Overview of methods for image reconstruction from projections in emission computed tomography
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paper_title: Evaluation of rotating slant-hole SPECT mammography using Monte-Carlo simulation methods

paper_content:

Multi-segment slant-hole (MSSH) collimators provide much higher detection efficiency of breast lesions at a single view than conventional low-energy high-resolution (LEHR) parallel-hole collimators. The purpose of this study is to evaluate rotating MSSH (RMSSH) SPECT mammography using Monte Carlo Simulation methods. We modified the SIMIND Monte Carlo code to generate RMSSH projection data including the effects of attenuation, collimator-detector response and scatter from a realistic NCAT phantom with an attached breast and a lesion placed at locations at the center and at the near and far walls of the breast. The projection data were reconstructed using analytic and iterative image reconstruction methods. For comparison, we examine an LEHR collimator with a resolution of 1 cm at 15.2 cm, which is the same as that of the MSSH at the center of the common volume-of-view. Also, we simulated the inclusion of a lead shield that covers the body to keep photons emitted from the other organs from reaching the MSSH collimator. Our results indicated that RMSSH SPECT provides greater breast lesion contrast than that of a planar image taken using a conventional LEHR collimator at all lesion locations. Depending on the lesion location, the spatial resolution of the RMSSH SPECT images may be comparable or slightly poorer than that of the planar scintimammographic images obtained using a LEHR collimator. Shielding of photon emissions from the other organs substantially improves the RMSSH SPECT image quality. We conclude that the Monte Carlo simulation methods are useful in evaluating the application of RMSSH SPECT to breast imaging as compared to conventional planar scintimammography and that RMSSH SPECT may enhance lesion detectability.
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paper_title: Spect: Single-Photon Emission Computed Tomography : A Primer

paper_content:

This book aims to assist nuclear medicine technologists in expanding their knowledge of nuclear medicine to include SPECT. The text of this primer is written with the assumption that the reader is proficient in most elements of nuclear medicine technology; therefore, the information is limited to data that will answer the basic questions of single-photon emission computed tomography .... The authors' goal is to bring the basics of this material together in a manner that would answer the technologist's fundamental questions. The authors have designed this primer in a generic manner to be used as an extension of the manufacturer's operating manual .... A glossary is included which contains some of the terminology relevant to the specialty, and reading lists are provided at the end of each chapter to direct the reader to more comprehensive text on specific subjects.
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paper_title: A Natural Pixel Decomposition for Two-Dimensional Image Reconstruction

paper_content:

In two-dimensional image reconstruction from line integrals using maximum likelihood, Bayesian, or minimum variance algorithms, the x-y plane on which the object estimate is defined is decomposed into nonoverlapping regions, or "pixels." This decomposition of an otherwise continuous structure results in significant errors, or model noise, which can exceed the effects of the fundamental measurement noise.
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paper_title: Geophysical Inverse Theory

paper_content:

PrefaceCh. 1Mathematical PrecursorCh. 2Linear Problems with Exact DataCh. 3Linear Problems with Uncertain DataCh. 4Resolution and InferenceCh. 5Nonlinear ProblemsAppendix A: The Dilogarithm FunctionAppendix B: Table for 1-norm MisfitsReferencesIndex
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paper_title: Random Point Processes in Time and Space

paper_content:

This senior graduate level textbook is the second revised edition of the textbook "Random Point Processes", written by D.L.Snyder and published in 1975. Its main subject area is point processes on multidimensional spaces, especially processes in two dimensions, reflecting the increase that has taken place in the use of point-process models for the description of data from which images of objects of interest are formed in a wide variety of scientific and engineering disciplines. Research done by the authors at the Biomedical Computer Laboratory at Washington University has resulted in newly developed models for position emission tomography and electron-microscopic autoradiography. All the applications are examples of nonparametric density estimation, which help provide new results concerning constrained estimation techniques. For these applications, the use of unconstrained maximum-likelihood estimation fails because the estimates are not consistent in the statistical sense; they do not converge, with increasing amounts of data, towards the quantity being estimated and regularization of the estimates is therefore considered essential. This graduate textbook on communications, signal processing and numerical mathematics is intended for the use of graduate students, researchers and practising engineers.
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paper_title: Improvement of image resolution and quantitative accuracy in clinical single photon emission computed tomography

paper_content:

Abstract Clinical Single Photon Emission Computed Tomography (SPECT) is a scanning technique which acquires gamma-camera images (‘projections’) over a range of angles around a patient. These projections allow the reconstruction of cross sectional (‘tomographic’) images of the gamma-radiating pharmaceutical distribution in the patient, thus providing interesting information about the functioning of organs and tissues. SPECT images are seriously affected by a variety of image degrading processes. Restrictions on the amount of radio-pharmaceutical that can be administered to a patient cause noise in the projections and the limited spatial resolution of the gamma-camera results in blurring of the projections. In addition to these image degradations, the reconstruction of cross-sections is complicated by Compton scattering of γ-photons in tissue, which causes attenuation of the photon flux received by the gamma-camera and causes improper detection of photons which have been scattered in tissue. This results in some additional blurring and loss of accuracy of the SPECT images in predicting activity concentrations. Tremendous efforts have been made to improve the quantitative accuracy and the spatial resolution of SPECT, and to reduce the noise in the reconstructed images. These efforts have resulted in corrective reconstruction algorithms, which are generally based on incorporation of accurate models of the main image degrading factors. Improvements of the data acquisition hardware can further increase image quality. In this paper, the image formation process of SPECT, including image-degrading factors, is explained. In addition, reconstruction algorithms and hardware modifications are reviewed, and their effects on image quality are illustrated with physical phantom and simulation experiments.
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paper_title: The X-ray transform

paper_content:

The k-plane integral transformations on the spaces L/sup 1/(R/sup n/) and L/sup 2/(R/sup n/) are studied. Such transformations arise naturally in electron microscopy, crystallography, biochemistry, molecular biology, aerodynamics, radio astronomy, radiography, and in various areas of pure mathematics such as partial differential equations, and integral geometry. (JFP)
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paper_title: Reconstruction of MR images from data acquired on a general nonregular grid by pseudoinverse calculation

paper_content:

A minimum-norm least-squares image-reconstruction method for the reconstruction of magnetic resonance images from non-Cartesian sampled data is proposed. The method is based on a general formalism for continuous-to-discrete mapping and pseudoinverse calculation. It does not involve any regridding or interpolation of the data and therefore the methodology differs fundamentally from existing regridding-based methods. Moreover, the method uses a continuous representation of objects in the image domain instead of a discretized representation. Simulations and experiments show the possibilities of the method in both radial and spiral imaging. Simulations revealed that minimum-norm least-squares image reconstruction can result in a drastic decrease of artifacts compared with regridding-based reconstruction. Besides, both in vivo and phantom experiments showed that minimum-norm least-squares image reconstruction leads to contrast improvement and increased signal-to-noise ratio compared with image reconstruction based on regridding. As an appendix, an analytical calculation of the raw data corresponding to the well-known Shepp and Logan software head phantom is presented.
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paper_title: Radiological Imaging The Theory Of Image Formation Detection And Processing

paper_content:

Thank you for reading radiological imaging the theory of image formation detection and processing. Maybe you have knowledge that, people have search numerous times for their favorite books like this radiological imaging the theory of image formation detection and processing, but end up in malicious downloads. Rather than reading a good book with a cup of coffee in the afternoon, instead they are facing with some malicious virus inside their laptop.
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paper_title: Implementation of an accelerated iterative algorithm for cone-beam SPECT

paper_content:

Describes the implementation of an accelerated iterative reconstruction algorithm for cone-beam projections using a single circular orbit in single-photon-emission computed tomography (SPECT). This algorithm is a modified maximum-likelihood-expectation-maximization (ML-EM) algorithm and several approaches have been used to accelerate the reconstruction process. These approaches include: (i) the use of ordered subsets; (ii) the use of active areas and volumes; and (iii) the storing in memory of the transition vector for a given ray (during the forward projection step). This algorithm, which compensates for collimator geometric sensitivity variation as a function of position and makes uniform attenuation corrections has been evaluated using experimentally acquired phantom data. The results demonstrate a two-orders-of-magnitude decrease of the computational time of this algorithm over the conventional ML-EM algorithm with similar convergence properties.
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paper_title: An SVD reconstruction algorithm using a natural pixel representation of the attenuated Radon transform

paper_content:

The purpose of this work was to investigate the use of natural pixels to reconstruct attenuation corrected emission data using attenuation factors calculated from a natural pixel representation of the reconstructed transmission data. The reconstruction algorithm for both transmission and emission reconstructions is based on the assumption that a continuous image can be obtained by backprojecting the discrete array q, which is the least squares solution to Mq=p, where p is the array of discrete measurements. The M matrix represents the operator obtained from the backprojection operator followed by the projection operator using the natural pixel basis. A singular value decomposition was used to solve the equation Mq=p, and the final image was obtained by sampling the continuous backprojection of the solution q at a discrete array of points. Computer simulations showed that accurate attenuation corrected reconstructions could be obtained using a natural pixel representation of the attenuated Radon transform for a variable attenuator.
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paper_title: Analytic determination of the pinhole collimator's point-spread function and RMS resolution with penetration

paper_content:

Pinhole collimators are widely used to image small organs and animals. The pinhole response function (PRF) of knife-edge pinhole collimators has been estimated previously using geometric constructions without considering penetration and using "roll-off" models that employ an exponential model for the flux. An exact expression for the PRF on the imaging plane that includes the effect of aperture penetration is derived in this paper by calculating the flux for photons passing through the aperture and those passing through the attenuating material. The PRF is then used to estimate the angular-dependent root-mean-square resolution in the directions parallel and perpendicular to the tilt of the point source. The formulas are then compared with experimental data.
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paper_title: Computed tomography : fundamentals, system technology, image quality, applications

paper_content:

System concepts System components Image reconstruction Spiral CT Multi-slice spiral CT Cone-beam CT Dynamic CT Quantitative CT Dual source CT Dual energy CT Flat detector CT Micro CT Image quality Spatial resolution Contrast Pixel noise Homogeneity Routine and special applications 3D displays Post-processing Quality assurance
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paper_title: Optimisation techniques for digital image reconstruction from their projections

paper_content:

A method is proposed for the digital reconstruction of images from their projections based on optimising specified performance criteria. The reconstruction problem is embedded into the framework of constrained optimisation and its solution is shown to lead to a relationship between the image and the one-dimensional Lagrange functions associated with each cost criterion. Two types of geometries (the parallel-beam and fan-beam systems) are considered for the acquisition of projection data and the constrained-optimisation problem is solved for both. The ensuing algorithms allow the reconstruction of multidimentional objects from one-dimensional functions only. For digital data a fast reconstruction algorithm is proposed which exploits the symmetries inherent in both a circular domain of image reconstruction and in projections obtained at equispaced angles. Computational complexity is significantly reduced by the use of fast-Fourier-transform techniques, as the underlying relationship between the available projection data and the associated Lagrange multipliers is shown to possess a block circulant matrix structure.
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paper_title: Principles of Computerized Tomographic Imaging

paper_content:

Tomography refers to the cross-sectional imaging of an object from either transmission or reflection data collected by illuminating the object from many different directions. The impact of tomography in diagnostic medicine has been revolutionary, since it has enabled doctors to view internal organs with unprecedented precision and safety to the patient. There are also numerous nonmedical imaging applications which lend themselves to methods of computerized tomography, such as mapping of underground resources...cross-sectional imaging of for nondestructive testing...the determination of the brightness distribution over a celestial sphere...three-dimensional imaging with electron microscopy. Principles of Computerized Tomographic Imaging provides a tutorial overview of topics in tomographic imaging covering mathematical principles and theory...how to apply the theory to problems in medical imaging and other fields...several variations of tomography that are currently being researched.
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paper_title: Statistical Approaches in Quantitative Positron Emission Tomography

paper_content:

Positron emission tomography is a medical imaging modality for producing 3D images of the spatial distribution of biochemical tracers within the human body. The images are reconstructed from data formed through detection of radiation resulting from the emission of positrons from radioisotopes tagged onto the tracer of interest. These measurements are approximate line integrals from which the image can be reconstructed using analytical inversion formulae. However these direct methods do not allow accurate modeling either of the detector system or of the inherent statistical fluctuations in the data. Here we review recent progress in developing statistical approaches to image estimation that can overcome these limitations. We describe the various components of the physical model and review different formulations of the inverse problem. The wide range of numerical procedures for solving these problems are then reviewed. Finally, we describe recent work aimed at quantifying the quality of the resulting images, both in terms of classical measures of estimator bias and variance, and also using measures that are of more direct clinical relevance.
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paper_title: The Mathematics of Computerized Tomography

paper_content:

The Mathematics of Computerized Tomography covers the relevant mathematical theory of the Radon transform and related transforms and also studies more practical questions such as stability, sampling, resolution, and accuracy. Quite a bit of attention is given to the derivation, analysis, and practical examination of reconstruction algorithm, for both standard problems and problems with incomplete data.
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paper_title: Iterative reconstruction algorithms

paper_content:

Abstract In this paper a survey of recent results on iterative reconstruction algorithms is given. These results, many of which have not yet appeared elsewhere, are applicable to a very general formulation of the reconstruction problem based on the series expansion approach. A set of optimization criteria and a number of iterative reconstruction algorithms are stated, together with theorems on the convergence of the algorithms to optimum images. The efficacy of the algorithms is compared to that of the convolution method. In particular, the falseness of the claim that ART and the backprojection method are the same is demonstrated.
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paper_title: Quantitative single-photon emission computed tomography: basics and clinical considerations.

paper_content:

Although quantitative single-photon emission computed tomography (SPECT) has been the goal of much research effort for a number of years, only recently has it received wide interest, especially for clinical applications. It has been increasingly recognized that the achievement of quantitative SPECT will increase the accuracy of measurements, such as dimensions of specific regions of interest, absolute amount of radioactivity, and dosimetry calculations, and substantially reduce reconstruction image artifacts and distortions, thus, greatly improving clinical diagnosis. This article provides a review of the definition of terms, major factors affecting SPECT quantitation and their degrading effects on SPECT image quality, and methods to compensate for these effects. Compensation methods include those that make certain approximations for ease of implementation and those that provide more accurate compensation by modeling the imaging process more exactly, usually at the cost of increased complexity and computational requirements. Different reconstruction and compensation methods may be compared through the use of phantom cardiac and brain SPECT studies. The clinical efficacy of the methods may be demonstrated by applying them to a clinical thallium-201 myocardial perfusion SPECT study. The results clearly demonstrate that, by modeling the imaging process and/or image degrading factors three-dimensionally, quantitative reconstruction and compensation methods provide the best image quality and quantitative accuracy. Important research efforts and developmental work being conducted currently to bring quantitative SPECT into routine clinical use are also discussed.
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paper_title: Ridge functions, natural pixels and minimal norm reconstruction

paper_content:

In this work reconstruction from a finite number of projections using ridge functions within the framework of parallel beam geometry is considered. Theoretically, ridge functions are continuous solutions of a system of integral equations and their sum results in the minimal norm solution of the reconstruction problem. In practice, discretized versions of ridge functions are considered to obtain reconstruction from real data. Discrete projection data are modeled with the help of natural pixels. Analytical inversion formulas based both on ridge functions and natural pixels are derived. Results of numerical experiments are discussed.
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paper_title: A filtered backprojection algorithm for axial head motion correction in fan-beam SPECT.

paper_content:

The authors present an approximate, but practical, three-dimensional filtered backprojection (FBP) reconstruction algorithm in fan-beam SPECT to correct for axial motion (both translation and rotation). A one-dimensional filter kernel was applied to the projections. It is assumed that the object is rigid and that its axial motion can be characterized by three components: one-dimensional translation and yaw and pitch rotations. It is further assumed that the motions that have occurred during the SPECT acquisition have been determined separately. The determined angular-view-dependent translation/rotation parameters were incorporated into the proposed FBP algorithm to correct for multiple axial head motions. The proposed axial head motion correction algorithm was evaluated using simulated three-dimensional Hoffman brain phantom data. Projections both with axial translation and with axial rotation, and with their combinations were generated. Images of a Hoffman brain phantom reconstructed using the proposed FBP algorithm and the conventional FBP algorithm were compared. Artefacts were observed in images without motion correction, but the artefacts were greatly reduced using the proposed reconstruction algorithm.
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paper_title: High-resolution image reconstruction method for time-of-flight positron emission tomography.

paper_content:

We present a new image reconstruction method for time-of-flight positron emission tomography (TOF-PET). The TOF-PET measurement system is modelled using the continuous-discrete mapping model, and images are reconstructed using an algebraic technique. The proposed method can produce images with better spatial resolution than conventional methods based on the filtered backprojection method. Numerical simulation results show that accurate modelling of the measurement system improves the spatial resolution and the contrast recovery, while the utilization of TOF information improves the signal-to-noise ratios of images.
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paper_title: Image-correction techniques in SPECT

paper_content:

This overview takes a look at different correction techniques for Single Photon Emission Computed Tomography (SPECT). We discuss the influence of the detection system followed by the scatter and attenuation caused by the object of investigation. When possible we describe how the correction methods for the different physical effects can be incorporated in the reconstruction method, being either filtered backprojection or iterative reconstruction.
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paper_title: Iterative and analytical reconstruction algorithms for varying-focal-length cone-beam projections.

paper_content:

In single photon emission computed tomography (SPECT), a varying-focal-length cone-beam collimator can be used to reduce the truncation problem and to maintain sensitivity when imaging the organ of interest. The collimator is constructed so that the collimator holes focus to a circular symmetric, spatially varying, focal point function. The focal length increases radially from the shortest focal length at the centre to the longest focal length at the periphery of the collimator. This paper describes a 3D backprojection-filtering (BF) algorithm for this varying-focal-length cone-beam geometry. The proposed algorithm is compared to an iterative ML-EM (maximum likelihood-expectation maximization) algorithm. The 3D Hoffman brain phantom and Defrise phantom are used in computer simulations. Since the maximum tilt angle of the projection rays is small for most realistic imaging geometries, the proposed algorithm provides a good approximation. When a circular orbit is used, the BF algorithm gives an exact reconstruction of the central slice.
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paper_title: Generalized approach to inverse problems in tomography: Image reconstruction for spatially variant systems using natural pixels

paper_content:

A major limitation in tomographic inverse problems is inadequate computation speed, which frequently impedes the application of engineering ideas and principles in medical science more than in the physical and engineering sciences. Medical problems are computationally taxing because a minimum description of the system often involves 5 dimensions (3 space, 1 energy, 1 time), with the range of each space coordinate requiring up to 512 samples. The computational tasks for this problem can be simply expressed by posing the problem as one in which the tomograph system response function is spatially invariant, and the noise is additive and Gaussian. Under these assumptions, a number of reconstruction methods have been implemented with generally satisfactory results for general medical imaging purposes. However, if the system response function of the tomograph is assumed more realistically to be spatially variant and the noise to be Poisson, the computational problem becomes much more difficult. Some of the algorithms being studied to compensate for position dependent resolution and statistical fluctuations in the data acquisition process, when expressed in canonical form, are not practical for clinical applications because the number of computations necessary exceeds the capabilities of high performance computer systems currently available. Reconstruction methods based onmore » natural pixels, specifically orthonormal natural pixels, preserve symmetries in the data acquisition process. Fast implementations of orthonormal natural pixel algorithms can achieve orders of magnitude speedup relative to general implementations. Thus, specialized thought in algorithm development can lead to more significant increases in performance than can be achieved through hardware improvements alone.« less
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paper_title: Correction of Nonuniform Response in the Reconstruction for Single Photon Emission Computed Tomography with a Spatially Varying Focal Length Collimator

paper_content:

SPECT (single photon emission computed tomography) images can visualize physiological functions directly and are diagnostically useful. The reconstruction methods based on the Continuous-Discrete (C-D) mapping model have an immediate effect on SPECT imaging because they employ photon detection kernels describing the photon transport from the body to the detector. However, in a spatially varying focal length geometry, the concentration of the sensitivity around the center of rotation of the detector causes a change in the response from the original image to the reconstructed image. As a result, a false hot spot often appears around the center of rotation. This paper presents a convenient method to repress the change by correcting the resolving kernels. Numerical simulations show that the method can reduce the false hot spots.
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paper_title: Mean and Variance of Coincidence Counting with Deadtime

paper_content:

Abstract We analyze the first and second moments of the coincidence-counting process for a system affected by paralyzable (extendable) deadtime with (possibly unequal) deadtimes in each singles channel. We consider both “accidental” and “genuine” coincidences, and derive exact analytical expressions for the first and second moments of the number of recorded coincidence events under various scenarios. The results include an exact form for the coincidence rate under the combined effects of decay, background, and deadtime. The analysis confirms that coincidence counts are not exactly Poisson, but suggests that the Poisson statistical model that is used for positron emission tomography image reconstruction is a reasonable approximation since the mean and variance are nearly equal.
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paper_title: Attenuation correction for rotating slant-hole (RSH) SPECT using exact rebinning

paper_content:

We have recently derived an analytic method of image reconstruction for the rotating slant-hole (RSH) SPECT scanner for cardiac imaging. The method assumes that the emission activity lies in a region of constant attenuation. It operates by rebinning the three-dimensional (3-D) measured projections into the geometry of a stack of two-dimensional (2-D) slices and performing 2-D reconstruction in a second step. An important consequence of this method is a mathematical proof that opposing projections are not necessary for full tomographic reconstruction in the presence of attenuation. We have now tested our algorithm in the presence of noisy projection data and with projections which violates the constant attenuation assumption. We present results from simulated and real phantom data demonstrating effective attenuation correction in the presence of a uniform low-level of background in the torso, with truncated liver activity, and without opposing views. Our conclusions are that attenuation correction can be performed for RSH SPECT without measuring opposing views, and that the effect of background activity (which violates the assumptions required for exponential data) and truncated activity of a nearby liver are small and easily dominated by typical noise levels in SPECT imaging.
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paper_title: Time-of-flight PET.

paper_content:

Time-of-Flight (TOF) positron emission tomographs (PET) were developed during the 1980s and early 1990s. Initial investigations into the potential improvements in signal-to-noise ratios if TOF information was utilized were published in 1980. By 1982, two groups (Washington University and Commissariat á l'Energie Atomique-Laboratorie d'Electronique et de L'Informatique [CEA-LETI]) were designing and building the first TOF tomographs. A third group at the University of Texas also designed and built a TOF system. These systems were optimized for high count-rate imaging of short-lived radiotracers for applications such as cardiac blood flow. The first system put into operation for patient scans was the Super PETT I built at Washington University by Michel Ter-Pogossian and his colleagues. The Washington University group went on to design two additional versions of TOF systems and the CEA-LETI group developed two basic tomograph designs. As Bismuth Germanate (BGO)-based scanners were refined, it became clear that the TOF systems could not provide the same high spatial resolution as offered by the newer systems. The use of the fast scintillators required for TOF systems also resulted in lower intrinsic sensitivity that was only partially compensated for by the effective gain in sensitivity offered by TOF image reconstruction techniques. Further development of TOF systems was suspended in the early 1990s. With the development of new scintillators that provide more light output and are denser than those available in the 1980s and considerably faster than BGO, there is new interest in the application of TOF techniques for future tomograph designs.
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paper_title: A combined half-cone beam and parallel hole collimation system for SPECT brain imaging

paper_content:

The authors' research group has recently examined new types of collimator designs with the goal of improving sensitivity and lesion detection for head imaging. One of these collimator designs is a half-cone beam collimator. However, the sensitivity is reduced as objects are removed from the focal line and it does not satisfy Tuy's sufficiency condition. Parallel hole collimation does not have this problem with axial blurring, however, values for sensitivity are not as large as for half-cone collimators. A potential configuration is to use one parallel hole collimator in conjunction with two half-cone beam collimators on a triple camera single photon emission computed tomography (SPECT) system. This might produce results that represent the best of both collimation systems. The authors acquired projection data with a Defrise disk phantom filled with Tc-99m. A half-cone beam collimator with a focal length of 50 cm was placed on one head of a triple camera SPECT system. A low energy super high resolution (LESR) parallel hole collimator was placed on a second head. Different projection data sets were combined to model acquisition in the three headed gamma camera with three half-cone beam collimators, two half-cone beam and one parallel beam collimators, one half-cone beam and two parallel beam collimators and three parallel beam collimators. Image reconstruction used a modified maximum likelihood maximization-expectation (ML-EM) algorithm. For the reconstruction with three half-cone beam collimators, the authors observe axial blurring. This is largely reduced with two half-cone beam and one parallel beam collimators. Graphs of the image profiles demonstrate that the blurring along the axial direction is decreased with the addition of parallel hole collimators.
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paper_title: Inverse problems as statistics

paper_content:

What mathematicians, scientists, engineers, and statisticians mean by “inverse problem” differs. For a statistician, an inverse problem is an inference or estimation problem. The data are finite in number and contain errors, as they do in classical estimation or inference problems, and the unknown typically is infinite-dimensional, as it is in nonparametric regression. The additional complication in an inverse problem is that the data are only indirectly related to the unknown. Standard statistical concepts, questions, and considerations such as bias, variance, mean-squared error, identifiability, consistency, efficiency, and various forms of optimality apply to inverse problems. This article discusses inverse problems as statistical estimation and inference problems, and points to the literature for a variety of techniques and results.
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paper_title: A unified reconstruction framework for both parallel-beam and variable focal-length fan-beam collimators by a Cormack-type inversion of exponential Radon transform

paper_content:

A variety of inversions of exponential Radon transform has been derived based on the circular harmonic transform in Fourier space by several research groups. However, these inversions cannot be directly applied to deal with the reconstruction for fan-beam or variable-focal-length fan-beam collimator geometries in single photon emission computed tomography (SPECT). In this paper, the authors derived a Cormack-type inversion of the exponential Radon transform by employing the circular harmonic transform directly in the projection space and the image space instead of the Fourier space. Thus, a unified reconstruction framework is established for parallel-, fan-, and variable-focal-length fan-beam collimator geometries. Compared to many existing algorithms, the presented one greatly mitigates the difficulty of image reconstruction due to the complicated collimator geometry and significantly reduces the computational burden of the special functions, such as Chebyshev or Bessel functions. By the well-established fast-Fourier transform (FFT), the authors' algorithm is very efficient, as demonstrated by several numerical simulations.
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paper_title: MENT: A maximum entropy algorithm for reconstructing a source from projection data

paper_content:

Abstract This paper describes an iterative algorithm, MENT, which produces a maximum entropy solution to the problem of reconstructing a source from a discrete set of projection data. Whereas the MART algorithm considered by Lent in (“1976 Society of Photographic Scientists and Engineers Conference Proceedings,” SPSE, Washington, D.C., 1977) uses a rectangular grid to represent the source, MENT uses a discretization which is better suited to the problem. Unlike MART, MENT does not require the evaluation of logarithms or exponentials. The storage requirements are also lower for MENT. Numerical examples are given of a two-dimensional reconstruction from five views. MENT and MART are compared with regard to convergence rates, artifact formation, and stability against noise errors in the data. A three-dimensional version of the algorithm is also considered; we give an example of a direct three-dimensional reconstruction from four views.
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paper_title: Positron-emission tomography

paper_content:

We review positron-emission tomography (PET), which has inherent advantages that avoid the shortcomings of other nuclear medicine imaging methods. PET image reconstruction methods with origins in signal and image processing are discussed, including the potential problems of these methods. A summary of statistical image reconstruction methods, which can yield improved image quality, is also presented.
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paper_title: Image reconstruction algorithm for a rotating slat collimator

paper_content:

A slat collimator in single photon emission computed tomography consists of a set of parallel slats. As the collimator spins, the detector measures a one-dimensional projection data set. A complete data set can be obtained by rotating the detector/collimator assembly around the object (patient) while the collimator spins continuously. The measured projection data are assumed to be weighted planar integrals of the object. This paper describes the development of an approximate three-dimensional image reconstruction algorithm for a rotating/spinning slat collimator. This algorithm is in filtered backprojection form. Computer simulations were performed to verify the effectiveness of the algorithm.
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paper_title: Pinhole collimation for ultra-high-resolution, small-field-of-view SPECT.

paper_content:

The objective of this investigation was to evaluate small-field-of-view, ultra-high-resolution pinhole collimation for a rotating-camera SPECT system that could be used to image small laboratory animals. Pinhole collimation offers distinct advantages over conventional parallel-hole collimation when used to image small objects. Since geometric sensitivity increases markedly for points close to the pinhole, small-diameter and high-magnification pinhole geometries may be useful for selected imaging tasks when used with large-field-of-view scintillation cameras. The use of large magnifications can minimize the loss of system resolution caused by the intrinsic resolution of the scintillation camera. A pinhole collimator has been designed and built that can be mounted on one of the scintillation cameras of a triple-head SPECT system. Three pinhole inserts with approximate aperture diameters of 0.6, 1.2 and 2.0 mm have been built and can be mounted individually on the collimator housing. When a ramp filter is used with a three-dimensional (3D) filtered backprojection (FBP) algorithm, the three apertures have in-plane SPECT spatial resolutions (FWHM) at 4 cm of 1.5, 1.9 and 2.8 mm, respectively. In-air point source sensitivities at 4 cm from the apertures are 0.9, 2.6 and 5.7 counts s(-1) microCi(-1) (24, 70 and 154 counts s(-1) MBq(-1)) for the 0.6, 1.2 and 2.0 mm apertures, respectively. In vitro image quality was evaluated with a micro-cold-rod phantom and a micro-Defrise phantom using both the 3D FBP algorithm and a 3D maximum likelihood-expectation maximization (ML-EM) algorithm. In vivo image quality was evaluated using two (315 and 325 g) rats. Ultra-high-resolution pinhole SPECT is an inexpensive and simple approach for imaging small animals that can be used with existing rotating-camera SPECT system.
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paper_title: Introduction to inverse problems in imaging

paper_content:

Part 1 Image deconvolution: some mathematical tools examples of image blurring the ill-posedness of image deconvolution regularization methods iterative regularization methods statistical methods. Part 2 Linear inverse problems: examples of linear inverse problems singular value decomposition (SVD) inversion methods revisited Fourier based methods for specific problems comments and concluding remarks. Part 3 Mathematical appendices. References.
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paper_title: Geophysical Inverse Theory

paper_content:

PrefaceCh. 1Mathematical PrecursorCh. 2Linear Problems with Exact DataCh. 3Linear Problems with Uncertain DataCh. 4Resolution and InferenceCh. 5Nonlinear ProblemsAppendix A: The Dilogarithm FunctionAppendix B: Table for 1-norm MisfitsReferencesIndex
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paper_title: Computed tomography : fundamentals, system technology, image quality, applications

paper_content:

