, geology and soils, inland and coastal waters, the atmosphere, snow and ice hydrology, biomass burning, environmental hazards, satellite simulation and calibration, commercial applications, spectral algorithms, human infrastructure, as well as spectral modeling.
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paper_title: Bayesian linear unmixing of hyperspectral images corrupted by colored Gaussian noise with unknown covariance matrix

paper_content:

This paper addresses the problem of unmixing hyperspectral images contamined by additive colored noise. Each pixel of the image is modeled as a linear combination of pure materials (denoted as end-members) corrupted by an additive zero mean Gaussian noise sequence with unknown covariance matrix. Appropriate priors are defined ensuring positivity and additivity constraints on the mixture coefficients (denoted as abundances). These coefficients as well as the noise covariance matrix are then estimated from their joint posterior distribution. A Gibbs sampling strategy generates abundances and noise covariance matrices distributed according to the joint posterior. These samples are then averaged for minimum mean square error estimation.
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paper_title: Unsupervised signature extraction and separation in hyperspectral images: a noise-adjusted fast independent component analysis approach

paper_content:

Multispectral/hyperspectral imaging spectrometry in earth re- mote sensing applications mostly focuses on determining the identity and abundance of materials in a geographic area of interest. Without any prior knowledge, however, it is generally very difficult to identify and determine how many endmembers reside in a scene. We cope with this limitation by estimating the number of endmembers using a noise- adjusted version of the transformed Gerschgorin disk approach (NATGD). This estimated result is then applied to a noise-adjusted ver- sion of fast independent component analysis (NAFICA). Experimental results indicate that NAFICA offers a new approach for unsupervised signature extraction and separation in hyperspectral images. © 2000 So- ciety of Photo-Optical Instrumentation Engineers. (S0091-3286(00)02004-3)
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paper_title: Signal processing for hyperspectral image exploitation

paper_content:

Electro-optical remote sensing involves the acquisition of information about an object or scene without coming into physical contact with it. This is achieved by exploiting the fact that the materials comprising the various objects in a scene reflect, absorb, and emit electromagnetic radiation in ways characteristic of their molecular composition and shape. If the radiation arriving at the sensor is measured at each wavelength over a sufficiently broad spectral band, the resulting spectral signature, or simply spectrum, can be used (in principle) to uniquely characterize and identify any given material. An important function of hyperspectral signal processing is to eliminate the redundancy in the spectral and spatial sample data while preserving the high-quality features needed for detection, discrimination, and classification. This dimensionality reduction is implemented in a scene-dependent (adaptive) manner and may be implemented as a distinct step in the processing or as an integral part of the overall algorithm. The most widely used algorithm for dimensionality reduction is principal component analysis (PCA) or, equivalently, Karhunen-Loeve transformation.
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paper_title: Hyperspectral image data analysis

paper_content:

The fundamental basis for space-based remote sensing is that information is potentially available from the electromagnetic energy field arising from the Earth's surface and, in particular, from the spatial, spectral, and temporal variations in that field. Rather than focusing on the spatial variations, which imagery perhaps best conveys, why not move on to look at how the spectral variations might be used. The idea was to enlarge the size of a pixel until it includes an area that is characteristic from a spectral response standpoint for the surface cover to be discriminated. The article includes an example of an image space representation, using three bands to simulate a color IR photograph of an airborne hyperspectral data set over the Washington, DC, mall.
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paper_title: Fuzzy Spectral and Spatial Feature Integration for Classification of Nonferrous Materials in Hyperspectral Data

paper_content:

Hyperspectral data allows the construction of more elaborate models to sample the properties of the nonferrous materials than the standard RGB color representation. In this paper, the nonferrous waste materials are studied as they cannot be sorted by classical procedures due to their color, weight and shape similarities. The experimental results presented in this paper reveal that factors such as the various levels of oxidization of the waste materials and the slight differences in their chemical composition preclude the use of the spectral features in a simplistic manner for robust material classification. To address these problems, the proposed FUSSER (fuzzy spectral and spatial classifier) algorithm detailed in this paper merges the spectral and spatial features to obtain a combined feature vector that is able to better sample the properties of the nonferrous materials than the single pixel spectral features when applied to the construction of multivariate Gaussian distributions. This approach allows the implementation of statistical region merging techniques in order to increase the performance of the classification process. To achieve an efficient implementation, the dimensionality of the hyperspectral data is reduced by constructing bio-inspired spectral fuzzy sets that minimize the amount of redundant information contained in adjacent hyperspectral bands. The experimental results indicate that the proposed algorithm increased the overall classification rate from 44% using RGB data up to 98% when the spectral-spatial features are used for nonferrous material classification.
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paper_title: Atomic Decomposition by Basis Pursuit

paper_content:

The time-frequency and time-scale communities have recently developed a large number of overcomplete waveform dictionaries --- stationary wavelets, wavelet packets, cosine packets, chirplets, and warplets, to name a few. Decomposition into overcomplete systems is not unique, and several methods for decomposition have been proposed, including the method of frames (MOF), Matching pursuit (MP), and, for special dictionaries, the best orthogonal basis (BOB). ::: Basis Pursuit (BP) is a principle for decomposing a signal into an "optimal" superposition of dictionary elements, where optimal means having the smallest l1 norm of coefficients among all such decompositions. We give examples exhibiting several advantages over MOF, MP, and BOB, including better sparsity and superresolution. BP has interesting relations to ideas in areas as diverse as ill-posed problems, in abstract harmonic analysis, total variation denoising, and multiscale edge denoising. ::: BP in highly overcomplete dictionaries leads to large-scale optimization problems. With signals of length 8192 and a wavelet packet dictionary, one gets an equivalent linear program of size 8192 by 212,992. Such problems can be attacked successfully only because of recent advances in linear programming by interior-point methods. We obtain reasonable success with a primal-dual logarithmic barrier method and conjugate-gradient solver.
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paper_title: On the Uniqueness of Nonnegative Sparse Solutions to Underdetermined Systems of Equations

paper_content:

An underdetermined linear system of equations Ax = b with nonnegativity constraint x ges 0 is considered. It is shown that for matrices A with a row-span intersecting the positive orthant, if this problem admits a sufficiently sparse solution, it is necessarily unique. The bound on the required sparsity depends on a coherence property of the matrix A. This coherence measure can be improved by applying a conditioning stage on A, thereby strengthening the claimed result. The obtained uniqueness theorem relies on an extended theoretical analysis of the lscr0 - lscr1 equivalence developed here as well, considering a matrix A with arbitrary column norms, and an arbitrary monotone element-wise concave penalty replacing the lscr1-norm objective function. Finally, from a numerical point of view, a greedy algorithm-a variant of the matching pursuit-is presented, such that it is guaranteed to find this sparse solution. It is further shown how this algorithm can benefit from well-designed conditioning of A .
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paper_title: Least Angle Regression

paper_content:

The purpose of model selection algorithms such as All Subsets, Forward ::: Selection and Backward Elimination is to choose a linear model on the basis of the same set of data to which the model will be applied. Typically we have available a large collection of possible covariates from which we hope to select a parsimonious set for the efficient prediction of a response variable. Least Angle Regression (LARS), a new model selection algorithm, is a useful and less greedy version of traditional forward selection methods. ::: Three main properties are derived: (1) A simple modification of the LARS algorithm implements the Lasso, an attractive version of ordinary least squares that constrains the sum of the absolute regression coefficients; the LARS modification calculates all possible Lasso estimates for a given problem, using an order of magnitude less computer time than previous methods. ::: (2) A different LARS modification efficiently implements Forward Stagewise linear regression, another promising new model selection method;
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paper_title: Orthogonal matching pursuit: recursive function approximation with applications to wavelet decomposition

paper_content:

We describe a recursive algorithm to compute representations of functions with respect to nonorthogonal and possibly overcomplete dictionaries of elementary building blocks e.g. affine (wavelet) frames. We propose a modification to the matching pursuit algorithm of Mallat and Zhang (1992) that maintains full backward orthogonality of the residual (error) at every step and thereby leads to improved convergence. We refer to this modified algorithm as orthogonal matching pursuit (OMP). It is shown that all additional computation required for the OMP algorithm may be performed recursively. >
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paper_title: Learning Sparse Codes for Hyperspectral Imagery

paper_content:

The spectral features in hyperspectral imagery (HSI) contain significant structure that, if properly characterized, could enable more efficient data acquisition and improved data analysis. Because most pixels contain reflectances of just a few materials, we propose that a sparse coding model is well-matched to HSI data. Sparsity models consider each pixel as a combination of just a few elements from a larger dictionary, and this approach has proven effective in a wide range of applications. Furthermore, previous work has shown that optimal sparse coding dictionaries can be learned from a dataset with no other a priori information (in contrast to many HSI “endmember” discovery algorithms that assume the presence of pure spectra or side information). We modified an existing unsupervised learning approach and applied it to HSI data (with significant ground truth labeling) to learn an optimal sparse coding dictionary. Using this learned dictionary, we demonstrate three main findings: 1) the sparse coding model learns spectral signatures of materials in the scene and locally approximates nonlinear manifolds for individual materials; 2) this learned dictionary can be used to infer HSI-resolution data with very high accuracy from simulated imagery collected at multispectral-level resolution, and 3) this learned dictionary improves the performance of a supervised classification algorithm, both in terms of the classifier complexity and generalization from very small training sets.
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paper_title: Spectral unmixing

paper_content:

Spectral unmixing using hyperspectral data represents a significant step in the evolution of remote decompositional analysis that began with multispectral sensing. It is a consequence of collecting data in greater and greater quantities and the desire to extract more detailed information about the material composition of surfaces. Linear mixing is the key assumption that has permitted well-known algorithms to be adapted to the unmixing problem. In fact, the resemblance of the linear mixing model to system models in other areas has permitted a significant legacy of algorithms from a wide range of applications to be adapted to unmixing. However, it is still unclear whether the assumption of linearity is sufficient to model the mixing process in every application of interest. It is clear, however, that the applicability of models and techniques is highly dependent on the variety of circumstances and factors that give rise to mixed pixels. The outputs of spectral unmixing, endmember, and abundance estimates are important for identifying the material composition of mixtures.
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paper_title: Compressed sensing

paper_content:

Suppose x is an unknown vector in Ropfm (a digital image or signal); we plan to measure n general linear functionals of x and then reconstruct. If x is known to be compressible by transform coding with a known transform, and we reconstruct via the nonlinear procedure defined here, the number of measurements n can be dramatically smaller than the size m. Thus, certain natural classes of images with m pixels need only n=O(m1/4log5/2(m)) nonadaptive nonpixel samples for faithful recovery, as opposed to the usual m pixel samples. More specifically, suppose x has a sparse representation in some orthonormal basis (e.g., wavelet, Fourier) or tight frame (e.g., curvelet, Gabor)-so the coefficients belong to an lscrp ball for 0<ples1. The N most important coefficients in that expansion allow reconstruction with lscr2 error O(N1/2-1p/). It is possible to design n=O(Nlog(m)) nonadaptive measurements allowing reconstruction with accuracy comparable to that attainable with direct knowledge of the N most important coefficients. Moreover, a good approximation to those N important coefficients is extracted from the n measurements by solving a linear program-Basis Pursuit in signal processing. The nonadaptive measurements have the character of "random" linear combinations of basis/frame elements. Our results use the notions of optimal recovery, of n-widths, and information-based complexity. We estimate the Gel'fand n-widths of lscrp balls in high-dimensional Euclidean space in the case 0<ples1, and give a criterion identifying near- optimal subspaces for Gel'fand n-widths. We show that "most" subspaces are near-optimal, and show that convex optimization (Basis Pursuit) is a near-optimal way to extract information derived from these near-optimal subspaces
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paper_title: On the use of spectral libraries to perform sparse unmixing of hyperspectral data

paper_content:

In recent years, the increasing availability of spectral libraries has opened a new path toward solving the hyperspec-tral unmixing problem in a semi-supervised fashion. The spectrally pure constituent materials (called endmembers) can be derived from a (potentially very large) spectral library and used for unmixing purposes. The advantage of this approach is that the results of the unmixing process do not depend on the availability of pure pixels in the original hyperspectral data nor on the ability of an endmember extraction algorithm to identify such endmembers. However, resulting from the fact that spectral libraries are usually very large, this approach generally results in a sparse solution. In this paper, we investigate the sensitivity of sparse unmixing techniques to certain characteristics of real and synthetic spectral libraries, including parameters such as mutual coherence and spectral similarity between the signatures contained in the library. Our main goal is to illustrate, via detailed experimental assessment, the potential of using spectral libraries to solve the spectral unmixing problem.
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paper_title: L1 unmixing and its application to hyperspectral image enhancement

paper_content:

Because hyperspectral imagery is generally low resolution, it is possible for one pixel in the image to contain ::: several materials. The process of determining the abundance of representative materials in a single pixel is called ::: spectral unmixing. We discuss the L1 unmixing model and fast computational approaches based on Bregman ::: iteration. We then use the unmixing information and Total Variation (TV) minimization to produce a higher ::: resolution hyperspectral image in which each pixel is driven towards a "pure" material. This method produces ::: images with higher visual quality and can be used to indicate the subpixel location of features.
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paper_title: Matching pursuits with time-frequency dictionaries

paper_content:

The authors introduce an algorithm, called matching pursuit, that decomposes any signal into a linear expansion of waveforms that are selected from a redundant dictionary of functions. These waveforms are chosen in order to best match the signal structures. Matching pursuits are general procedures to compute adaptive signal representations. With a dictionary of Gabor functions a matching pursuit defines an adaptive time-frequency transform. They derive a signal energy distribution in the time-frequency plane, which does not include interference terms, unlike Wigner and Cohen class distributions. A matching pursuit isolates the signal structures that are coherent with respect to a given dictionary. An application to pattern extraction from noisy signals is described. They compare a matching pursuit decomposition with a signal expansion over an optimized wavepacket orthonormal basis, selected with the algorithm of Coifman and Wickerhauser see (IEEE Trans. Informat. Theory, vol. 38, Mar. 1992). >
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paper_title: Image Denoising Via Sparse and Redundant Representations Over Learned Dictionaries

paper_content:

We address the image denoising problem, where zero-mean white and homogeneous Gaussian additive noise is to be removed from a given image. The approach taken is based on sparse and redundant representations over trained dictionaries. Using the K-SVD algorithm, we obtain a dictionary that describes the image content effectively. Two training options are considered: using the corrupted image itself, or training on a corpus of high-quality image database. Since the K-SVD is limited in handling small image patches, we extend its deployment to arbitrary image sizes by defining a global image prior that forces sparsity over patches in every location in the image. We show how such Bayesian treatment leads to a simple and effective denoising algorithm. This leads to a state-of-the-art denoising performance, equivalent and sometimes surpassing recently published leading alternative denoising methods
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paper_title: Sparse Unmixing of Hyperspectral Data

paper_content:

Linear spectral unmixing is a popular tool in remotely sensed hyperspectral data interpretation. It aims at estimating the fractional abundances of pure spectral signatures (also called as endmembers) in each mixed pixel collected by an imaging spectrometer. In many situations, the identification of the end-member signatures in the original data set may be challenging due to insufficient spatial resolution, mixtures happening at different scales, and unavailability of completely pure spectral signatures in the scene. However, the unmixing problem can also be approached in semisupervised fashion, i.e., by assuming that the observed image signatures can be expressed in the form of linear combinations of a number of pure spectral signatures known in advance (e.g., spectra collected on the ground by a field spectroradiometer). Unmixing then amounts to finding the optimal subset of signatures in a (potentially very large) spectral library that can best model each mixed pixel in the scene. In practice, this is a combinatorial problem which calls for efficient linear sparse regression (SR) techniques based on sparsity-inducing regularizers, since the number of endmembers participating in a mixed pixel is usually very small compared with the (ever-growing) dimensionality (and availability) of spectral libraries. Linear SR is an area of very active research, with strong links to compressed sensing, basis pursuit (BP), BP denoising, and matching pursuit. In this paper, we study the linear spectral unmixing problem under the light of recent theoretical results published in those referred to areas. Furthermore, we provide a comparison of several available and new linear SR algorithms, with the ultimate goal of analyzing their potential in solving the spectral unmixing problem by resorting to available spectral libraries. Our experimental results, conducted using both simulated and real hyperspectral data sets collected by the NASA Jet Propulsion Laboratory's Airborne Visible Infrared Imaging Spectrometer and spectral libraries publicly available from the U.S. Geological Survey, indicate the potential of SR techniques in the task of accurately characterizing the mixed pixels using the library spectra. This opens new perspectives for spectral unmixing, since the abundance estimation process no longer depends on the availability of pure spectral signatures in the input data nor on the capacity of a certain endmember extraction algorithm to identify such pure signatures.
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paper_title: Iterative Spectral Unmixing for Optimizing Per-Pixel Endmember Sets

paper_content:

Fractional abundances predicted for a given pixel using spectral mixture analysis (SMA) are most accurate when only the endmembers that comprise it are used, with larger errors occurring if inappropriate endmembers are included in the unmixing process. This paper presents an iterative implementation of SMA (ISMA) to determine optimal per-pixel endmember sets from the image endmember set using two steps: 1) an iterative unconstrained unmixing, which removes one endmember per iteration based on minimum abundance and 2) analysis of the root-mean-square error as a function of iteration to locate the critical iteration defining the optimal endmember set. The ISMA was tested using simulated data at various signal-to-noise ratios (SNRs), and the results were compared with those of published unmixing methods. The ISMA method correctly selected the optimal endmember set 96% of the time for SNR of 100 : 1. As a result, per-pixel errors in fractional abundances were lower than for unmixing each pixel using the full endmember set. ISMA was also applied to Airborne Visible/Infrared Imaging Spectrometer hyperspectral data of Cuprite, NV. Results show that the ISMA is effective in obtaining abundance fractions that are physically realistic (sum close to one and nonnegative) and is more effective at selecting endmembers that occur within a pixel as opposed to those that are simply used to improve the goodness of fit of the model but not part of the mixture
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paper_title: Sparsity and Incoherence in Compressive Sampling

paper_content:

We consider the problem of reconstructing a sparse signal x 0 2 R n from a limited number of linear measurements. Given m randomly selected samples of Ux 0 , where U is an orthonormal matrix, we show that ‘1 minimization recovers x 0 exactly when the number of measurements exceeds m Const ·µ 2 (U) ·S · logn, where S is the number of nonzero components in x 0 , and µ is the largest entry in U properly normalized: µ(U) = p n · maxk,j |Uk,j|. The smaller µ, the fewer samples needed. The result holds for “most” sparse signals x 0 supported on a fixed (but arbitrary) set T. Given T, if the sign of x 0 for each nonzero entry on T and the observed values of Ux 0 are drawn at random, the signal is recovered with overwhelming probability. Moreover, there is a sense in which this is nearly optimal since any method succeeding with the same probability would require just about this many samples.
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paper_title: Stable signal recovery from incomplete and inaccurate measurements

paper_content:

Suppose we wish to recover an n-dimensional real-valued vector x_0 (e.g. a digital signal or image) from incomplete and contaminated observations y = A x_0 + e; A is a n by m matrix with far fewer rows than columns (n << m) and e is an error term. Is it possible to recover x_0 accurately based on the data y? ::: To recover x_0, we consider the solution x* to the l1-regularization problem min \|x\|_1 subject to \|Ax-y\|_2 <= epsilon, where epsilon is the size of the error term e. We show that if A obeys a uniform uncertainty principle (with unit-normed columns) and if the vector x_0 is sufficiently sparse, then the solution is within the noise level \|x* - x_0\|_2 \le C epsilon. ::: As a first example, suppose that A is a Gaussian random matrix, then stable recovery occurs for almost all such A's provided that the number of nonzeros of x_0 is of about the same order as the number of observations. Second, suppose one observes few Fourier samples of x_0, then stable recovery occurs for almost any set of p coefficients provided that the number of nonzeros is of the order of n/[\log m]^6. In the case where the error term vanishes, the recovery is of course exact, and this work actually provides novel insights on the exact recovery phenomenon discussed in earlier papers. The methodology also explains why one can also very nearly recover approximately sparse signals.
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paper_title: Alternating direction algorithms for constrained sparse regression: Application to hyperspectral unmixing

paper_content:

Convex optimization problems are common in hyperspectral unmixing. Examples include: the constrained least squares (CLS) and the fully constrained least squares (FCLS) problems, which are used to compute the fractional abundances in linear mixtures of known spectra; the constrained basis pursuit (CBP) problem, which is used to find sparse (i.e., with a small number of non-zero terms) linear mixtures of spectra from large libraries; the constrained basis pursuit denoising (CBPDN) problem, which is a generalization of BP that admits modeling errors. In this paper, we introduce two new algorithms to efficiently solve these optimization problems, based on the alternating direction method of multipliers, a method from the augmented Lagrangian family. The algorithms are termed SUnSAL (sparse unmixing by variable splitting and augmented Lagrangian) and C-SUnSAL (constrained SUnSAL). C-SUnSAL solves the CBP and CBPDN problems, while SUnSAL solves CLS and FCLS, as well as a more general version thereof, called constrained sparse regression (CSR). C-SUnSAL and SUnSAL are shown to outperform off-the-shelf methods in terms of speed and accuracy.
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paper_title: Sparse Approximate Solutions to Linear Systems

paper_content:

The following problem is considered: given a matrix $A$ in ${\bf R}^{m \times n}$, ($m$ rows and $n$ columns), a vector $b$ in ${\bf R}^m$, and ${\bf \epsilon} > 0$, compute a vector $x$ satisfying $\| Ax - b \|_2 \leq {\bf \epsilon}$ if such exists, such that $x$ has the fewest number of non-zero entries over all such vectors. It is shown that the problem is NP-hard, but that the well-known greedy heuristic is good in that it computes a solution with at most $\left\lceil 18 \mbox{ Opt} ({\bf \epsilon}/2) \|{\bf A}^+\|^2_2 \ln(\|b\|_2/{\bf \epsilon}) \right\rceil$ non-zero entries, where $\mbox{Opt}({\bf \epsilon}/2)$ is the optimum number of nonzero entries at error ${\bf \epsilon}/2$, ${\bf A}$ is the matrix obtained by normalizing each column of $A$ with respect to the $L_2$ norm, and ${\bf A}^+$ is its pseudo-inverse.
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paper_title: Robust hyperspectral data unmixing with spatial and spectral regularized NMF

paper_content:

This paper considers the problem of unsupervised hyperspectral data unmixing under the linear spectral mixing model assumption (LSMM). The aim is to recover both end member spectra and abundances fractions. The problem is ill-posed and needs some additional information to be solved. We consider here the Non-negative Matrix Factorization (NMF), which is degenerated on its own, but has the advantage of low complexity and the ability to easily include physical constraints. In addition with abundances sum-to-one constraint, we propose to introduce relevant information within spatial and spectral regularization for the NMF, derived from the analysis of the hyperspectral data. We use an alternate projected gradient to minimize the regularized error reconstruction function. This algorithm, called MDMD-NMF for Minimum Spectral Dispersion Maximum Spatial Dispersion NMF, allows to simultaneously estimate the number of end members, the abundances fractions, and accurately recover more than 10 end members without any pure pixel in the scene.
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paper_title: Piece-wise convex spatial-spectral unmixing of hyperspectral imagery using possibilistic and fuzzy clustering

paper_content:

Imaging spectroscopy refers to methods for identifying materials in a scene using cameras that digitize light into hundreds of spectral bands. Each pixel in these images consists of vectors representing the amount of light reflected in the different spectral bands from the physical location corresponding to the pixel. Images of this type are called hyperspectral images. Hyperspectral image analysis differs from traditional image analysis in that, in addition to the spatial information inherent in an image, there is abundant spectral information at the pixel or sub-pixel level that can be used to identify materials in the scene. Spectral unmixing techniques attempt to identify the material spectra in a scene down to the sub-pixel level. In this paper, a piece-wise convex hyperspectral unmixing algorithm using both spatial and spectral image information is presented. The proposed method incorporates possibilistic and fuzzy clustering methods. The typicality and membership estimates from those methods can be combined with traditional material proportion estimates to produce more meaningful proportion estimates than obtained with previous spectral unmixing algorithms. An analysis of the utility of using all three estimates produce a better estimate is given using real hyperspectral imagery.
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paper_title: SVM- and MRF-Based Method for Accurate Classification of Hyperspectral Images

paper_content:

The high number of spectral bands acquired by hyperspectral sensors increases the capability to distinguish physical materials and objects, presenting new challenges to image analysis and classification. This letter presents a novel method for accurate spectral-spatial classification of hyperspectral images. The proposed technique consists of two steps. In the first step, a probabilistic support vector machine pixelwise classification of the hyperspectral image is applied. In the second step, spatial contextual information is used for refining the classification results obtained in the first step. This is achieved by means of a Markov random field regularization. Experimental results are presented for three hyperspectral airborne images and compared with those obtained by recently proposed advanced spectral-spatial classification techniques. The proposed method improves classification accuracies when compared to other classification approaches.
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paper_title: Integration of spatial-spectral information for the improved extraction of endmembers

paper_content:

Spectral-based image endmember extraction methods hinge on the ability to discriminate between pixels based on spectral characteristics alone. Endmembers with distinct spectral features (high spectral contrast) are easy to select, whereas those with minimal unique spectral information (low spectral contrast) are more problematic. Spectral contrast, however, is dependent on the endmember assemblage, such that as the assemblage changes so does the “relative” spectral contrast of each endmember to all other endmembers. It is then possible for an endmember to have low spectral contrast with respect to the full image, but have high spectral contrast within a subset of the image. The spatial–spectral endmember extraction tool (SSEE) works by analyzing a scene in parts (subsets), such that we increase the spectral contrast of low contrast endmembers, thus improving the potential for these endmembers to be selected. The SSEE method comprises three main steps: 1) application of singular value decomposition (SVD) to determine a set of basis vectors that describe most of the spectral variance for subsets of the image; 2) projection of the full image data set onto the locally defined basis vectors to determine a set of candidate endmember pixels; and, 3) imposing spatial constraints for averaging spectrally similar endmembers, allowing for separation of endmembers that are spectrally similar, but spatially independent. The SSEE method is applied to two real hyperspectral data sets to demonstrate the effects of imposing spatial constraints on the selection of endmembers. The results show that the SSEE method is an effective approach to extracting image endmembers. Specific improvements include the extraction of physically meaningful, low contrast endmembers that occupy unique image regions.
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paper_title: Spatially-smooth piece-wise convex endmember detection

paper_content:

An endmember detection and spectral unmixing algorithm that uses both spatial and spectral information is presented. This method, Spatial Piece-wise Convex Multiple Model Endmember Detection (Spatial P-COMMEND), autonomously estimates multiple sets of endmembers and performs spectral unmixing for input hyperspectral data. Spatial P-COMMEND does not restrict the estimated endmembers to define a single convex region during spectral unmixing. Instead, a piece-wise convex representation is used that can effectively represent non-convex hyperspectral data. Spatial P-COMMEND drives neighboring pixels to be unmixed by the same set of endmembers encouraging spatially-smooth unmixing results.
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paper_title: Hyperspectral Image Segmentation Using a New Bayesian Approach With Active Learning

paper_content:

This paper introduces a new supervised Bayesian approach to hyperspectral image segmentation with active learning, which consists of two main steps. First, we use a multinomial logistic regression (MLR) model to learn the class posterior probability distributions. This is done by using a recently introduced logistic regression via splitting and augmented Lagrangian algorithm. Second, we use the information acquired in the previous step to segment the hyperspectral image using a multilevel logistic prior that encodes the spatial information. In order to reduce the cost of acquiring large training sets, active learning is performed based on the MLR posterior probabilities. Another contribution of this paper is the introduction of a new active sampling approach, called modified breaking ties, which is able to provide an unbiased sampling. Furthermore, we have implemented our proposed method in an efficient way. For instance, in order to obtain the time-consuming maximum a posteriori segmentation, we use the α-expansion min-cut-based integer optimization algorithm. The state-of-the-art performance of the proposed approach is illustrated using both simulated and real hyperspectral data sets in a number of experimental comparisons with recently introduced hyperspectral image analysis methods.
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paper_title: Semi-Supervised Linear Spectral Unmixing Using a Hierarchical Bayesian Model for Hyperspectral Imagery

paper_content:

This paper proposes a hierarchical Bayesian model that can be used for semi-supervised hyperspectral image unmixing. The model assumes that the pixel reflectances result from linear combinations of pure component spectra contaminated by an additive Gaussian noise. The abundance parameters appearing in this model satisfy positivity and additivity constraints. These constraints are naturally expressed in a Bayesian context by using appropriate abundance prior distributions. The posterior distributions of the unknown model parameters are then derived. A Gibbs sampler allows one to draw samples distributed according to the posteriors of interest and to estimate the unknown abundances. An extension of the algorithm is finally studied for mixtures with unknown numbers of spectral components belonging to a know library. The performance of the different unmixing strategies is evaluated via simulations conducted on synthetic and real data.
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paper_title: Spectral and Spatial Classification of Hyperspectral Data Using SVMs and Morphological Profiles

paper_content:

Classification of hyperspectral data with high spatial resolution from urban areas is discussed. An approach has been proposed which is based on using several principal components from the hyperspectral data and build morphological profiles. These profiles can be used all together in one extended morphological profile. A shortcoming of the approach is that it is primarily designed for classification of urban structures and it does not fully utilize the spectral information in the data. Similarly, a pixel-wise classification solely based on the spectral content can be performed, but it lacks information on the structure of the features in the image. An extension is proposed in this paper in order to overcome these dual problems. The proposed method is based on the data fusion of the morphological information and the original hyperspectral data: the two vectors of attributes are concatenated. After a reduction of the dimensionality using Decision Boundary Feature Extraction, the final classification is achieved using a Support Vector Machines classifier. The proposed approach is tested in experiments on ROSIS data from urban areas. Significant improvements are achieved in terms of accuracies when compared to results of approaches based on the use of morphological profiles based on PCs only and conventional spectral classification.
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paper_title: Spectral and Spatial Complexity-Based Hyperspectral Unmixing

paper_content:

Hyperspectral unmixing, which decomposes pixel spectra into a collection of constituent spectra, is a preprocessing step for hyperspectral applications like target detection and classification. It can be considered as a blind source separation (BSS) problem. Independent component analysis, which is a widely used method for performing BSS, models a mixed pixel as a linear mixture of its constituent spectra weighted by the correspondent abundance fractions (sources). The sources are assumed to be independent and stationary. However, in many instances, this assumption is not valid. In this paper, a complexity-based BSS algorithm is introduced, which studies the complexity of sources instead of the independence. We extend the 1-D temporal complexity, which is called complexity pursuit that was proposed by Stone, to the 2-D spatial complexity, which is named spatial complexity BSS (SCBSS), to describe the spatial autocorrelation of each abundance fraction. Further, the temporal complexity of spectrum is combined into SCBSS to account for the spectral smoothness, which is termed spectral and spatial complexity BSS. More importantly, a strict theoretic interpretation is given, showing that the complexity-based BSS is very suitable for hyperspectral unmixing. Experimental results on synthetic and real hyperspectral data demonstrate the advantages of the proposed two algorithms with respect to other methods.
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paper_title: Spectral–Spatial Classification of Hyperspectral Imagery Based on Partitional Clustering Techniques

paper_content:

A new spectral-spatial classification scheme for hyperspectral images is proposed. The method combines the results of a pixel wise support vector machine classification and the segmentation map obtained by partitional clustering using majority voting. The ISODATA algorithm and Gaussian mixture resolving techniques are used for image clustering. Experimental results are presented for two hyperspectral airborne images. The developed classification scheme improves the classification accuracies and provides classification maps with more homogeneous regions, when compared to pixel wise classification. The proposed method performs particularly well for classification of images with large spatial structures and when different classes have dissimilar spectral responses and a comparable number of pixels.
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paper_title: L1 unmixing and its application to hyperspectral image enhancement

paper_content:

Because hyperspectral imagery is generally low resolution, it is possible for one pixel in the image to contain ::: several materials. The process of determining the abundance of representative materials in a single pixel is called ::: spectral unmixing. We discuss the L1 unmixing model and fast computational approaches based on Bregman ::: iteration. We then use the unmixing information and Total Variation (TV) minimization to produce a higher ::: resolution hyperspectral image in which each pixel is driven towards a "pure" material. This method produces ::: images with higher visual quality and can be used to indicate the subpixel location of features.
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paper_title: Enhancing hyperspectral image unmixing with spatial correlations

paper_content:

This paper describes a new algorithm for hyperspectral image unmixing. Most unmixing algorithms proposed in the literature do not take into account the possible spatial correlations between the pixels. In this paper, a Bayesian model is introduced to exploit these correlations. The image to be unmixed is assumed to be partitioned into regions (or classes) where the statistical properties of the abundance coefficients are homogeneous. A Markov random field, is then proposed to model the spatial dependencies between the pixels within any class. Conditionally upon a given class, each pixel is modeled by using the classical linear mixing model with additive white Gaussian noise. For this model, the posterior distributions of the unknown parameters and hyperparameters allow the parameters of interest to be inferred. These parameters include the abundances for each pixel, the means and variances of the abundances for each class, as well as a classification map indicating the classes of all pixels in the image. To overcome the complexity of the posterior distribution, we consider a Markov chain Monte Carlo method that generates samples asymptotically distributed according to the posterior. The generated samples are then used for parameter and hyperparameter estimation. The accuracy of the proposed algorithms is illustrated on synthetic and real data.
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paper_title: Spectral–Spatial Hyperspectral Image Segmentation Using Subspace Multinomial Logistic Regression and Markov Random Fields

paper_content:

This paper introduces a new supervised segmentation algorithm for remotely sensed hyperspectral image data which integrates the spectral and spatial information in a Bayesian framework. A multinomial logistic regression (MLR) algorithm is first used to learn the posterior probability distributions from the spectral information, using a subspace projection method to better characterize noise and highly mixed pixels. Then, contextual information is included using a multilevel logistic Markov-Gibbs Markov random field prior. Finally, a maximum a posteriori segmentation is efficiently computed by the min-cut-based integer optimization algorithm. The proposed segmentation approach is experimentally evaluated using both simulated and real hyperspectral data sets, exhibiting state-of-the-art performance when compared with recently introduced hyperspectral image classification methods. The integration of subspace projection methods with the MLR algorithm, combined with the use of spatial-contextual information, represents an innovative contribution in the literature. This approach is shown to provide accurate characterization of hyperspectral imagery in both the spectral and the spatial domain.
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paper_title: Semisupervised Hyperspectral Image Segmentation Using Multinomial Logistic Regression With Active Learning

paper_content:

This paper presents a new semisupervised segmentation algorithm, suited to high-dimensional data, of which remotely sensed hyperspectral image data sets are an example. The algorithm implements two main steps: 1) semisupervised learning of the posterior class distributions followed by 2) segmentation, which infers an image of class labels from a posterior distribution built on the learned class distributions and on a Markov random field. The posterior class distributions are modeled using multinomial logistic regression, where the regressors are learned using both labeled and, through a graph-based technique, unlabeled samples. Such unlabeled samples are actively selected based on the entropy of the corresponding class label. The prior on the image of labels is a multilevel logistic model, which enforces segmentation results in which neighboring labels belong to the same class. The maximum a posteriori segmentation is computed by the α-expansion min-cut-based integer optimization algorithm. Our experimental results, conducted using synthetic and real hyperspectral image data sets collected by the Airborne Visible/Infrared Imaging Spectrometer system of the National Aeronautics and Space Administration Jet Propulsion Laboratory over the regions of Indian Pines, IN, and Salinas Valley, CA, reveal that the proposed approach can provide classification accuracies that are similar or higher than those achieved by other supervised methods for the considered scenes. Our results also indicate that the use of a spatial prior can greatly improve the final results with respect to a case in which only the learned class densities are considered, confirming the importance of jointly considering spatial and spectral information in hyperspectral image segmentation.
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paper_title: Bayesian Hyperspectral Image Segmentation With Discriminative Class Learning

paper_content:

This paper introduces a new supervised technique to segment hyperspectral images: the Bayesian segmentation based on discriminative classification and on multilevel logistic (MLL) spatial prior. The approach is Bayesian and exploits both spectral and spatial information. Given a spectral vector, the posterior class probability distribution is modeled using multinomial logistic regression (MLR) which, being a discriminative model, allows to learn directly the boundaries between the decision regions and, thus, to successfully deal with high-dimensionality data. To control the machine complexity and, thus, its generalization capacity, the prior on the multinomial logistic vector is assumed to follow a componentwise independent Laplacian density. The vector of weights is computed via the fast sparse multinomial logistic regression (FSMLR), a variation of the sparse multinomial logistic regression (SMLR), conceived to deal with large data sets beyond the reach of the SMLR. To avoid the high computational complexity involved in estimating the Laplacian regularization parameter, we have also considered the Jeffreys prior, as it does not depend on any hyperparameter. The prior probability distribution on the class-label image is an MLL Markov-Gibbs distribution, which promotes segmentation results with equal neighboring class labels. The -expansion optimization algorithm, a powerful graph-cut-based integer optimization tool, is used to compute the maximum a posteriori segmentation. The effectiveness of the proposed methodology is illustrated by comparing its performance with the state-of-the-art methods on synthetic and real hyperspectral image data sets. The reported results give clear evidence of the relevance of using both spatial and spectral information in hyperspectral image segmentation.
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paper_title: Spatial Preprocessing for Endmember Extraction

paper_content:

Endmember extraction is the process of selecting a collection of pure signature spectra of the materials present in a remotely sensed hyperspectral scene. These pure signatures are then used to decompose the scene into abundance fractions by means of a spectral unmixing algorithm. Most techniques available in the endmember extraction literature rely on exploiting the spectral properties of the data alone. As a result, the search for endmembers in a scene is conducted by treating the data as a collection of spectral measurements with no spatial arrangement. In this paper, we propose a novel strategy to incorporate spatial information into the traditional spectral-based endmember search process. Specifically, we propose to estimate, for each pixel vector, a scalar spatially derived factor that relates to the spectral similarity of pixels lying within a certain spatial neighborhood. This scalar value is then used to weigh the importance of the spectral information associated to each pixel in terms of its spatial context. Two key aspects of the proposed methodology are given as follows: 1) No modification of existing image spectral-based endmember extraction methods is necessary in order to apply the proposed approach. 2) The proposed preprocessing method enhances the search for image spectral endmembers in spatially homogeneous areas. Our experimental results, which were obtained using both synthetic and real hyperspectral data sets, indicate that the spectral endmembers obtained after spatial preprocessing can be used to accurately model the original hyperspectral scene using a linear mixture model. The proposed approach is suitable for jointly combining spectral and spatial information when searching for image-derived endmembers in highly representative hyperspectral image data sets.
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paper_title: Spatial/spectral endmember extraction by multidimensional morphological operations

paper_content:

Spectral mixture analysis provides an efficient mechanism for the interpretation and classification of remotely sensed multidimensional imagery. It aims to identify a set of reference signatures (also known as endmembers) that can be used to model the reflectance spectrum at each pixel of the original image. Thus, the modeling is carried out as a linear combination of a finite number of ground components. Although spectral mixture models have proved to be appropriate for the purpose of large hyperspectral dataset subpixel analysis, few methods are available in the literature for the extraction of appropriate endmembers in spectral unmixing. Most approaches have been designed from a spectroscopic viewpoint and, thus, tend to neglect the existing spatial correlation between pixels. This paper presents a new automated method that performs unsupervised pixel purity determination and endmember extraction from multidimensional datasets; this is achieved by using both spatial and spectral information in a combined manner. The method is based on mathematical morphology, a classic image processing technique that can be applied to the spectral domain while being able to keep its spatial characteristics. The proposed methodology is evaluated through a specifically designed framework that uses both simulated and real hyperspectral data.
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paper_title: Total variation regulatization in sparse hyperspectral unmixing

paper_content:

Hyperspectral unmixing has recently been addressed as a sparse regression problem by using predefined spectral libraries instead of image-derived endmembers in the unmixing process. This new approach has attracted much attention, as it sidesteps well known obstacles met in endmember extraction, such as the stopping criteria for the extraction process (represented by the number of endmembers needed to explain the observed scene) and the fact that the scene might not contain pure pixels. It happens, however, that in many applications the spectral libraries contain highly correlated signatures, which limits the success of sparse regression applied to mixtures with a very small number of materials. In this paper, we mitigate this limitation by adding the total variation regularization to the classical sparse regression, thus, exploiting the spatial contextual information present in the hyperspectral images. The effectiveness of the new approach is illustrated in experiments carried out on simulated data sets.
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paper_title: Total Variation Spatial Regularization for Sparse Hyperspectral Unmixing

paper_content:

Spectral unmixing aims at estimating the fractional abundances of pure spectral signatures (also called endmembers) in each mixed pixel collected by a remote sensing hyperspectral imaging instrument. In recent work, the linear spectral unmixing problem has been approached in semisupervised fashion as a sparse regression one, under the assumption that the observed image signatures can be expressed as linear combinations of pure spectra, known a priori and available in a library. It happens, however, that sparse unmixing focuses on analyzing the hyperspectral data without incorporating spatial information. In this paper, we include the total variation (TV) regularization to the classical sparse regression formulation, thus exploiting the spatial-contextual information present in the hyperspectral images and developing a new algorithm called sparse unmixing via variable splitting augmented Lagrangian and TV. Our experimental results, conducted with both simulated and real hyperspectral data sets, indicate the potential of including spatial information (through the TV term) on sparse unmixing formulations for improved characterization of mixed pixels in hyperspectral imagery.
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paper_title: Spatial classification using fuzzy membership models

paper_content:

In the usual statistical approach to spatial classification, it is assumed that each pixel belongs to precisely one of a small number of known groups. This framework is extended to include mixed-pixel data; then, only a proportion of each pixel belongs to each group. Two models based on multivariate Gaussian random fields are proposed to model this fuzzy membership process. The problems of predicting the group membership and estimating the parameters are discussed. Some simulations are presented to study the properties of this approach, and an example is given using Landsat remote-sensing data. >
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paper_title: Spatial-spectral unmixing using fuzzy local information

paper_content:

Hyperspectral unmixing estimates the proportions of materials represented within a spectral signature. The overwhelming majority of hyperspectral unmixing algorithms are based entirely on the spectral signatures of each individual pixel and do not incorporate the spatial information found in a hyperspectral data cube. In this work, a spectral unmixing algorithm, the Local Information Proportion estimation (LIP) algorithm, is presented. The proposed LIP algorithm incorporates spatial information while determining the proportions of materials found within a spectral signature. Spatial information is incorporated through the addition of a spatial term that regularizes proportion value estimates based on the weighted proportion values of neighboring pixels. Results are shown in the AVIRIS Indian Pines hyperspectral data set.
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paper_title: Multiple Spectral–Spatial Classification Approach for Hyperspectral Data

paper_content:

A new multiple-classifier approach for spectral-spatial classification of hyperspectral images is proposed. Several classifiers are used independently to classify an image. For every pixel, if all the classifiers have assigned this pixel to the same class, the pixel is kept as a marker, i.e., a seed of the spatial region with a corresponding class label. We propose to use spectral-spatial classifiers at the preliminary step of the marker-selection procedure, each of them combining the results of a pixelwise classification and a segmentation map. Different segmentation methods based on dissimilar principles lead to different classification results. Furthermore, a minimum spanning forest is built, where each tree is rooted on a classification-driven marker and forms a region in the spectral-spatial classification map. Experimental results are presented for two hyperspectral airborne images. The proposed method significantly improves classification accuracies when compared with previously proposed classification techniques.
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paper_title: Hyperspectral Image Unmixing via Alternating Projected Subgradients

paper_content:

We consider the problem of factorizing a hyperspectral image into the product of two nonnegative matrices, which represent nonnegative bases for image spectra and mixing coefficients, respectively. This spectral unmixing problem is a nonconvex optimization problem, which is very difficult to solve exactly. We present a simple heuristic for approximately solving this problem based on the idea of alternating projected subgradient descent. Finally, we present the results of applying this method on the 1990 AVIRIS image of Cuprite, Nevada and show that our results are in agreement with similar studies on the same data.
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paper_title: Hyperspectral Image Unmixing Using a Multiresolution Sticky HDP

paper_content:

This paper is concerned with joint Bayesian endmember extraction and linear unmixing of hyperspectral images using a spatial prior on the abundance vectors. We propose a generative model for hyperspectral images in which the abundances are sampled from a Dirichlet distribution (DD) mixture model, whose parameters depend on a latent label process. The label process is then used to enforces a spatial prior which encourages adjacent pixels to have the same label. A Gibbs sampling framework is used to generate samples from the posterior distributions of the abundances and the parameters of the DD mixture model. The spatial prior that is used is a tree-structured sticky hierarchical Dirichlet process (SHDP) and, when used to determine the posterior endmember and abundance distributions, results in a new unmixing algorithm called spatially constrained unmixing (SCU). The directed Markov model facilitates the use of scale-recursive estimation algorithms, and is therefore more computationally efficient as compared to standard Markov random field (MRF) models. Furthermore, the proposed SCU algorithm estimates the number of regions in the image in an unsupervised fashion. The effectiveness of the proposed SCU algorithm is illustrated using synthetic and real data.
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paper_title: C-HiLasso: A Collaborative Hierarchical Sparse Modeling Framework

paper_content:

Sparse modeling is a powerful framework for data analysis and processing. Traditionally, encoding in this framework is performed by solving an l1-regularized linear regression problem, commonly referred to as Lasso or Basis Pursuit. In this work we combine the sparsity-inducing property of the Lasso at the individual feature level, with the block-sparsity property of the Group Lasso, where sparse groups of features are jointly encoded, obtaining a sparsity pattern hierarchically structured. This results in the Hierarchical Lasso (HiLasso), which shows important practical advantages. We then extend this approach to the collaborative case, where a set of simultaneously coded signals share the same sparsity pattern at the higher (group) level, but not necessarily at the lower (inside the group) level, obtaining the collaborative HiLasso model (C-HiLasso). Such signals then share the same active groups, or classes, but not necessarily the same active set. This model is very well suited for applications such as source identification and separation. An efficient optimization procedure, which guarantees convergence to the global optimum, is developed for these new models. The underlying presentation of the framework and optimization approach is complemented by experimental examples and theoretical results regarding recovery guarantees.
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paper_title: Parallel heterogeneous CBIR system for efficient hyperspectral image retrieval using spectral mixture analysis

paper_content:

The purpose of content-based image retrieval (CBIR) is to retrieve, from real data stored in a database, information that is relevant to a query. In remote sensing applications, the wealth of spectral information provided by latest-generation (hyperspectral) instruments has quickly introduced the need for parallel CBIR systems able to effectively retrieve features of interest from ever-growing data archives. To address this need, this paper develops a new parallel CBIR system that has been specifically designed to be run on heterogeneous networks of computers (HNOCs). These platforms have soon become a standard computing architecture in remote sensing missions due to the distributed nature of data repositories. The proposed heterogeneous system first extracts an image feature vector able to characterize image content with sub-pixel precision using spectral mixture analysis concepts, and then uses the obtained feature as a search reference. The system is validated using a complex hyperspectral image database, and implemented on several networks of workstations and a Beowulf cluster at NASA's Goddard Space Flight Center. Our experimental results indicate that the proposed parallel system can efficiently retrieve hyperspectral images from complex image databases by efficiently adapting to the underlying parallel platform on which it is run, regardless of the heterogeneity in the compute nodes and communication links that form such parallel platform. Copyright © 2009 John Wiley & Sons, Ltd.
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paper_title: Parallel unmixing of remotely sensed hyperspectral images on commodity graphics processing units

paper_content:

Hyperspectral imaging instruments are capable of collecting hundreds of images, corresponding to different wavelength channels, for the same area on the surface of the Earth. One of the main problems in the analysis of hyperspectral data cubes is the presence of mixed pixels, which arise when the spatial resolution of the sensor is not enough to separate spectrally distinct materials. Hyperspectral unmixing is one of the most popular techniques to analyze hyperspectral data. It comprises two stages: (i) automatic identification of pure spectral signatures (endmembers) and (ii) estimation of the fractional abundance of each endmember in each pixel. The spectral unmixing process is quite expensive in computational terms, mainly due to the extremely high dimensionality of hyperspectral data cubes. Although this process maps nicely to high performance systems such as clusters of computers, these systems are generally expensive and difficult to adapt to real-time data processing requirements introduced by several applications, such as wildland fire tracking, biological threat detection, monitoring of oil spills, and other types of chemical contamination. In this paper, we develop an implementation of the full hyperspectral unmixing chain on commodity graphics processing units (GPUs). The proposed methodology has been implemented, using the CUDA (compute device unified architecture), and tested on three different GPU architectures: NVidia Tesla C1060, NVidia GeForce GTX 275, and NVidia GeForce 9800 GX2, achieving near real-time unmixing performance in some configurations tested when analyzing two different hyperspectral images, collected over the World Trade Center complex in New York City and the Cuprite mining district in Nevada. Copyright © 2011 John Wiley & Sons, Ltd.
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paper_title: Spectral unmixing

paper_content:

Spectral unmixing using hyperspectral data represents a significant step in the evolution of remote decompositional analysis that began with multispectral sensing. It is a consequence of collecting data in greater and greater quantities and the desire to extract more detailed information about the material composition of surfaces. Linear mixing is the key assumption that has permitted well-known algorithms to be adapted to the unmixing problem. In fact, the resemblance of the linear mixing model to system models in other areas has permitted a significant legacy of algorithms from a wide range of applications to be adapted to unmixing. However, it is still unclear whether the assumption of linearity is sufficient to model the mixing process in every application of interest. It is clear, however, that the applicability of models and techniques is highly dependent on the variety of circumstances and factors that give rise to mixed pixels. The outputs of spectral unmixing, endmember, and abundance estimates are important for identifying the material composition of mixtures.
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paper_title: Recent Developments in High Performance Computing for Remote Sensing: A Review

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Remote sensing data have become very widespread in recent years, and the exploitation of this technology has gone from developments mainly conducted by government intelligence agencies to those carried out by general users and companies. There is a great deal more to remote sensing data than meets the eye, and extracting that information turns out to be a major computational challenge. For this purpose, high performance computing (HPC) infrastructure such as clusters, distributed networks or specialized hardware devices provide important architectural developments to accelerate the computations related with information extraction in remote sensing. In this paper, we review recent advances in HPC applied to remote sensing problems; in particular, the HPC-based paradigms included in this review comprise multiprocessor systems, large-scale and heterogeneous networks of computers, grid and cloud computing environments, and hardware systems such as field programmable gate arrays (FPGAs) and graphics processing units (GPUs). Combined, these parts deliver a snapshot of the state-of-the-art and most recent developments in those areas, and offer a thoughtful perspective of the potential and emerging challenges of applying HPC paradigms to remote sensing problems.
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paper_title: High Performance Computing in Remote Sensing

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Solutions for Time-Critical Remote Sensing Applications The recent use of latest-generation sensors in airborne and satellite platforms is producing a nearly continual stream of high-dimensional data, which, in turn, is creating new processing challenges. To address the computational requirements of time-critical applications, researchers have begun incorporating high performance computing (HPC) models in remote sensing missions. High Performance Computing in Remote Sensing is one of the first volumes to explore state-of-the-art HPC techniques in the context of remote sensing problems. It focuses on the computational complexity of algorithms that are designed for parallel computing and processing. A Diverse Collection of Parallel Computing Techniques and Architectures The book first addresses key computing concepts and developments in remote sensing. It also covers application areas not necessarily related to remote sensing, such as multimedia andvideo processing. Each subsequent chapter illustrate s a specific parallel computing paradigm, including multiprocessor (cluster-based) systems, large-scale and heterogeneous networks of computers, grid computing platforms, and specialized hardware architectures for remotely sensed data analysis and interpretation. An Interdisciplinary Forum to Encourage Novel Ideas The extensive reviews of current and future developments combined with thoughtful perspectives on the potential challenges of adapting HPC paradigms to remote sensing problems will undoubtedly foster collaboration and development among many fields.
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paper_title: FPGA Implementation of Abundance Estimation for Spectral Unmixing of Hyperspectral Data Using the Image Space Reconstruction Algorithm

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One of the most popular and widely used techniques for analyzing remotely sensed hyperspectral data is spectral unmixing, which relies on two stages: (i) identification of pure spectral signatures (endmembers) in the data, and (ii) estimation of the abundance of each endmember in each (possibly mixed) pixel. Due to the high dimensionality of the hyperspectral data, spectral unmixing is a very time-consuming task. With recent advances in reconfigurable computing, especially using field programmable gate arrays (FPGAs), hyperspectral image processing algorithms can now be accelerated for on-board exploitation using compact hardware components with small size and cost. Although in previous work several efforts have been directed towards FPGA implementation of endmember extraction algorithms, the abundance estimation step has received comparatively much less attention. In this work, we develop a parallel FPGA-based design of the image space reconstruction algorithm (ISRA), a technique for solving linear inverse problems with positive constraints that has been used to estimate the abundance of each endmember in each pixel of a hyperspectral image. It is an iterative algorithm that guarantees convergence (after a certain number of iterations) and positive values in the results of the abundances (an important consideration in unmixing applications). Our system includes a direct memory access (DMA) module and implements a pre-fetching technique to hide the latency of the input/output communications. The method has been implemented on a Virtex-4 XC4VFX60 FPGA (a model that is similar to radiation-hardened FPGAs certified for space operation) and tested using real hyperspectral data sets collected by the Airborne Visible Infra-Red Imaging Spectrometer (AVIRIS) over the Cuprite mining district in Nevada and the Jasper Ridge Biological Preserve in California. Experimental results demonstrate that our hardware version can significantly outperform an equivalent software version, thus being able to provide abundance estimation results in near real-time, which makes our reconfigurable system appealing for on-board hyperspectral data processing.
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paper_title: FPGA Implementation of the N-FINDR Algorithm for Remotely Sensed Hyperspectral Image Analysis

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Hyperspectral remote sensing attempts to identify features in the surface of the Earth using sensors that generally provide large amounts of data. The data are usually collected by a satellite or an airborne instrument and sent to a ground station that processes it. The main bottleneck of this approach is the (often reduced) bandwidth connection between the satellite and the station, which drastically limits the information that can be sent and processed in real time. A possible way to overcome this problem is to include onboard computing resources able to preprocess the data, reducing its size by orders of magnitude. Reconfigurable field-programmable gate arrays (FPGAs) are a promising platform that allows hardware/software codesign and the potential to provide powerful onboard computing capability and flexibility at the same time. Since FPGAs can implement custom hardware solutions, they can reach very high performance levels. Moreover, using run-time reconfiguration, the functionality of the FPGA can be updated at run time as many times as needed to perform different computations. Hence, the FPGA can be reused for several applications reducing the number of computing resources needed. One of the most popular and widely used techniques for analyzing hyperspectral data is linear spectral unmixing, which relies on the identification of pure spectral signatures via a so-called endmember extraction algorithm. In this paper, we present the first FPGA design for N-FINDR, a widely used endmember extraction algorithm in the literature. Our system includes a direct memory access module and implements a prefetching technique to hide the latency of the input/output communications. The proposed method has been implemented on a Virtex-4 XC4VFX60 FPGA (a model that is similar to radiation-hardened FPGAs certified for space operation) and tested using real hyperspectral data collected by NASA's Earth Observing-1 Hyperion (a satellite instrument) and the Airborne Visible Infra-Red Imaging Spectrometer over the Cuprite mining district in Nevada and the Jasper Ridge Biological Preserve in California. Experimental results demonstrate that our hardware version of the N-FINDR algorithm can significantly outperform an equivalent software version and is able to provide accurate results in near real time, which makes our reconfigurable system appealing for onboard hyperspectral data processing.
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Title: Hyperspectral Unmixing Overview: Geometrical, Statistical, and Sparse Regression-Based Approaches

Section 1: INTRODUCTION
Description 1: Provide an introduction to hyperspectral cameras and their applications, the concept of hyperspectral unmixing (HU), and an overview of linear and nonlinear mixing models.

Section 2: Linear and Nonlinear Mixing Models
Description 2: Discuss linear mixing and nonlinear mixing models in detail, incidents of light interaction with materials, and the applicability of each model in real scenarios.

Section 3: Brief Overview of Nonlinear Approaches
Description 3: Present an overview of various nonlinear unmixing approaches, including radiative transfer theory, kernel methods, and machine learning strategies.

Section 4: Hyperspectral Unmixing Processing Chain
Description 4: Detail the processing steps typically involved in hyperspectral unmixing including atmospheric correction, data reduction, endmember determination, and inversion.

Section 5: LINEAR MIXTURE MODEL
Description 5: Describe the linear mixture model (LMM), including the mathematical representation and characterization of the spectral unmixing inverse problem.

Section 6: SIGNAL SUBSPACE IDENTIFICATION
Description 6: Explain signal subspace identification, including various methods such as band selection/extraction, projection techniques, and model order inference.

Section 7: GEOMETRICAL BASED APPROACHES TO LINEAR SPECTRAL UNMIXING
Description 7: Discuss geometrical-based approaches divided into two main categories: Pure Pixel (PP) based algorithms and Minimum Volume (MV) based algorithms.

Section 8: STATISTICAL METHODS
Description 8: Explore statistical methods for hyperspectral unmixing, including Bayesian approaches and independent component analysis (ICA).

Section 9: SPARSE REGRESSION BASED UNMIXING
Description 9: Discuss sparse regression-based unmixing, the use of spectral libraries, and learning dictionaries directly from the dataset.

Section 10: SPATIAL-SPECTRAL CONTEXTUAL INFORMATION
Description 10: Address the integration of spatial and spectral information for unmixing purposes and various methods to exploit contextual information in hyperspectral images.

Section 11: SUMMARY
Description 11: Summarize the main points covered in the paper, including the state-of-the-art techniques in hyperspectral unmixing and future directions.