System concepts System components Image reconstruction Spiral CT Multi-slice spiral CT Cone-beam CT Dynamic CT Quantitative CT Dual source CT Dual energy CT Flat detector CT Micro CT Image quality Spatial resolution Contrast Pixel noise Homogeneity Routine and special applications 3D displays Post-processing Quality assurance
---
paper_title: Principles of Computerized Tomographic Imaging

paper_content:

Tomography refers to the cross-sectional imaging of an object from either transmission or reflection data collected by illuminating the object from many different directions. The impact of tomography in diagnostic medicine has been revolutionary, since it has enabled doctors to view internal organs with unprecedented precision and safety to the patient. There are also numerous nonmedical imaging applications which lend themselves to methods of computerized tomography, such as mapping of underground resources...cross-sectional imaging of for nondestructive testing...the determination of the brightness distribution over a celestial sphere...three-dimensional imaging with electron microscopy. Principles of Computerized Tomographic Imaging provides a tutorial overview of topics in tomographic imaging covering mathematical principles and theory...how to apply the theory to problems in medical imaging and other fields...several variations of tomography that are currently being researched.
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paper_title: The Mathematics of Computerized Tomography

paper_content:

The Mathematics of Computerized Tomography covers the relevant mathematical theory of the Radon transform and related transforms and also studies more practical questions such as stability, sampling, resolution, and accuracy. Quite a bit of attention is given to the derivation, analysis, and practical examination of reconstruction algorithm, for both standard problems and problems with incomplete data.
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paper_title: Introduction to inverse problems in imaging

paper_content:

Part 1 Image deconvolution: some mathematical tools examples of image blurring the ill-posedness of image deconvolution regularization methods iterative regularization methods statistical methods. Part 2 Linear inverse problems: examples of linear inverse problems singular value decomposition (SVD) inversion methods revisited Fourier based methods for specific problems comments and concluding remarks. Part 3 Mathematical appendices. References.
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paper_title: Spect: Single-Photon Emission Computed Tomography : A Primer

paper_content:

This book aims to assist nuclear medicine technologists in expanding their knowledge of nuclear medicine to include SPECT. The text of this primer is written with the assumption that the reader is proficient in most elements of nuclear medicine technology; therefore, the information is limited to data that will answer the basic questions of single-photon emission computed tomography .... The authors' goal is to bring the basics of this material together in a manner that would answer the technologist's fundamental questions. The authors have designed this primer in a generic manner to be used as an extension of the manufacturer's operating manual .... A glossary is included which contains some of the terminology relevant to the specialty, and reading lists are provided at the end of each chapter to direct the reader to more comprehensive text on specific subjects.
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paper_title: Quantitative single-photon emission computed tomography: basics and clinical considerations.

paper_content:

Although quantitative single-photon emission computed tomography (SPECT) has been the goal of much research effort for a number of years, only recently has it received wide interest, especially for clinical applications. It has been increasingly recognized that the achievement of quantitative SPECT will increase the accuracy of measurements, such as dimensions of specific regions of interest, absolute amount of radioactivity, and dosimetry calculations, and substantially reduce reconstruction image artifacts and distortions, thus, greatly improving clinical diagnosis. This article provides a review of the definition of terms, major factors affecting SPECT quantitation and their degrading effects on SPECT image quality, and methods to compensate for these effects. Compensation methods include those that make certain approximations for ease of implementation and those that provide more accurate compensation by modeling the imaging process more exactly, usually at the cost of increased complexity and computational requirements. Different reconstruction and compensation methods may be compared through the use of phantom cardiac and brain SPECT studies. The clinical efficacy of the methods may be demonstrated by applying them to a clinical thallium-201 myocardial perfusion SPECT study. The results clearly demonstrate that, by modeling the imaging process and/or image degrading factors three-dimensionally, quantitative reconstruction and compensation methods provide the best image quality and quantitative accuracy. Important research efforts and developmental work being conducted currently to bring quantitative SPECT into routine clinical use are also discussed.
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paper_title: Penalized weighted least-squares image reconstruction for positron emission tomography

paper_content:

This paper presents an image reconstruction method for positron-emission tomography (PET) based on a penalized, weighted least-squares (PWLS) objective. For PET measurements that are precorrected for accidental coincidences, we argue statistically that a least-squares objective function is as appropriate, if not more so, than the popular Poisson likelihood objective. We propose a simple data-based method for determining the weights that accounts for attenuation and detector efficiency. A nonnegative successive over-relaxation (+SOR) algorithm converges rapidly to the global minimum of the PWLS objective. Quantitative simulation results demonstrate that the bias/variance tradeoff of the PWLS+SOR method is comparable to the maximum-likelihood expectation-maximization (ML-EM) method (but with fewer iterations), and is improved relative to the conventional filtered backprojection (FBP) method. Qualitative results suggest that the streak artifacts common to the FBP method are nearly eliminated by the PWLS+SOR method, and indicate that the proposed method for weighting the measurements is a significant factor in the improvement over FBP.
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paper_title: Maximum likelihood from incomplete data via the EM algorithm

paper_content:

Vibratory power unit for vibrating conveyers and screens comprising an asynchronous polyphase motor, at least one pair of associated unbalanced masses disposed on the shaft of said motor, with the first mass of a pair of said unbalanced masses being rigidly fastened to said shaft and with said second mass of said pair being movably arranged relative to said first mass, means for controlling and regulating the conveying rate during conveyer operation by varying the rotational speed of said motor between predetermined minimum and maximum values, said second mass being movably outwardly by centrifugal force against the pressure of spring means, said spring means being prestressed in such a manner that said second mass is, at rotational motor speeds lower than said minimum speed, held in its initial position, and at motor speeds between said lower and upper values in positions which are radially offset with respect to the axis of said motor to an extent depending on the value of said rotational motor speed.
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paper_title: Application of the row action maximum likelihood algorithm with spherical basis functions to clinical PET imaging

paper_content:

Three-dimensional (3D) reconstructions from fully 3D positron emission tomography (PET) data can yield high-quality images but at a high computational cost. The 3D row action maximum likelihood algorithm (3D RAMLA) with spherically-symmetric basis functions (blobs) has recently been modified to reconstruct multi-slice 2D PET data after Fourier rebinning (FORE) but still using 3D basis functions (2.5D RAMLA. In this study 2.5D RAMLA and 3D RAMLA were applied to several patient and phantom PET datasets to assess their clinical performance. RAMLA performance was compared to that for the reconstruction techniques in routine clinical use on the authors' PET scanners. Torso phantom and whole-body patient scans acquired on the C-PET scanner were reconstructed after FORE with filtered back-projection (FORE+FBP), the ordered subsets expectation maximization algorithm (FORE+OSEM), and FORE+2.5D RAMLA for various reconstruction parameters. The 3D Hoffman brain phantom scanned on the HEAD Penn-PET scanner was reconstructed with the 3D reprojection algorithm (3DRP) and 3D RAMLA, as well, as FORE+FBP, FORE+OSEM, and FORE+2.5D RAMLA. The authors' results demonstrate improvement of 3D and 2.5D RAMLA with blob basis functions, compared to the reconstruction methods currently in clinical use, in terms of contrast recovery and noise, especially in regions of limited statistics.
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paper_title: A row-action alternative to the EM algorithm for maximizing likelihood in emission tomography.

paper_content:

The maximum likelihood (ML) approach to estimating the radioactive distribution in the body cross section has become very popular among researchers in emission computed tomography (ECT) since it has been shown to provide very good images compared to those produced with the conventional filtered backprojection (FBP) algorithm. The expectation maximization (EM) algorithm is an often-used iterative approach for maximizing the Poisson likelihood in ECT because of its attractive theoretical and practical properties. Its major disadvantage is that, due to its slow rate of convergence, a large amount of computation is often required to achieve an acceptable image. Here, the authors present a row-action maximum likelihood algorithm (RAMLA) as an alternative to the EM algorithm for maximizing the Poisson likelihood in ECT. The authors deduce the convergence properties of this algorithm and demonstrate by way of computer simulations that the early iterates of RAMLA increase the Poisson likelihood in ECT at an order of magnitude faster that the standard EM algorithm. Specifically, the authors show that, from the point of view of measuring total radionuclide uptake in simulated brain phantoms, iterations 1, 2, 3, and 4 of RAMLA perform at least as well as iterations 45, 60, 70, and 80, respectively, of EM. Moreover, the authors show that iterations 1, 2, 3, and 4 of RAMLA achieve comparable likelihood values as iterations 45, 60, 70, and 80, respectively, of EM. The authors also present a modified version of a recent fast ordered subsets EM (OS-EM) algorithm and show that RAMLA is a special case of this modified OS-EM. Furthermore, the authors show that their modification converges to a ML solution whereas the standard OS-EM does not.
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paper_title: A Statistical Model for Positron Emission Tomography

paper_content:

Abstract Positron emission tomography (PET)—still in its research stages—is a technique that promises to open new medical frontiers by enabling physicians to study the metabolic activity of the body in a pictorial manner. Much as in X-ray transmission tomography and other modes of computerized tomography, the quality of the reconstructed image in PET is very sensitive to the mathematical algorithm to be used for reconstruction. In this article, we tailor a mathematical model to the physics of positron emissions, and we use the model to describe the basic image reconstruction problem of PET as a standard problem in statistical estimation from incomplete data. We describe various estimation procedures, such as the maximum likelihood (ML) method (using the EM algorithm), the method of moments, and the least squares method. A computer simulation of a PET experiment is then used to demonstrate the ML and the least squares reconstructions. The main purposes of this article are to report on what we believe is an...
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paper_title: Maximum Likelihood Reconstruction for Emission Tomography

paper_content:

Previous models for emission tomography (ET) do not distinguish the physics of ET from that of transmission tomography. We give a more accurate general mathematical model for ET where an unknown emission density ? = ?(x, y, z) generates, and is to be reconstructed from, the number of counts n*(d) in each of D detector units d. Within the model, we give an algorithm for determining an estimate ? of ? which maximizes the probability p(n*|?) of observing the actual detector count data n* over all possible densities ?. Let independent Poisson variables n(b) with unknown means ?(b), b = 1, ···, B represent the number of unobserved emissions in each of B boxes (pixels) partitioning an object containing an emitter. Suppose each emission in box b is detected in detector unit d with probability p(b, d), d = 1, ···, D with p(b, d) a one-step transition matrix, assumed known. We observe the total number n* = n*(d) of emissions in each detector unit d and want to estimate the unknown ? = ?(b), b = 1, ···, B. For each ?, the observed data n* has probability or likelihood p(n*|?). The EM algorithm of mathematical statistics starts with an initial estimate ?0 and gives the following simple iterative procedure for obtaining a new estimate ?new, from an old estimate ?old, to obtain ?k, k = 1, 2, ···, ?new(b)= ?old(b) ?Dd=1 n*(d)p(b,d)/??old(b?)p(b?,d),b=1,···B.
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paper_title: Reconstruction of MR images from data acquired on a general nonregular grid by pseudoinverse calculation

paper_content:

A minimum-norm least-squares image-reconstruction method for the reconstruction of magnetic resonance images from non-Cartesian sampled data is proposed. The method is based on a general formalism for continuous-to-discrete mapping and pseudoinverse calculation. It does not involve any regridding or interpolation of the data and therefore the methodology differs fundamentally from existing regridding-based methods. Moreover, the method uses a continuous representation of objects in the image domain instead of a discretized representation. Simulations and experiments show the possibilities of the method in both radial and spiral imaging. Simulations revealed that minimum-norm least-squares image reconstruction can result in a drastic decrease of artifacts compared with regridding-based reconstruction. Besides, both in vivo and phantom experiments showed that minimum-norm least-squares image reconstruction leads to contrast improvement and increased signal-to-noise ratio compared with image reconstruction based on regridding. As an appendix, an analytical calculation of the raw data corresponding to the well-known Shepp and Logan software head phantom is presented.
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paper_title: Penalized-likelihood estimators and noise analysis for randoms-precorrected PET transmission scans

paper_content:

This paper analyzes and compares image reconstruction methods based on practical approximations to the exact log likelihood of randoms precorrected positron emission tomography (PET) measurements. The methods apply to both emission and transmission tomography, however, in this paper the authors focus on transmission tomography. The results of experimental PET transmission scans and variance approximations demonstrate that the shifted Poisson (SP) method avoids the systematic bias of the conventional data-weighted least squares (WLS) method and leads to significantly lower variance than conventional statistical methods based on the log likelihood of the ordinary Poisson (OF) model. The authors develop covariance approximations to analyze the propagation of noise from attenuation maps into emission images via the attenuation correction factors (ACF's). Empirical pixel and region variances from real transmission data agree closely with the analytical predictions. Both the approximations and the empirical results show that the performance differences between the OP model and SP model are even larger, when considering noise propagation from the transmission images into the final emission images, than the differences in the attenuation maps themselves.
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paper_title: Fully 3D Bayesian image reconstruction for the ECAT EXACT HR

paper_content:

A fully 3D Bayesian method is described for high resolution reconstruction of images from the Siemens/CTI ECAT EXACT HR+ whole body positron emission tomography (PET) scanner. To maximize resolution recovery from the system the authors model depth dependent geometric efficiency, intrinsic detector efficiency, photon pair non-colinearity, crystal penetration and inter-crystal scatter. They also explicitly model the effects of axial rebinning and angular mashing on the detection probability or system matrix. By fully exploiting sinogram symmetries and using a factored system matrix and automated indexing schemes, the authors are able to achieve substantial savings in both the storage size and time required to compute forward and backward projections. Reconstruction times are further reduced using multi-threaded programming on a four processor Unix server. Bayesian reconstructions are computed using a Huber prior and a shifted-Poisson likelihood model that accounts for the effects of randoms subtraction and scatter. Reconstructions of phantom data show that the 3D Bayesian method can achieve improved FWHM resolution and contrast recovery ratios at matched background noise levels compared to both the 3D reprojection method and an OSEM method based on the shifted-Poisson model.
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paper_title: Accelerated Image Reconstruction using Ordered Subsets of Projection Data

paper_content:

The authors define ordered subset processing for standard algorithms (such as expectation maximization, EM) for image restoration from projections. Ordered subsets methods group projection data into an ordered sequence of subsets (or blocks). An iteration of ordered subsets EM is defined as a single pass through all the subsets, in each subset using the current estimate to initialize application of EM with that data subset. This approach is similar in concept to block-Kaczmarz methods introduced by Eggermont et al. (1981) for iterative reconstruction. Simultaneous iterative reconstruction (SIRT) and multiplicative algebraic reconstruction (MART) techniques are well known special cases. Ordered subsets EM (OS-EM) provides a restoration imposing a natural positivity condition and with close links to the EM algorithm. OS-EM is applicable in both single photon (SPECT) and positron emission tomography (PET). In simulation studies in SPECT, the OS-EM algorithm provides an order-of-magnitude acceleration over EM, with restoration quality maintained. >
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paper_title: Penalized weighted least-squares image reconstruction for positron emission tomography

paper_content:

This paper presents an image reconstruction method for positron-emission tomography (PET) based on a penalized, weighted least-squares (PWLS) objective. For PET measurements that are precorrected for accidental coincidences, we argue statistically that a least-squares objective function is as appropriate, if not more so, than the popular Poisson likelihood objective. We propose a simple data-based method for determining the weights that accounts for attenuation and detector efficiency. A nonnegative successive over-relaxation (+SOR) algorithm converges rapidly to the global minimum of the PWLS objective. Quantitative simulation results demonstrate that the bias/variance tradeoff of the PWLS+SOR method is comparable to the maximum-likelihood expectation-maximization (ML-EM) method (but with fewer iterations), and is improved relative to the conventional filtered backprojection (FBP) method. Qualitative results suggest that the streak artifacts common to the FBP method are nearly eliminated by the PWLS+SOR method, and indicate that the proposed method for weighting the measurements is a significant factor in the improvement over FBP.
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paper_title: The EM algorithm and extensions

paper_content:

The only single-source--now completely updated and revised--to offer a unified treatment of the theory, methodology, and applications of the EM algorithm Complete with updates that capture developments from the past decade, The EM Algorithm and Extensions, Second Edition successfully provides a basic understanding of the EM algorithm by describing its inception, implementation, and applicability in numerous statistical contexts. In conjunction with the fundamentals of the topic, the authors discuss convergence issues and computation of standard errors, and, in addition, unveil many parallels and connections between the EM algorithm and Markov chain Monte Carlo algorithms. Thorough discussions on the complexities and drawbacks that arise from the basic EM algorithm, such as slow convergence and lack of an in-built procedure to compute the covariance matrix of parameter estimates, are also presented. While the general philosophy of the First Edition has been maintained, this timely new edition has been updated, revised, and expanded to include: New chapters on Monte Carlo versions of the EM algorithm and generalizations of the EM algorithm New results on convergence, including convergence of the EM algorithm in constrained parameter spaces Expanded discussion of standard error computation methods, such as methods for categorical data and methods based on numerical differentiation Coverage of the interval EM, which locates all stationary points in a designated region of the parameter space Exploration of the EM algorithm's relationship with the Gibbs sampler and other Markov chain Monte Carlo methods Plentiful pedagogical elements-chapter introductions, lists of examples, author and subject indices, computer-drawn graphics, and a related Web site The EM Algorithm and Extensions, Second Edition serves as an excellent text for graduate-level statistics students and is also a comprehensive resource for theoreticians, practitioners, and researchers in the social and physical sciences who would like to extend their knowledge of the EM algorithm.
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paper_title: Application of the row action maximum likelihood algorithm with spherical basis functions to clinical PET imaging

paper_content:

Three-dimensional (3D) reconstructions from fully 3D positron emission tomography (PET) data can yield high-quality images but at a high computational cost. The 3D row action maximum likelihood algorithm (3D RAMLA) with spherically-symmetric basis functions (blobs) has recently been modified to reconstruct multi-slice 2D PET data after Fourier rebinning (FORE) but still using 3D basis functions (2.5D RAMLA. In this study 2.5D RAMLA and 3D RAMLA were applied to several patient and phantom PET datasets to assess their clinical performance. RAMLA performance was compared to that for the reconstruction techniques in routine clinical use on the authors' PET scanners. Torso phantom and whole-body patient scans acquired on the C-PET scanner were reconstructed after FORE with filtered back-projection (FORE+FBP), the ordered subsets expectation maximization algorithm (FORE+OSEM), and FORE+2.5D RAMLA for various reconstruction parameters. The 3D Hoffman brain phantom scanned on the HEAD Penn-PET scanner was reconstructed with the 3D reprojection algorithm (3DRP) and 3D RAMLA, as well, as FORE+FBP, FORE+OSEM, and FORE+2.5D RAMLA. The authors' results demonstrate improvement of 3D and 2.5D RAMLA with blob basis functions, compared to the reconstruction methods currently in clinical use, in terms of contrast recovery and noise, especially in regions of limited statistics.
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paper_title: Multiresolution Markov models for signal and image processing

paper_content:

Reviews a significant component of the rich field of statistical multiresolution (MR) modeling and processing. These MR methods have found application and permeated the literature of a widely scattered set of disciplines, and one of our principal objectives is to present a single, coherent picture of this framework. A second goal is to describe how this topic fits into the even larger field of MR methods and concepts-in particular, making ties to topics such as wavelets and multigrid methods. A third goal is to provide several alternate viewpoints for this body of work, as the methods and concepts we describe intersect with a number of other fields. The principle focus of our presentation is the class of MR Markov processes defined on pyramidally organized trees. The attractiveness of these models stems from both the very efficient algorithms they admit and their expressive power and broad applicability. We show how a variety of methods and models relate to this framework including models for self-similar and 1/f processes. We also illustrate how these methods have been used in practice.
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paper_title: Fast gradient-based methods for Bayesian reconstruction of transmission and emission PET images

paper_content:

The authors describe conjugate gradient algorithms for reconstruction of transmission and emission PET images. The reconstructions are based on a Bayesian formulation, where the data are modeled as a collection of independent Poisson random variables and the image is modeled using a Markov random field. A conjugate gradient algorithm is used to compute a maximum a posteriori (MAP) estimate of the image by maximizing over the posterior density. To ensure nonnegativity of the solution, a penalty function is used to convert the problem to one of unconstrained optimization. Preconditioners are used to enhance convergence rates. These methods generally achieve effective convergence in 15-25 iterations. Reconstructions are presented of an /sup 18/FDG whole body scan from data collected using a Siemens/CTI ECAT931 whole body system. These results indicate significant improvements in emission image quality using the Bayesian approach, in comparison to filtered backprojection, particularly when reprojections of the MAP transmission image are used in place of the standard attenuation correction factors. >
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paper_title: A concave prior penalizing relative differences for maximum-a-posteriori reconstruction in emission tomography

paper_content:

A well-known problem with maximum likelihood reconstruction in emission tomography is the excessive noise propagation. To prevent this, the objective function is often extended with a Gibbs prior favoring smooth solutions. We hypothesize that the following three requirements should produce a useful and conservative Gibbs prior for emission tomography: 1) the prior function should be concave to ensure that the posterior has a unique maximum; 2) the prior should penalize relative differences rather than absolute differences; 3) the prior should be tolerant for "large" differences between neighboring pixels. The second requirement should avoid tuning problems caused by the large dynamic range of activity values in the reconstructed image. A simple function has been derived that meets these three requirements. Our initial evaluations indicate that the prior behaves as intended.
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paper_title: Improvement of image resolution and quantitative accuracy in clinical single photon emission computed tomography

paper_content:

Abstract Clinical Single Photon Emission Computed Tomography (SPECT) is a scanning technique which acquires gamma-camera images (‘projections’) over a range of angles around a patient. These projections allow the reconstruction of cross sectional (‘tomographic’) images of the gamma-radiating pharmaceutical distribution in the patient, thus providing interesting information about the functioning of organs and tissues. SPECT images are seriously affected by a variety of image degrading processes. Restrictions on the amount of radio-pharmaceutical that can be administered to a patient cause noise in the projections and the limited spatial resolution of the gamma-camera results in blurring of the projections. In addition to these image degradations, the reconstruction of cross-sections is complicated by Compton scattering of γ-photons in tissue, which causes attenuation of the photon flux received by the gamma-camera and causes improper detection of photons which have been scattered in tissue. This results in some additional blurring and loss of accuracy of the SPECT images in predicting activity concentrations. Tremendous efforts have been made to improve the quantitative accuracy and the spatial resolution of SPECT, and to reduce the noise in the reconstructed images. These efforts have resulted in corrective reconstruction algorithms, which are generally based on incorporation of accurate models of the main image degrading factors. Improvements of the data acquisition hardware can further increase image quality. In this paper, the image formation process of SPECT, including image-degrading factors, is explained. In addition, reconstruction algorithms and hardware modifications are reviewed, and their effects on image quality are illustrated with physical phantom and simulation experiments.
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paper_title: Internet2-based 3D PET image reconstruction using a PC cluster

paper_content:

We describe an approach to fast iterative reconstruction from fully three-dimensional (3D) PET data using a network of PentiumIII PCs configured as a Beowulf cluster. To facilitate the use of this system, we have developed a browser-based interface using Java. The system compresses PET data on the user's machine, sends these data over a network, and instructs the PC cluster to reconstruct the image. The cluster implements a parallelized version of our preconditioned conjugate gradient method for fully 3D MAP image reconstruction. We report on the speed-up factors using the Beowulf approach and the impacts of communication latencies in the local cluster network and the network connection between the user's machine and our PC cluster.
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paper_title: A three-dimensional ray-driven attenuation, scatter and geometric response correction technique for SPECT in inhomogeneous media

paper_content:

The qualitative and quantitative accuracy of SPECT images is degraded by physical factors of attenuation, Compton scatter and spatially varying collimator geometric response. This paper presents a 3D ray-tracing technique for modelling attenuation, scatter and geometric response for SPECT imaging in an inhomogeneous attenuating medium. The model is incorporated into a three-dimensional projector-backprojector and used with the maximum-likelihood expectation-maximization algorithm for reconstruction of parallel-beam data. A transmission map is used to define the inhomogeneous attenuating and scattering object being imaged. The attenuation map defines the probability of photon attenuation between the source and the scattering site, the scattering angle at the scattering site and the probability of attenuation of the scattered photon between the scattering site and the detector. The probability of a photon being scattered through a given angle and being detected in the emission energy window is approximated using a Gaussian function. The parameters of this Gaussian function are determined using physical measurements of parallel-beam scatter line spread functions from a non-uniformly attenuating phantom. The 3D ray-tracing scatter projector-backprojector produces the scatter and primary components. Then, a 3D ray-tracing projector-backprojector is used to model the geometric response of the collimator. From Monte Carlo and physical phantom experiments, it is shown that the best results are obtained by simultaneously correcting attenuation, scatter and geometric response, compared with results obtained with only one or two of the three corrections. It is also shown that a 3D scatter model is more accurate than a 2D model. A transmission map is useful for obtaining measurements of attenuation and scatter in SPECT data, which can be used together with a model of the geometric response of the collimator to obtain corrected images with quantitative and diagnostically accurate information.
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paper_title: Advances in nuclear medicine instrumentation: considerations in the design and selection of an imaging system

paper_content:

Nuclear medicine remains a vibrant and dynamic medical specialty because it so adeptly marries advances in basic science research, technology, and medical practice in attempting to solve patients' problems. As a physicist, it is my responsibility to identify or design new instrumentation and techniques, and to implement, validate, and help apply these new approaches in the practice of nuclear medicine. At Johns Hopkins, we are currently in the process of purchasing both a single-photon/coincidence tomographic imaging system and a dedicated positron emission tomography (PET) scanner. Given the exciting advances that have been made, but the conflicting opinions of manufacturers and colleagues alike regarding "best" choices, it seemed useful to review what is new now, and what is on the horizon, to help identify all of the important considerations in the design and selection of an imaging system. It is important to note that many of the "advances" described here are in an early stage of development, and may never make it to routine clinical practice. Further, not all of the advances are of equal importance, or have the same degree of general clinical applicability. Please also note that the references contained herein are for illustrative purposes and are not all-inclusive; no implication that those chosen are "better" than others not mentioned is intended.
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paper_title: Maximum-likelihood expectation-maximization reconstruction of sinograms with arbitrary noise distribution using NEC-transformations

paper_content:

The maximum-likelihood (ML) expectation-maximization (EM) [ML-EM] algorithm is being widely used for image reconstruction in positron emission tomography. The algorithm is strictly valid if the data are Poisson distributed. However, it is also often applied to processed sinograms that do not meet this requirement. This may sometimes lead to suboptimal results: streak artifacts appear and the algorithm converges toward a lower likelihood value. As a remedy, the authors propose two simple pixel-by-pixel methods [noise equivalent counts (NEC)-scaling and NEC-shifting] in order to transform arbitrary sinogram noise into noise which is approximately Poisson distributed (the first and second moments of the distribution match those of the Poisson distribution). The convergence speed associated with both transformation methods is compared, and the NEC-scaling method is validated with both simulations and clinical data. These new methods extend the ML-EM algorithm to a general purpose nonnegative reconstruction algorithm.
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paper_title: Edge-preserving tomographic reconstruction with nonlocal regularization

paper_content:

We propose a new objective function for the image reconstruction problem, where the image is comprised of piecewise smooth regions separated by sharp boundaries. We use alternating minimization to minimize our objective function. We use the level set technique to minimize with regard to the boundary. The advantage of this new approach is shown through the bias/variance analysis of a hot spot.
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paper_title: Quantitative SPECT imaging: A review and recommendations by the focus committee of the society of nuclear medicine Computer and Instrumentation Council

paper_content:

This article is a review of the physics principles, instrumentation and reconstruction methods behind SPECT imaging. Particular attention is paid to the mechanisms that can significantly affect the accuracy of a SPECT image. We describe instrumentation advances and reconstruction methods used to correct images to improve image quality and produce quantitative images. The clinical importance of improved image quality and quantitation are also reviewed.
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paper_title: Statistical Approaches in Quantitative Positron Emission Tomography

paper_content:

Positron emission tomography is a medical imaging modality for producing 3D images of the spatial distribution of biochemical tracers within the human body. The images are reconstructed from data formed through detection of radiation resulting from the emission of positrons from radioisotopes tagged onto the tracer of interest. These measurements are approximate line integrals from which the image can be reconstructed using analytical inversion formulae. However these direct methods do not allow accurate modeling either of the detector system or of the inherent statistical fluctuations in the data. Here we review recent progress in developing statistical approaches to image estimation that can overcome these limitations. We describe the various components of the physical model and review different formulations of the inverse problem. The wide range of numerical procedures for solving these problems are then reviewed. Finally, we describe recent work aimed at quantifying the quality of the resulting images, both in terms of classical measures of estimator bias and variance, and also using measures that are of more direct clinical relevance.
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paper_title: Quantitative single-photon emission computed tomography: basics and clinical considerations.

paper_content:

Although quantitative single-photon emission computed tomography (SPECT) has been the goal of much research effort for a number of years, only recently has it received wide interest, especially for clinical applications. It has been increasingly recognized that the achievement of quantitative SPECT will increase the accuracy of measurements, such as dimensions of specific regions of interest, absolute amount of radioactivity, and dosimetry calculations, and substantially reduce reconstruction image artifacts and distortions, thus, greatly improving clinical diagnosis. This article provides a review of the definition of terms, major factors affecting SPECT quantitation and their degrading effects on SPECT image quality, and methods to compensate for these effects. Compensation methods include those that make certain approximations for ease of implementation and those that provide more accurate compensation by modeling the imaging process more exactly, usually at the cost of increased complexity and computational requirements. Different reconstruction and compensation methods may be compared through the use of phantom cardiac and brain SPECT studies. The clinical efficacy of the methods may be demonstrated by applying them to a clinical thallium-201 myocardial perfusion SPECT study. The results clearly demonstrate that, by modeling the imaging process and/or image degrading factors three-dimensionally, quantitative reconstruction and compensation methods provide the best image quality and quantitative accuracy. Important research efforts and developmental work being conducted currently to bring quantitative SPECT into routine clinical use are also discussed.
---
paper_title: Fully 3D Bayesian image reconstruction for the ECAT EXACT HR

paper_content:

A fully 3D Bayesian method is described for high resolution reconstruction of images from the Siemens/CTI ECAT EXACT HR+ whole body positron emission tomography (PET) scanner. To maximize resolution recovery from the system the authors model depth dependent geometric efficiency, intrinsic detector efficiency, photon pair non-colinearity, crystal penetration and inter-crystal scatter. They also explicitly model the effects of axial rebinning and angular mashing on the detection probability or system matrix. By fully exploiting sinogram symmetries and using a factored system matrix and automated indexing schemes, the authors are able to achieve substantial savings in both the storage size and time required to compute forward and backward projections. Reconstruction times are further reduced using multi-threaded programming on a four processor Unix server. Bayesian reconstructions are computed using a Huber prior and a shifted-Poisson likelihood model that accounts for the effects of randoms subtraction and scatter. Reconstructions of phantom data show that the 3D Bayesian method can achieve improved FWHM resolution and contrast recovery ratios at matched background noise levels compared to both the 3D reprojection method and an OSEM method based on the shifted-Poisson model.
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paper_title: Image-correction techniques in SPECT

paper_content:

This overview takes a look at different correction techniques for Single Photon Emission Computed Tomography (SPECT). We discuss the influence of the detection system followed by the scatter and attenuation caused by the object of investigation. When possible we describe how the correction methods for the different physical effects can be incorporated in the reconstruction method, being either filtered backprojection or iterative reconstruction.
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paper_title: High-resolution 3D Bayesian image reconstruction using the microPET small-animal scanner

paper_content:

A Bayesian method is described for reconstruction of high-resolution 3D images from the microPET small-animal scanner. Resolution recovery is achieved by explicitly modelling the depth dependent geometric sensitivity for each voxel in combination with an accurate detector response model that includes factors due to photon pair non-collinearity and inter-crystal scatter and penetration. To reduce storage and computational costs we use a factored matrix in which the detector response is modelled using a sinogram blurring kernel. Maximum a posteriori (MAP) images are reconstructed using this model in combination with a Poisson likelihood function and a Gibbs prior on the image. Reconstructions obtained from point source data using the accurate system model demonstrate a potential for near-isotropic FWHM resolution of approximately 1.2 mm at the center of the field of view compared with approximately 2 mm when using an analytic 3D reprojection (3DRP) method with a ramp filter. These results also show the ability of the accurate system model to compensate for resolution loss due to crystal penetration producing nearly constant radial FWHM resolution of 1 mm out to a 4 mm radius. Studies with a point source in a uniform cylinder indicate that as the resolution of the image is reduced to control noise propagation the resolution obtained using the accurate system model is superior to that obtained using 3DRP at matched background noise levels. Additional studies using pie phantoms with hot and cold cylinders of diameter 1-2.5 mm and 18FDG animal studies appear to confirm this observation.
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paper_title: Generalization of median root prior reconstruction

paper_content:

Penalized iterative algorithms for image reconstruction in emission tomography contain conditions on which kind of images are accepted as solutions. The penalty term has commonly been a function of pairwise pixel differences in the activity in a local neighborhood, such that smooth images are favored. Attempts to ensure better edge and detail preservation involve difficult tailoring of parameter values or the penalty function itself. The previously introduced median root prior (MRP) favors locally monotonic images. MRP preserves sharp edges while reducing locally nonmonotonic noise at the same time. Quantitative properties of MRP are good, because differences in the neighboring pixel values are not penalized as such. The median is used as an estimate for a penalty reference, against which the pixel value is compared when setting the penalty. In order to generalize the class of MRP-type of priors, the standard median was replaced by other order statistic operations, the L and finite-impulse-response median hybrid (FMH) filters. They allow for smoother appearance as they apply linear weighting together with robust nonlinear operations. The images reconstructed using the new MRP-L and MRP-FMH priors are visually more conventional. Good quantitative properties of MRP are not significantly altered by the new priors.
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paper_title: Bayesian Reconstruction and Use of Anatomical A Priori Information for Emission Tomography

paper_content:

A Bayesian method is presented for simultaneously segmenting and reconstructing emission computed tomography (ECT) images and for incorporating high-resolution, anatomical information into those reconstructions. The anatomical information is often available from other imaging modalities such as computed tomography (CT) or magnetic resonance imaging (MRI). The Bayesian procedure models the ECT radiopharmaceutical distribution as consisting of regions, such that radiopharmaceutical activity is similar throughout each region. It estimates the number of regions, the mean activity of each region, and the region classification and mean activity of each voxel. Anatomical information is incorporated by assigning higher prior probabilities to ECT segmentations in which each ECT region stays within a single anatomical region. This approach is effective because anatomical tissue type often strongly influences radiopharmaceutical uptake. The Bayesian procedure is evaluated using physically acquired single-photon emission computed tomography (SPECT) projection data and MRI for the three-dimensional (3-D) Hoffman brain phantom. A clinically realistic count level is used. A cold lesion within the brain phantom is created during the SPECT scan but not during the MRI to demonstrate that the estimation procedure can detect ECT structure that is not present anatomically.
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paper_title: Efficient 3D grids for image reconstruction using spherically-symmetric volume elements

paper_content:

Incorporation of spherically-symmetric volume elements (blobs), instead of the conventional voxels, into iterative image reconstruction algorithms, has been found in the authors' previous studies to lead to significant improvement in the quality of the reconstructed images. Furthermore, for 3D positron emission tomography, the 3D algebraic reconstruction technique using blobs can reach comparable or even better duality than the 3D filtered backprojection method after only one cycle through the projection data. The only shortcoming of the blob reconstruction is an increased computational demand, because of the overlapping nature of the blobs. These encouraging results mere obtained in the authors' previous studies for the case when the blobs were placed on the same 3D simple cubic grid used for voxel basis functions. For basis functions which are spherically-symmetric, there are more advantageous arrangements of the 3D grid, enabling a more isotropic distribution of the spherical functions in the 3D space and a, better packing efficiency of the image spectrum. A good arrangement is the body centered cubic grid. The authors' studies confirmed that, when using this type of 3D grid, the number of grid points can be effectively reduced, decreasing the computational and memory demands while preserving the quality of the reconstructed images. >
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paper_title: Comparison of 3-D maximum a posteriori and filtered backprojection algorithms for high-resolution animal imaging with microPET

paper_content:

The authors have evaluated the performance of two three-dimensional (3-D) reconstruction algorithms with data acquired from microPET, a high resolution tomograph dedicated to small animal imaging. The first was a linear filtered-backprojection algorithm (FBP) with reprojection of the missing data, and the second was a statistical maximum a posteriori probability algorithm (MAP). The two algorithms were evaluated in terms of their resolution performance, both in phantoms and in vivo. Sixty independent realizations of a phantom simulating the brain of a baby monkey were acquired, each containing three million counts. Each of these realizations was reconstructed independently with both algorithms. The ensemble of the 60 reconstructed realizations was used to estimate the standard deviation as a measure of the noise for each reconstruction algorithm. More detail was recovered in the MAP reconstruction without an increase in noise relative to FBP. Studies in a simple cylindrical compartment phantom demonstrated improved recovery of known activity ratios with MAP. Finally, in vivo studies also demonstrated a clear improvement in spatial resolution using the MAP algorithm. The quantitative accuracy of the MAP reconstruction was also evaluated by comparison with autoradiography and direct well counting of tissue samples and was shown to be superior.
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paper_title: Positron-emission tomography

paper_content:

We review positron-emission tomography (PET), which has inherent advantages that avoid the shortcomings of other nuclear medicine imaging methods. PET image reconstruction methods with origins in signal and image processing are discussed, including the potential problems of these methods. A summary of statistical image reconstruction methods, which can yield improved image quality, is also presented.
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paper_title: Multidimensional digital image representations using generalized Kaiser-Bessel window functions.

paper_content:

Inverse problems that require the solution of integral equations are inherent in a number of indirect imaging applications, such as computerized tomography. Numerical solutions based on discretization of the mathematical model of the imaging process, or on discretization of analytic formulas for iterative inversion of the integral equations, require a discrete representation of an underlying continuous image. This paper describes discrete image representations, in n-dimensional space, that are constructed by the superposition of shifted copies of a rotationally symmetric basis function. The basis function is constructed using a generalization of the Kaiser-Bessel window function of digital signal processing. The generalization of the window function involves going from one dimension to a rotationally symmetric function in n dimensions and going from the zero-order modified Bessel function of the standard window to a function involving the modified Bessel function of order m. Three methods are given for the construction, in n-dimensional space, of basis functions having a specified (finite) number of continuous derivatives, and formulas are derived for the Fourier transform, the x-ray transform, the gradient, and the Laplacian of these basis functions. Properties of the new image representations using these basis functions are discussed, primarily in the context of two-dimensional and three-dimensional image reconstruction from line-integral data by iterative inversion of the x-ray transform. Potential applications to three-dimensional image display are also mentioned.
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paper_title: Iterative reconstruction methods for high-throughput PET tomographs

paper_content:

A fast iterative method is described for processing clinical PET scans acquired in three dimensions, that is, with no inter-plane septa, using standard computers to replace dedicated processors used until the late 1990s. The method is based on sinogram resampling, Fourier rebinning, Monte Carlo scatter simulation and iterative reconstruction using the attenuation-weighted OSEM method and a projector based on a Gaussian pixel model. Resampling of measured sinogram values occurs before Fourier rebinning, to minimize parallax and geometric distortions due to the circular geometry, and also to reduce the size of the sinogram. We analyse the geometrical and statistical effects of resampling, showing that the lines of response are positioned correctly and that resampling is equivalent to about 4 mm of post-reconstruction filtering. We also present phantom and patient results. In this approach, multi-bed clinical oncology scans can be ready for diagnosis within minutes.
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paper_title: Introduction to inverse problems in imaging

paper_content:

Part 1 Image deconvolution: some mathematical tools examples of image blurring the ill-posedness of image deconvolution regularization methods iterative regularization methods statistical methods. Part 2 Linear inverse problems: examples of linear inverse problems singular value decomposition (SVD) inversion methods revisited Fourier based methods for specific problems comments and concluding remarks. Part 3 Mathematical appendices. References.
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paper_title: Image Reconstruction from List-Mode Data in an Emission Tomography System Having Time-of-Flight Measurements

paper_content:

List-mode data collected in a positron-emission tomography system having time-of-flight measurements are three dimensional, but all algorithms which have been published to date operate on two-dimensional data derived from these three-dimensional data. We argue in this paper that the additional information present in the three-dimensional data is useful for improving reconstructions of images.
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paper_title: LMIRA: list mode iterative reconstruction algorithm for SPECT

paper_content:

The goal of this project is to investigate a new method for using list-mode data instead of binned projection data for SPECT reconstruction. List-mode acquisition was once standardly available for planar scintigraphy, and is now becoming available for SPECT acquisition. The authors implemented maximum likelihood iterative reconstruction with list-mode data for SPECT using a projector/backprojector pair which employ an intermediate stage to separate the source and camera specific components. The algorithm uses the emission radiance distribution, calculated from the estimated source distribution, to obtain the probabilities used in the iterative reconstruction method. The model takes into account distance-dependent resolution and non-uniform attenuation. Results illustrate how the individual steps of the algorithm work. LMIRA (List-Mode Iterative Reconstruction Algorithm) is compared to the standard MLEM iterative reconstruction method which employs binned data. The authors conclude that the finer sampling provided by list-mode data, and the noise characteristics of list-mode reconstruction result in superior image quality. In further research they plan to investigate the inclusion of the intrinsic spatial resolution, the energy resolution, and the modeling of scatter in LMIRA.
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paper_title: Spatiotemporal reconstruction of list-mode PET data

paper_content:

We describe a method for computing a continuous time estimate of tracer density using list-mode positron emission tomography data. The rate function in each voxel is modeled as an inhomogeneous Poisson process whose rate function can be represented using a cubic B-spline basis. The rate functions are estimated by maximizing the likelihood of the arrival times of detected photon pairs over the control vertices of the spline, modified by quadratic spatial and temporal smoothness penalties and a penalty term to enforce nonnegativity. Randoms rate functions are estimated by assuming independence between the spatial and temporal randoms distributions. Similarly, scatter rate functions are estimated by assuming spatiotemporal independence and that the temporal distribution of the scatter is proportional to the temporal distribution of the trues. A quantitative evaluation was performed using simulated data and the method is also demonstrated in a human study using 11C-raclopride.
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paper_title: Scatter Correction for Positron Emission Mammography

paper_content:

In this paper we present a scatter correction method for a regularized list mode maximum likelihood reconstruction algorithm for the positron emission mammograph (PEM) that is being developed at our laboratory. The scatter events inside the object are modelled as additive Poisson random variables in the forward model of the reconstruction algorithm. The mean scatter sinogram is estimated using a Monte Carlo simulation program. With the assumption that the background activity is nearly uniform, the Monte Carlo scatter simulation only needs to run once for each PEM configuration. This saves computation time. The crystal scatters are modelled as a shift-invariant blurring in image domain because they are more localized. Thus, the useful information in the crystal scatters can be deconvolved in high-resolution reconstructions. The propagation of the noise from the estimated scatter sinogram into the reconstruction is analysed theoretically. The results provide an easy way to calculate the required number of events in the Monte Carlo scatter simulation for a given noise level in the image. The analysis is also applicable to other scatter estimation methods, provided that the covariance of the estimated scatter sinogram is available.
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paper_title: Resolution recovery for list-mode reconstruction in SPECT.

paper_content:

The purpose of the study was to evaluate the resolution recovery in the list-mode iterative reconstruction algorithm (LMIRA) for SPECT. In this study we compare the performance of the proposed method with other iterative resolution recovery methods for different noise levels. We developed an iterative reconstruction method which uses list-mode data instead of binned data. The new algorithm makes use of a more accurate model of the collimator structure. We compared the SPECT list-mode reconstruction with MLEM, OSEM and RBI, all including resolution recovery. For the evaluation we used Gaussian shaped sources with different FWHM at three different locations and three noise levels. For these distributions we calculated the reconstructed images for a different number of iterations. The absolute error for the reconstructed images was used to evaluate the performance. The performance of all four methods is comparable for the sources located in the centre of the field of view. For the sources located out of the centre, the error of the list-mode method is significantly lower than that of the other methods. Splitting the system model into a separate object-dependent and detector-dependent module gives us a flexible reconstruction method. With this we can very easily adapt the resolution recovery to different collimator types.
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paper_title: Penalised least squares image reconstruction from backprojection space for 3D PET

paper_content:

Image reconstruction directly from backprojected images has two advantages over projection-data based reconstruction for 3D PET: (i) 3D backprojected images offer data compression compared to large 4D projection data sets and (ii) full spatial sampling accuracy of the scanner is retained (unlike with projection-data mashing). This second advantage can be useful for scanners with large numbers of possible lines of response (LORs), as a backprojected image preserves the positional accuracy of the LORs (which can otherwise be compromised by binning into projections). This work presents an algorithm for penalised least squares (PLS) reconstruction directly from 3D backprojected images. All the data can be used, and the shift-variant 3D point response function of the scanner can be accounted for. The proposed algorithm was compared with the ISRA and BPF reconstruction algorithms, and was found to converge significantly more quickly than ISRA and to offer some improvements in noise-contrast behaviour in cold regions when compared to BPF. However, with current computational limitations, the algorithm can only practically be applied to images up to a maximum size of 64/spl times/64/spl times/64 voxels.
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paper_title: Continuous time dynamic PET imaging using list mode data

paper_content:

We describe a method for computing a continuous time estimate of dynamic changes in tracer density using list mode PET data. The tracer density in each voxel is modeled as an inhomogeneous Poisson process whose rate function can be represented using a cubic B-spline basis. An estimate of these rate functions is obtained by maximizing the likelihood of the arrival times of each detected photon pair over the control vertices of the spline. By resorting the list mode data into a standard sinogram plus a "timogram" that retains the arrival times of each of the events, we are able to perform efficient computation that exploits the symmetry inherent in the ordered sinogram. The maximum likelihood estimator uses quadratic temporal and spatial smoothness penalties and an additional penalty term to enforce non-negativity. Corrections for scatter and randoms are described and the results of studies using simulated and human data are included.
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paper_title: Image fusion using an integrated, dual-head coincidence camera with X-ray tube-based attenuation maps.

paper_content:

UNLABELLED ::: The purpose of this study was to characterize a dual-head gamma camera capable of FDG imaging using coincidence detection and equipped with an integrated x-ray transmission system for attenuation correction, anatomic mapping, and image fusion. ::: ::: ::: METHODS ::: Radiation dose (425 mrads skin dose) and tissue contrast (0.7% deviation from expected values) were assessed for the x-ray system. Registration of transmission and emission scans was validated using a hot sphere phantom and was verified in selected patient studies. ::: ::: ::: RESULTS ::: Fusion of anatomic maps and FDG images allowed precise anatomic localization of lesions identified using dual-head coincidence imaging. ::: ::: ::: CONCLUSION ::: The combined approach of x-ray attenuation, anatomic mapping, and image fusion with scintigraphic studies provides a new diagnostic tool for nuclear medicine and fertile ground for future research.
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paper_title: A Matheematical Model for Positron-Emission Tomography Systems Having Time-of-Flight Measurements

paper_content:

Improvements in high speed electronics and scintillation-crystal technology now permit usable differential time-of-flight measurements to be made in tomography systems that employ coincidence detection of the annihilation photons created with positron emitting radionuclides. A mathematical model for these new measurements is developed in this paper. Reconstruction algorithms and their signal-to-noise ratio performance are given.
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paper_title: Improved modeling of system response in list mode EM reconstruction of Compton scatter camera images

paper_content:

An improved List Mode EM method for reconstructing Compton scattering camera images has been developed. First, an approximate method for computation of the spatial variation in the detector sensitivity has been derived and validated by Monte Carlo computation. A technique for estimating the relative weight of system matrix coefficients for each gamma in the list has also been employed, as has a method for determining the relative probabilities of emission having some from pixels tallied in each list-mode back-projection. Finally, a technique has been developed for modeling the effects of Doppler broadening and finite detector energy resolution on the relative weights for pixels neighbor to those intersected by the back-projection, based on values for the FWHM of the spread in the cone angle computed by Monte Carlo. Memory issues typically associated with list mode reconstruction are circumvented by storing only a list of the pixels intersected by the back-projections, and computing the weights of the neighboring pixels at each iteration step. Simulated projection data has been generated for a representative Compton camera system (CSPRINT) for several source distributions and reconstructions performed. Reconstructions have also been performed for experimental data for distributed sources.
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paper_title: Gamma camera-mounted anatomical X-ray tomography: technology, system characteristics and first images

paper_content:

Scintigraphic diagnosis, based on functional image interpretation, becomes more accurate and meaningful when supported by corresponding anatomical data. In order to produce anatomical images that are inherently registered with images of emission computerised tomography acquired with a gamma camera, an X-ray transmission system was mounted on the slip-ring gantry of a GEMS Millennium VG gamma camera. The X-ray imaging system is composed of an X-ray tube and a set of detectors located on opposite sides of the gantry rotor that moves around the patient along with the nuclear detectors. A cross-sectional anatomical transmission map is acquired as the system rotates around the patient in a manner similar to a third-generation computerised tomography (CT) system. Following transmission, single-photon emission tomography (SPET) or positron emission tomography (PET) coincidence detection images are acquired and the resultant emission images are thus inherently registered to the anatomical maps. Attenuation correction of the emission images is performed with the same anatomical maps to generate transmission maps. Phantom experiments of system performance and examples of first SPET and coincidence detection patient images are presented. Despite limitations of the system when compared with a state of the art CT scanner, the transmission anatomical maps allow for precise anatomical localisation and for attenuation correction of the emission images.
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paper_title: Adaptive correction of scatter and random events for 3-D backprojected PET data

paper_content:

Backprojected images offer compact and accurate storage of three-dimensional (3-D) positron emission tomography (PET) data and so are potentially very suited to high-resolution 3-D PET scanners. However, there are very few methods available for correcting backprojected images for the effects of scatter and random events. Scatter correction methods usually involve experimental measurement of scatter response functions or accurate simulations, and are thus limited by the accuracy of the model used. In contrast, the approach proposed here uniquely adapts a general form of a combined scatter and random response function. This method inherently does not rely upon accurate determination of the scatter and random response functions but does, however, rely upon knowledge of the attenuating object's outline. The technique is able to adapt to different imaging situations and account for detector scatter, and in particular is readily implemented in backprojection space. The method has been implemented for a small-volume HIDAC (II) PET scanner.
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paper_title: Fast accurate iterative reconstruction for low-statistics positron volume imaging.

paper_content:

A fast accurate iterative reconstruction (FAIR) method suitable for low-statistics positron volume imaging has been developed. The method, based on the expectation maximization-maximum likelihood (EM-ML) technique, operates on list-mode data rather than histogrammed projection data and can, in just one pass through the data, generate images with the same characteristics as several ML iterations. Use of list-mode data preserves maximum sampling accuracy and implicitly ignores lines of response (LORs) in which no counts were recorded. The method is particularly suited to systems where sampling accuracy can be lost by histogramming events into coarse LOR bins, and also to sparse data situations such as fast whole-body and dynamic imaging where sampling accuracy may be compromised by storage requirements and where reconstruction time can be wasted by including LORs with no counts. The technique can be accelerated by operating on subsets of list-mode data which also allows scope for simultaneous data acquisition and iterative reconstruction. The method is compared with a standard implementation of the EM-ML technique and is shown to offer improved resolution, contrast and noise properties as a direct result of using improved spatial sampling, limited only by hardware specifications.
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paper_title: The radon transform and local tomography

paper_content:

Introduction Brief Description of New Results and the Aims of the Book Review of Some Applications of the Radon Transform Properties of the Radon Transform and Inversion Formulas Definitions and Properties of the Radon Transform and Related Transforms Inversion Formulas for R Singular Value Decomposition of the Radon Transform Estimates in Sobolev Spaces Inversion Formulas for the Backprojection Operator Inversion Formulas for X-Ray Transform Uniqueness Theorems for the Radon and X-Ray Transforms Attenuated and Exponential Radon Transforms Convergence Properties of the Inversion Formulas on Various Classes of Functions Range Theorems and Reconstruction Algorithms Range Functions for R on Smooth Functions Range Functions for R on Sobolev Spaces Range Theorems for R* Range Theorem for X-Ray Transform Numerical Solution of the Equation Rf = g with Noisy Data Filtered Backprojection Algorithm Other Reconstruction Algorithms Singularities of the Radon Transform Introduction Singular Support of the Radon Transform The Relation Between S and S (WE NEED A "HAT" OVER THE LAST S. See hard copy of toc for details) The Envelopes and the Duality Law Asymptotics of Rf Near S Singularities of the Radon Transform: An Alternative Approach Asymptotics of the Fourier Transform Wave Front Sets Singularities of X-Ray Transform Stable Calculation of the Legendre Transform Local Tomography Introduction A Family of Local Tomography Functions Optimization of Noise Stability Algorithm for Finding Values of Jumps of a Function Using Local Tomography Numerical Implementation Local Tomography for the Exponential Radon Transform Local Tomography for the Generalized Radon Transform Local Tomography for the Limited-Angle Data Asymptotics of Pseudodifferential Operators, Acting on a Piecewise-Smooth Function, f, Near the Singular Support of f Pseudolocal Tomography Introduction Definition of a Pseudolocal Tomography Function Investigation of the Convergence frc(x) Ae f(x) as r Ae 0 More Results on Functions frc, fr, and on convergence frc Ae f A Family of Pseudolocal Tomography Functions Numerical Implementation of Pseudolocal Tomography Pseudolocal Tomography for the Exponential Radon Transform Geometric Tomography Basic Idea Description of the Algorithm and Numerical Experiments Inversion of Incomplete Tomographic Data Inversion of Incomplete Fourier Transform Data Filtered Backprojection Method for Inversion of the Limited-Angle Tomographic Data The Extrapolation Problem The Davison-Grunbaum Algorithm Inversion of Cone-Beam Data Inversion of the Complete Cone-Beam Data Inversion of Incomplete Cone-Beam Data An Exact Algorithm for the Cone-Beam Circle Geometry g-Ray Tomography Radon Transform of Distributions Main Definitions Properties of the Test Function Spaces Examples Range Theorem for the Radon Transform on e' A Definition Based on Spherical Harmonics Expansion When Does the Radon Transform on Distributions Coincide with the Classical Radon Transform? The Dual Radon Transform on Distributions Abel-Type Integral Equation The Classical Abel Equation Abel-Type Equations Reduction of the Equation to a More Stable One Finding Locations and Values of Jumps of the Solution to the Abel Equation Multidimensional Algorithm for Finding Discontinuities of Signals from Noisy Discrete Data Introduction Edge Detection Algorithm Thin Line Detection Algorithm Generalization of the Algorithms Justification of the Edge Detection Algorithm Justification of the Algorithm for Thin Line Detection Justification of the General Scheme Numerical Experiments Proof of Auxiliary Results Test of Randomness and Its Applications Introduction Consistency of Rank Test Against Change Points (Change Surfaces) Alternative Consistency of Rank Test Against Trend in Location Auxiliary Results Abstract and Functional Spaces Distribution Theory Pseudodifferential and Fourier Integral Operators Special Functions Asymptotic Expansions Linear Equations in Banach Spaces Ill-Posed Problems Examples of Regularization of Ill-Posed Problems Radon Transform and PDE Statistics Research Problems Bibliographical Notes References Index List of Notations
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paper_title: Past and future directions in x-ray computed tomography (CT)

paper_content:

We give a short account of the history of CT from motion tomography in the early 1930's to sprial CT. We discuss the physical and the mathematical background. Finally we give an outlook on possible future developments of CT and related techniques. © 2002 Wiley Periodicals, Inc. Int J Imaging Syst Technol 12, 175–187, 2002; Published online in Wiley InterScience (www.interscience.wiley.com). DOI 10.1002/ima.10021
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paper_title: BICAV: a block-iterative parallel algorithm for sparse systems with pixel-related weighting

paper_content:

Component averaging (CAV) was recently introduced by Censor, Gordon, and Gordon as a new iterative parallel technique suitable for large and sparse unstructured systems of linear equations. Based on earlier work of Byrne and Censor, it uses diagonal weighting matrices, with pixel-related weights determined by the sparsity of the system matrix. CAV is inherently parallel (similar to the very slowly converging Cimmino method) but its practical convergence on problems of image reconstruction from projections is similar to that of the algebraic reconstruction technique (ART). Parallel techniques are becoming more important for practical image reconstruction since they are relevant not only for supercomputers but also for the increasingly prevalent multiprocessor workstations. This paper reports on experimental results with a block-iterative version of component averaging (BICAV). When BICAV is optimized for block size and relaxation parameters, its very first iterates are far superior to those of CAV, and more or less on a par with ART. Similar to CAV, BICAV is also inherently parallel. The fast convergence is demonstrated on problems of image reconstruction from projections, using the SNARK93 image reconstruction software package. Detailed plots of various measures of convergence, and reconstructed images are presented.
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paper_title: Averaging Strings of Sequential Iterations for Convex Feasibility Problems

paper_content:

An algorithmic scheme for the solution of convex feasibility problems is proposed in which the end-points of strings of sequential projections onto the constraints are averaged. The scheme employing Bregman projections, is analyzed with the aid of an extended product space formalism. For the case of orthogonal projections we give also a relaxed version. Along with the well-known purely sequential fully simultaneous cases the new scheme includes many other inherently parallel algorithmic options epending on the choice of strings. Convergence in the consistent case is proven an application to optimization over linear inequalities is given.
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paper_title: Analytic and iterative reconstruction algorithms in SPECT.

paper_content:

Images of the inside of the human body can be obtained noninvasively using tomographic acquisition and processing techniques. In particular, these techniques are commonly used to obtain images of a gamma-emitter distribution after its administration in the human body. The reconstructed images are obtained given a set of their projections, acquired using rotating gamma cameras. A general overview of analytic and iterative methods of reconstruction in SPECT is presented with a special focus on filter backprojection (FBP), conjugate gradient, maximum likelihood expectation maximization, and maximum a posteriori expectation maximization algorithms. The FBP algorithm is faster than iterative algorithms, with the latter providing a framework for accurately modeling the emission and detection processes.
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paper_title: Parallel Optimization: Theory, Algorithms, and Applications

paper_content:

Foreword Preface Glossary of Symbols 1. Introduction Part I Theory 2. Generalized Distances and Generalized Projections 3. Proximal Minimization with D-Functions Part II Algorithms 4. Penalty Methods, Barrier Methods and Augmented Lagrangians 5. Iterative Methods for Convex Feasibility Problems 6. Iterative Algorithms for Linearly Constrained Optimization Problems 7. Model Decomposition Algorithms 8. Decompositions in Interior Point Algorithms Part III Applications 9. Matrix Estimation Problems 10. Image Reconsturction from Projections 11. The Inverse Problem in Radiation Therapy Treatment Planning 12. Multicommodity Network Flow Problems 13. Planning Under Uncertainty 14. Decompositions for Parallel Computing 15. Numerical Investigations
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paper_title: Three-Dimensional Reconstruction from Projections: A Review of Algorithms

paper_content:

The problem of reconstructing three-dimensional objects from a set of two-dimensional projected images has arisen and been solved independently in fields ranging from medicine and electron microscopy to holographic interferometry. By using a source of radiation external to the object, we obtain a transmission picture or projection of the three-dimensional object onto a two-dimensional surface such as the film of an ordinary electron micrograph or x-ray. The reconstruction problem is: Given a subset of all possible projections of an object, estimate its internal density distribution.© (1975) COPYRIGHT SPIE--The International Society for Optical Engineering. Downloading of the abstract is permitted for personal use only.
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paper_title: Iterative reconstruction algorithms in nuclear medicine.

paper_content:

Iterative reconstruction algorithms produce accurate images without streak artifacts as in filtered backprojection. They allow improved incorporation of important corrections for image degrading effects, such as attenuation, scatter and depth-dependent resolution. Only some corrections, which are important for accurate reconstruction in positron emission tomography and single photon emission computed tomography, can be applied to the data before filtered backprojection. The main limitation for introducing iterative algorithms in nuclear medicine has been computation time, which is much longer for iterative techniques than for filtered backprojection. Modern algorithms make use of acceleration techniques to speed up the reconstruction. These acceleration techniques and the development in computer processors have introduced iterative reconstruction in daily nuclear medicine routine. We give an overview of the most important iterative techniques and discuss the different corrections that can be incorporated to improve the image quality.
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paper_title: Internet2-based 3D PET image reconstruction using a PC cluster

paper_content:

We describe an approach to fast iterative reconstruction from fully three-dimensional (3D) PET data using a network of PentiumIII PCs configured as a Beowulf cluster. To facilitate the use of this system, we have developed a browser-based interface using Java. The system compresses PET data on the user's machine, sends these data over a network, and instructs the PC cluster to reconstruct the image. The cluster implements a parallelized version of our preconditioned conjugate gradient method for fully 3D MAP image reconstruction. We report on the speed-up factors using the Beowulf approach and the impacts of communication latencies in the local cluster network and the network connection between the user's machine and our PC cluster.
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paper_title: Proximity Function Minimization Using Multiple Bregman Projections, with Applications to Split Feasibility and Kullback–Leibler Distance Minimization

paper_content:

Problems in signal detection and image recovery can sometimes be formulated as a convex feasibility problem (CFP) of finding a vector in the intersection of a finite family of closed convex sets. Algorithms for this purpose typically employ orthogonal or generalized projections onto the individual convex sets. The simultaneous multiprojection algorithm of Censor and Elfving for solving the CFP, in which different generalized projections may be used at the same time, has been shown to converge for the case of nonempty intersection; still open is the question of its convergence when the intersection of the closed convex sets is empty.Motivated by the geometric alternating minimization approach of Csiszár and Tusnády and the product space formulation of Pierra, we derive a new simultaneous multiprojection algorithm that employs generalized projections of Bregman to solve the convex feasibility problem or, in the inconsistent case, to minimize a proximity function that measures the average distance from a point to all convex sets. We assume that the Bregman distances involved are jointly convex, so that the proximity function itself is convex. When the intersection of the convex sets is empty, but the closure of the proximity function has a unique global minimizer, the sequence of iterates converges to this unique minimizer. Special cases of this algorithm include the “Expectation Maximization Maximum Likelihood” (EMML) method in emission tomography and a new convergence result for an algorithm that solves the split feasibility problem.
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paper_title: Generalized matrix inverse reconstruction for SPECT using a weighted singular value spectrum

paper_content:

A method of weighting the singular value spectrum for SPECT image reconstruction with generalized matrix inverses (GMIs) has been developed. At each source voxel, spectral weights are computed to minimize the misfit of the resolution kernel with a delta function, subject to a constraint on the variance of the reconstructed source voxel activity. The method was applied to a Monte Carlo simulated Tc-99m myocardial perfusion study and a Tc-99m myocardial perfusion phantom study acquired on a clinical scanner. GMI reconstructions using weighted and truncated spectra were compared. Resolution kernels for a weighted spectrum are narrower and have smaller sidelobes than kernels for a truncated spectrum. As a result, the quality of reconstructed SPECT images is improved and the simulated myocardial wall thickness is better estimated with the use of a weighted spectrum. The use of a weighted singular value spectrum is an important tool for obtaining more localized source activity estimates for GMI reconstruction.
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paper_title: Reconstruction algorithms: Transform methods

paper_content:

Transform methods for image reconstruction from projections are based on analytic inversion formulas. In this tutorial paper, the inversion formula for the case of two-dimensional (2-D) reconstruction from line integrals is manipulated into a number of different forms, each of which may be discretized to obtain different algorithms for reconstruction from sampled data. For the convolution-backprojection algorithm and the direct Fourier algorithm the emphasis is placed on understanding the relationship between the discrete operations specified by the algorithm and the functional operations expressed by the inversion formula. The performance of the Fourier algorithm may be improved, with negligible extra computation, by interleaving two polar sampling grids in Fourier space. The convolution-backprojection formulas are adapted for the fan-beam geometry, and other reconstruction methods are summarized, including the rho-filtered layergram method, and methods involving expansions in angular harmonics. A standard mathematical process leads to a known formula for iterative reconstruction from projections at a finite number of angles. A new iterative reconstruction algorithm is obtained from this formula by introducing one-dimensional (1-D) and 2-D interpolating functions, applied to sampled projections and images, respectively. These interpolating functions are derived by the same Fourier approach which aids in the development and understanding of the more conventional transform methods.
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paper_title: On the determination of functions from their integral values along certain manifolds

paper_content:

When one integrates a function of two variables x,y - a point function f(P) in the plane - subject to suitable regularity conditions along an arbitrary straight line g then one obtains in the integral values F(g), a line function. In Part A of the present paper the problem which is solved is the inversion of this linear functional transformation, that is the following questions are answered: can every line function satisfying suitable regularity conditions be regarded as constructed in this way? If so, is f uniquely known from F and how can f be calculated? In Part B a solution of the dual problem of calculating a line function F(g) from its point mean values f(P) is solved in a certain sense. Finally, in Part C certain generalizations are discussed, prompted by consideration of non-Euclidean manifolds as well as higher dimensional spaces. The treatment of these problems, themselves of interest, gains enhanced importance through the numerous relationships that exist between this topic and the theory of logarithmic and Newtonian potentials. These are mentioned at appropriate places in the text.
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paper_title: Radiological Imaging The Theory Of Image Formation Detection And Processing

paper_content:

Thank you for reading radiological imaging the theory of image formation detection and processing. Maybe you have knowledge that, people have search numerous times for their favorite books like this radiological imaging the theory of image formation detection and processing, but end up in malicious downloads. Rather than reading a good book with a cup of coffee in the afternoon, instead they are facing with some malicious virus inside their laptop.
---
paper_title: A short reader's guide to 3D tomographic reconstruction.

paper_content:

This text summarizes the main technical problems related to 3D image reconstruction in PET, SPECT and CT, and provides references to a selection of key papers and to review papers.
---
paper_title: Finite series-expansion reconstruction methods

paper_content:

Series-expansion reconstruction methods made their first appearance in the scientific literature and in the CT scanner industry around 1970. Great research efforts have gone into them since but many questions still wait to be answered. These methods, synonymously known as algebraic methods, iterative algorithms, or optimization theory techniques, are based on the discretization of the image domain prior to any mathematical analysis and thus are rooted in a completely different branch of mathematics than the transform methods which are discussed in this issue by Lewitt [51]. How is the model set up? What is the methodology of the approach? Where does mathematical optimization theory enter? What do these reconstruction algorithms look like? How are quadratic optimization, entropy optimization, and Bayesian analysis used in image reconstruction? Finally, why study series expansion methods if transform methods are so much faster? These are some of the questions that are answered in this paper.
---
paper_title: Image reconstruction from finite numbers of projections

paper_content:

Several results are obtained appertaining to the reconstruction of a two- dimensional image from a finite number of projections. Several schemes are considered for interpolating between the given data. When a trigonometrical Fourier series is used for angular interpolation then one finds, firstly, a consistency condition whereby a posteriori estimates can be made of the errors in the given data, and secondly, a basic image which contains only that information common to all physically permissible interpolation schemes. This basic image is necessarily free of misleading artefacts but it is computationally slow. Several computationally rapid interpolation schemes (based on the fast Fourier transform algorithm) are found to give good quality images, provided the given number of projections is sufficient to resolve the major details of the true image. A computational example is presented showing that a contrived image can be accurately reconstructed from a single projection.
---
paper_title: Principles of Computerized Tomographic Imaging

paper_content:

Tomography refers to the cross-sectional imaging of an object from either transmission or reflection data collected by illuminating the object from many different directions. The impact of tomography in diagnostic medicine has been revolutionary, since it has enabled doctors to view internal organs with unprecedented precision and safety to the patient. There are also numerous nonmedical imaging applications which lend themselves to methods of computerized tomography, such as mapping of underground resources...cross-sectional imaging of for nondestructive testing...the determination of the brightness distribution over a celestial sphere...three-dimensional imaging with electron microscopy. Principles of Computerized Tomographic Imaging provides a tutorial overview of topics in tomographic imaging covering mathematical principles and theory...how to apply the theory to problems in medical imaging and other fields...several variations of tomography that are currently being researched.
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paper_title: The Radon Transform And Some Of Its Applications

paper_content:

the radon transform and some of its applications is available in our book collection an online access to it is set as public so you can get it instantly. Our digital library spans in multiple locations, allowing you to get the most less latency time to download any of our books like this one. Kindly say, the the radon transform and some of its applications is universally compatible with any devices to read.
---
paper_title: Data Acquisition and Image Reconstruction for 3D PET

paper_content:

The purpose of this chapter is to explain the underlying concepts of the most common image reconstruction methods. The question to be answered in this chapter is: how can we use the additional information from a 3D PET scan (as compared to a 2D scan) to improve the signal to noise ratio in the reconstructed image?
---
paper_title: Statistical Approaches in Quantitative Positron Emission Tomography

paper_content:

Positron emission tomography is a medical imaging modality for producing 3D images of the spatial distribution of biochemical tracers within the human body. The images are reconstructed from data formed through detection of radiation resulting from the emission of positrons from radioisotopes tagged onto the tracer of interest. These measurements are approximate line integrals from which the image can be reconstructed using analytical inversion formulae. However these direct methods do not allow accurate modeling either of the detector system or of the inherent statistical fluctuations in the data. Here we review recent progress in developing statistical approaches to image estimation that can overcome these limitations. We describe the various components of the physical model and review different formulations of the inverse problem. The wide range of numerical procedures for solving these problems are then reviewed. Finally, we describe recent work aimed at quantifying the quality of the resulting images, both in terms of classical measures of estimator bias and variance, and also using measures that are of more direct clinical relevance.
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paper_title: The Mathematics of Computerized Tomography

paper_content:

The Mathematics of Computerized Tomography covers the relevant mathematical theory of the Radon transform and related transforms and also studies more practical questions such as stability, sampling, resolution, and accuracy. Quite a bit of attention is given to the derivation, analysis, and practical examination of reconstruction algorithm, for both standard problems and problems with incomplete data.
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paper_title: Iterative reconstruction algorithms

paper_content:

Abstract In this paper a survey of recent results on iterative reconstruction algorithms is given. These results, many of which have not yet appeared elsewhere, are applicable to a very general formulation of the reconstruction problem based on the series expansion approach. A set of optimization criteria and a number of iterative reconstruction algorithms are stated, together with theorems on the convergence of the algorithms to optimum images. The efficacy of the algorithms is compared to that of the convolution method. In particular, the falseness of the claim that ART and the backprojection method are the same is demonstrated.
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paper_title: Image reconstruction--a tutorial.

paper_content:

This paper is written for physicians and presents basic principles of image reconstruction in nuclear medicine. Both analytical and iterative methods are discussed without rigorous mathematics.
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paper_title: Three-dimensional reconstruction in nuclear medicine emission imaging

paper_content:

The study presents application of methods of ascertaining the three-dimensional distribution of isotope concentration or density in nuclear medicine, and differs from previous three-dimensional reconstruction efforts of astrophysics, electron microscopy, and X-ray radiology in that statistically poor measurements and photon attenuation are taken into account by the algorithm. The methods discussed are applicable to photon or heavy ion transmission radiography as well as emission imaging.
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paper_title: HeinzelCluster: accelerated reconstruction for FORE and OSEM3D

paper_content:

We have investigated two strategies to speedup reconstruction of data from our high resolution PET scanner (HRRT) on a dedicated cluster (7 nodes with 4 PIII Xeon@700 MHz, switched Fast Ethernet and Myrinet networking, Windows NT). With the first strategy, FORE/FOREJ and a fast 2D reconstruction method, we were able to reduce the 2D reconstruction time from 38 minutes to approx. 100 seconds. The second approach was to accelerate OSEM3D with a two-stage parallelization scheme resulting in a drop of reconstruction time from about 8 hours on one CPU to approx. 35 minutes for one core iteration.
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paper_title: On projection algorithms for solving convex feasibility problems

paper_content:

Due to their extraordinary utility and broad applicability in many areas of classical mathematics and modern physical sciences (most notably, computerized tomography), algorithms for solving convex feasibility problems continue to receive great attention. To unify, generalize, and review some of these algorithms, a very broad and flexible framework is investigated. Several crucial new concepts which allow a systematic discussion of questions on behaviour in general Hilbert spaces and on the quality of convergence are brought out. Numerous examples are given.
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paper_title: Algorithms for limited-view computed tomography: an annotated bibliography and a challenge

paper_content:

In many applications of computed tomography, it may not be possible to acquire projection data at all angles, as required by the most commonly used algorithm of convolution backprojection. In such a limited-data situation, we face an ill-posed problem in attempting to reconstruct an image from an incomplete set of projections. Many techniques have been proposed to tackle this situation, employing diverse theories such as signal recovery, image restoration, constrained deconvolution, and constrained optimization, as well as novel schemes such as iterative object-dependent algorithms incorporating a priori knowledge and use of multispectral radiation. We present an overview of such techniques and offer a challenge to all readers to reconstruct images from a set of limited-view data provided here.
---
paper_title: Fast PET image reconstruction based on SVD decomposition of the system matrix

paper_content:

Data filtering based on matrix pseudo-inverse is a well known but not yet appreciated means of tomographic image reconstruction. Matrix pseudo-inversion step is very demanding in terms of numerical precision and necessary computing power. Ill conditioning of the system matrix in positron emission tomography (PET) results in solutions highly sensitive to noise in the experimental data. In the present work, the feasibility of image reconstruction based on singular value decomposition (SVD) of the system matrix for animal 2-D PET is demonstrated. Analytic detector response accounting for the non-invariant spatial system response is explicitly included into the system matrix. Regularization of the SVD-based solution with the singular spectrum truncation (TSVD solution) derived from spatial resolution analysis is proposed. TSVD reconstruction is fast except for the matrix decomposition step, which is performed once for a given scanner geometry. Reconstructed image quality and quantitation are compared to those obtained with filtered backprojection (FBP) and iterative maximum likelihood technique. TSVD image reconstruction may be a viable alternative to FBP for routine clinical applications.
---
paper_title: A tribute to Johann Radon

paper_content:

A brief biographical review of the life and works of Johann Radon is presented.
---
paper_title: Block-Iterative Algorithms with Diagonally Scaled Oblique Projections for the Linear Feasibility Problem

paper_content:

We formulate a block-iterative algorithmic scheme for the solution of systems of linear inequalities and/or equations and analyze its convergence. This study provides as special cases proofs of convergence of (i) the recently proposed component averaging (CAV) method of Censor, Gordon, and Gordon [Parallel Comput., 27 (2001), pp. 777--808], (ii) the recently proposed block-iterative CAV (BICAV) method of the same authors [IEEE Trans. Medical Imaging, 20 (2001), pp. 1050--1060], and (iii) the simultaneous algebraic reconstruction technique (SART) of Andersen and Kak [ Ultrasonic Imaging, 6 (1984), pp. 81--94] and generalizes them to linear inequalities. The first two algorithms are projection algorithms which use certain generalized oblique projections and diagonal weighting matrices which reflect the sparsity of the underlying matrix of the linear system. The previously reported experimental acceleration of the initial behavior of CAV and BICAV is thus complemented here by a mathematical study of the convergence of the algorithms.
---
paper_title: The Radon Transform

paper_content:

The first edition of this book has been out of print for some time and I have decided to follow the publisher's kind suggestion to prepare a new edition. Many examples with explicit inversion formulas and range theo- rems have been added, and the group-theoretic viewpoint emphasized. For example, the integral geometric viewpoint of the Poisson integral for the disk leads to interesting analogies with the X-ray transform in Euclidean 3-space. To preserve the introductory flavor of the book the short and self-contained Chapter Von Schwartz' distributions has been added. Here 5 provides proofs of the needed results about the Riesz potentials while 3-4 develop the tools from Fourier analysis following closely the account in Hormander's books (1963] and [1983]. There is some overlap with my books (1984] and [1994b] which however rely heavily on Lie group theory. The present book is much more elementary. I am indebted to Sine Jensen for a critical reading of parts of the manuscript and to Hilgert and Schlichtkrull for concrete contributions men- tioned at specific places in the text. Finally I thank Jan Wetzel and Bonnie Friedman for their patient and skillful preparation of the manuscript.
---
paper_title: Penalized weighted least-squares image reconstruction for positron emission tomography

paper_content:

This paper presents an image reconstruction method for positron-emission tomography (PET) based on a penalized, weighted least-squares (PWLS) objective. For PET measurements that are precorrected for accidental coincidences, we argue statistically that a least-squares objective function is as appropriate, if not more so, than the popular Poisson likelihood objective. We propose a simple data-based method for determining the weights that accounts for attenuation and detector efficiency. A nonnegative successive over-relaxation (+SOR) algorithm converges rapidly to the global minimum of the PWLS objective. Quantitative simulation results demonstrate that the bias/variance tradeoff of the PWLS+SOR method is comparable to the maximum-likelihood expectation-maximization (ML-EM) method (but with fewer iterations), and is improved relative to the conventional filtered backprojection (FBP) method. Qualitative results suggest that the streak artifacts common to the FBP method are nearly eliminated by the PWLS+SOR method, and indicate that the proposed method for weighting the measurements is a significant factor in the improvement over FBP.
---
paper_title: Positron-emission tomography

paper_content:

We review positron-emission tomography (PET), which has inherent advantages that avoid the shortcomings of other nuclear medicine imaging methods. PET image reconstruction methods with origins in signal and image processing are discussed, including the potential problems of these methods. A summary of statistical image reconstruction methods, which can yield improved image quality, is also presented.
---
paper_title: Strip Integration in Radio Astronomy

paper_content:

When a celestial source of radio waves is scanned with an aerial beam which is much longer than the source in one direction but suitably narrow in the other, the transformation from the true distribution to the measured value is referred to as strip integration. It is here treated as a special case of two-dimensional aerial smoothing in which the aerial beam is allowed to spin about its centre as it moves about the sky. It is shown that the resolution obtainable is set by the cross-sectional profile of the strip beam in the narrow dimension. Thus, when the strip reduces to a line, the resolution is complete and full reconstruction of the true distribution is possible; but scans must be made in all directions. In the general case it is shown that there is a principal solution, and that a finite number of scans suffices to determine it. A method is presented for reconstructing the principal solution from the observed data.
---
paper_title: Improvement of image resolution and quantitative accuracy in clinical single photon emission computed tomography

paper_content:

Abstract Clinical Single Photon Emission Computed Tomography (SPECT) is a scanning technique which acquires gamma-camera images (‘projections’) over a range of angles around a patient. These projections allow the reconstruction of cross sectional (‘tomographic’) images of the gamma-radiating pharmaceutical distribution in the patient, thus providing interesting information about the functioning of organs and tissues. SPECT images are seriously affected by a variety of image degrading processes. Restrictions on the amount of radio-pharmaceutical that can be administered to a patient cause noise in the projections and the limited spatial resolution of the gamma-camera results in blurring of the projections. In addition to these image degradations, the reconstruction of cross-sections is complicated by Compton scattering of γ-photons in tissue, which causes attenuation of the photon flux received by the gamma-camera and causes improper detection of photons which have been scattered in tissue. This results in some additional blurring and loss of accuracy of the SPECT images in predicting activity concentrations. Tremendous efforts have been made to improve the quantitative accuracy and the spatial resolution of SPECT, and to reduce the noise in the reconstructed images. These efforts have resulted in corrective reconstruction algorithms, which are generally based on incorporation of accurate models of the main image degrading factors. Improvements of the data acquisition hardware can further increase image quality. In this paper, the image formation process of SPECT, including image-degrading factors, is explained. In addition, reconstruction algorithms and hardware modifications are reviewed, and their effects on image quality are illustrated with physical phantom and simulation experiments.
---
paper_title: Computed tomography : fundamentals, system technology, image quality, applications

paper_content:

System concepts System components Image reconstruction Spiral CT Multi-slice spiral CT Cone-beam CT Dynamic CT Quantitative CT Dual source CT Dual energy CT Flat detector CT Micro CT Image quality Spatial resolution Contrast Pixel noise Homogeneity Routine and special applications 3D displays Post-processing Quality assurance
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paper_title: Image-correction techniques in SPECT

paper_content:

This overview takes a look at different correction techniques for Single Photon Emission Computed Tomography (SPECT). We discuss the influence of the detection system followed by the scatter and attenuation caused by the object of investigation. When possible we describe how the correction methods for the different physical effects can be incorporated in the reconstruction method, being either filtered backprojection or iterative reconstruction.
---
paper_title: Analytic 3D image reconstruction using all detected events

paper_content:

The authors present the results of testing an algorithm for three-dimensional image reconstruction that uses all gamma-ray coincidence events detected by a PET (positron emission tomography) volume imaging scanner. By using two iterations of an analytic filter-backprojection method, the algorithm is not constrained by the requirement of a spatially invariant detector point spread function, which limits normal analytic techniques. Removing this constraint allows the incorporation of all detected events, regardless of orientation, which improves the statistical quality of the final reconstructed image. >
---
paper_title: Data Acquisition and Image Reconstruction for 3D PET

paper_content:

The purpose of this chapter is to explain the underlying concepts of the most common image reconstruction methods. The question to be answered in this chapter is: how can we use the additional information from a 3D PET scan (as compared to a 2D scan) to improve the signal to noise ratio in the reconstructed image?
---
paper_title: Attenuation correction for rotating slant-hole (RSH) SPECT using exact rebinning

paper_content:

We have recently derived an analytic method of image reconstruction for the rotating slant-hole (RSH) SPECT scanner for cardiac imaging. The method assumes that the emission activity lies in a region of constant attenuation. It operates by rebinning the three-dimensional (3-D) measured projections into the geometry of a stack of two-dimensional (2-D) slices and performing 2-D reconstruction in a second step. An important consequence of this method is a mathematical proof that opposing projections are not necessary for full tomographic reconstruction in the presence of attenuation. We have now tested our algorithm in the presence of noisy projection data and with projections which violates the constant attenuation assumption. We present results from simulated and real phantom data demonstrating effective attenuation correction in the presence of a uniform low-level of background in the torso, with truncated liver activity, and without opposing views. Our conclusions are that attenuation correction can be performed for RSH SPECT without measuring opposing views, and that the effect of background activity (which violates the assumptions required for exponential data) and truncated activity of a nearby liver are small and easily dominated by typical noise levels in SPECT imaging.
---
paper_title: 3D-FRP: Direct Fourier reconstruction with Fourier reprojection for fully 3D PET

paper_content:

The direct Fourier method for 3D reconstruction of a 3D volume is based on the relationship between the 3D Fourier transform of the volume and the 2D FT of a parallel-ray projection of the volume. The direct Fourier method has the potential for very fast reconstruction, but a straightforward implementation of the method leads to unsatisfactory results. This paper presents an implementation (3D-FRP) of the direct Fourier method for fully 3D PET data with incomplete oblique projections that gives results as good as (or better than) those of a much slower method (3DRP), and in the same time as a fast but less accurate method (Fourier rebinning followed by slice-by-slice reconstruction). In common with filtered backprojection (3DRP), the Fourier method (3D-FRP) is based on a discretization of an inversion formula (so it is geometrically accurate for large oblique angles), and the method involves reprojection of an initial image. The critical two steps in the method are the estimations of the samples of the 3D transform of the image from the samples of the 2D transforms of the projections on the planes through the origin of Fourier space, and vice versa for reprojection. These steps use a gridding strategy, combined with new approaches for weighting in the transform and image domains. The authors' experimental results confirm that good image accuracy can be achieved together with a short reconstruction time.
---
paper_title: Weighted Back-Projection Methods

paper_content:

Traditionally, 3D reconstruction methods have been classified into two major groups, Fourier reconstruction methods and direct methods (e.g. Crowther et al., 1970; Gilbert, 1972). Fourier methods are defined as algorithms that restore the Fourier transform of the object from the Fourier transforms of their projections and then obtain the real-space distribution of the object by inverse Fourier transformation. Included in this group are also equivalent reconstruction schemes that use expansions of object and projections into orthogonal function systems (e.g. Cormack, 1963, 1964; Smith et al., 1973; Chapter 9 of this volume). In contrast, direct methods are defined as those that carry out all calculations in real space. These include the convolution back-projection algorithms (Bracewell and Riddle, 1967; Gilbert, 1972; Ramachandran and Lakshminarayanan, 1971) and iterative algorithms (Gordon et al., 1970; Colsher, 1977). Weighted back-projection methods are difficult to classify in this scheme, since they are equivalent to convolution back-projection algorithms, but work on the real-space data as well as the Fourier transform data of either the object or the projections. Both convolution back-projection and weighted back-projection algorithms are based on the same theory as Fourier reconstruction methods, whereas iterative methods normally do not take into account the Fourier relationships between object transform and projection transforms.
---
paper_title: Strip Integration in Radio Astronomy

paper_content:

When a celestial source of radio waves is scanned with an aerial beam which is much longer than the source in one direction but suitably narrow in the other, the transformation from the true distribution to the measured value is referred to as strip integration. It is here treated as a special case of two-dimensional aerial smoothing in which the aerial beam is allowed to spin about its centre as it moves about the sky. It is shown that the resolution obtainable is set by the cross-sectional profile of the strip beam in the narrow dimension. Thus, when the strip reduces to a line, the resolution is complete and full reconstruction of the true distribution is possible; but scans must be made in all directions. In the general case it is shown that there is a principal solution, and that a finite number of scans suffices to determine it. A method is presented for reconstructing the principal solution from the observed data.
---
paper_title: Three-Dimensional Electron Microscopy of Macromolecular Assemblies

paper_content:

Electron microscopy of macromolecular assemblies two-dimensional averaging techniques multivariate statistical analysis and classification of images three-dimensional reconstruction interpretation of three-dimensional images of macromolecules example of an application - calcium release channel. Appendices: software implementations macromolecular assemblies reconstructed from micrographs of single macromolecules.
---
paper_title: A Fast Sinc Function Gridding Algorithm for Fourier Inversion in Computer Tomography

paper_content:

The Fourier inversion method for reconstruction of images in computerized tomography has not been widely used owing to the perceived difficulty of interpolating from polar or other measurement grids to the Cartesian grid required for fast numerical Fourier inversion. Although the Fourier inversion method is recognized as being computationally faster than the back-projection method for parallel ray projection data, the artifacts resulting from inaccurate interpolation have generally limited application of the method. This paper presents a computationally efficient gridding algorithm which can be used with direct Fourier transformation to achieve arbitrarily small artifact levels. The method has potential for application to other measurement geometries such as fan-beam projections and diffraction tomography and NMR imaging.
---
paper_title: Analytic 3D image reconstruction using all detected events

paper_content:

The authors present the results of testing an algorithm for three-dimensional image reconstruction that uses all gamma-ray coincidence events detected by a PET (positron emission tomography) volume imaging scanner. By using two iterations of an analytic filter-backprojection method, the algorithm is not constrained by the requirement of a spatially invariant detector point spread function, which limits normal analytic techniques. Removing this constraint allows the incorporation of all detected events, regardless of orientation, which improves the statistical quality of the final reconstructed image. >
---
paper_title: Reconstruction algorithms: Transform methods

paper_content:

Transform methods for image reconstruction from projections are based on analytic inversion formulas. In this tutorial paper, the inversion formula for the case of two-dimensional (2-D) reconstruction from line integrals is manipulated into a number of different forms, each of which may be discretized to obtain different algorithms for reconstruction from sampled data. For the convolution-backprojection algorithm and the direct Fourier algorithm the emphasis is placed on understanding the relationship between the discrete operations specified by the algorithm and the functional operations expressed by the inversion formula. The performance of the Fourier algorithm may be improved, with negligible extra computation, by interleaving two polar sampling grids in Fourier space. The convolution-backprojection formulas are adapted for the fan-beam geometry, and other reconstruction methods are summarized, including the rho-filtered layergram method, and methods involving expansions in angular harmonics. A standard mathematical process leads to a known formula for iterative reconstruction from projections at a finite number of angles. A new iterative reconstruction algorithm is obtained from this formula by introducing one-dimensional (1-D) and 2-D interpolating functions, applied to sampled projections and images, respectively. These interpolating functions are derived by the same Fourier approach which aids in the development and understanding of the more conventional transform methods.
---
paper_title: Resampling of data between arbitrary grids using convolution interpolation

paper_content:

For certain medical applications resampling of data is required. In magnetic resonance tomography (MRT) or computer tomography (CT), e.g., data may be sampled on nonrectilinear grids in the Fourier domain. For the image reconstruction a convolution-interpolation algorithm, often called gridding, can be applied for resampling of the data onto a rectilinear grid. Resampling of data from a rectilinear onto a nonrectilinear grid are needed, e.g., if projections of a given rectilinear data set are to be obtained. In this paper the authors introduce the application of the convolution interpolation for resampling of data from one arbitrary grid onto another. The basic algorithm can be split into two steps. First, the data are resampled from the arbitrary input grid onto a rectilinear grid and second, the rectilinear data is resampled onto the arbitrary output grid. Furthermore, the authors like to introduce a new technique to derive the sampling density function needed for the first step of their algorithm. For fast, sampling-pattern-independent determination of the sampling density function the Voronoi diagram of the sample distribution is calculated. The volume of the Voronoi cell around each sample is used as a measure for the sampling density. It is shown that the introduced resampling technique allows fast resampling of data between arbitrary grids. Furthermore, it is shown that the suggested approach to derive the sampling density function is suitable even for arbitrary sampling patterns. Examples are given in which the proposed technique has been applied for the reconstruction of data acquired along spiral, radial, and arbitrary trajectories and for the fast calculation of projections of a given rectilinearly sampled image.
---
paper_title: Data Acquisition and Image Reconstruction for 3D PET

paper_content:

The purpose of this chapter is to explain the underlying concepts of the most common image reconstruction methods. The question to be answered in this chapter is: how can we use the additional information from a 3D PET scan (as compared to a 2D scan) to improve the signal to noise ratio in the reconstructed image?
---
paper_title: On the optimality of the gridding reconstruction algorithm

paper_content:

Gridding reconstruction is a method to reconstruct data onto a Cartesian grid from a set of nonuniformly sampled measurements. This method is appreciated for being robust and computationally fast. However, it lacks solid analysis and design tools to quantify or minimize the reconstruction error. Least squares reconstruction (LSR), on the other hand, is another method which is optimal in the sense that it minimizes the reconstruction error. This method is computationally intensive and, in many cases, sensitive to measurement noise. Hence, it is rarely used in practice. Despite their seemingly different approaches, the gridding and LSR methods are shown to be closely related. The similarity between these two methods is accentuated when they are properly expressed in a common matrix form. It is shown that the gridding algorithm can be considered an approximation to the least squares method. The optimal gridding parameters are defined as the ones which yield the minimum approximation error. These parameters are calculated by minimizing the norm of an approximation error matrix. This problem is studied and solved in the general form of approximation using linearly structured matrices. This method not only supports more general forms of the gridding algorithm, it can also be used to accelerate the reconstruction techniques from incomplete data. The application of this method to a case of two-dimensional (2-D) spiral magnetic resonance imaging shows a reduction of more than 4 dB in the average reconstruction error.
---
paper_title: The gridding method for image reconstruction by Fourier transformation

paper_content:

The authors explore a computational method for reconstructing an n-dimensional signal f from a sampled version of its Fourier transform f/spl circ/. The method involves a window function w/spl circ/ and proceeds in three steps. First, the convolution g/spl circ/=w/spl circ/*f/spl circ/ is computed numerically on a Cartesian grid, using the available samples of f/spl circ/. Then, g=wf is computed via the inverse discrete Fourier transform, and finally f is obtained as g/w. Due to the smoothing effect of the convolution, evaluating w/spl circ/*f/spl circ/ is much less error prone than merely interpolating f/spl circ/. The method was originally devised for image reconstruction in radio astronomy, but is actually applicable to a broad range of reconstructive imaging methods, including magnetic resonance imaging and computed tomography. In particular, it provides a fast and accurate alternative to the filtered backprojection. The basic method has several variants with other applications, such as the equidistant resampling of arbitrarily sampled signals or the fast computation of the Radon (Hough) transform. >
---
paper_title: Three-dimensional Reconstruction from Radiographs and Electron Micrographs: Application of Convolutions instead of Fourier Transforms

paper_content:

A new technique is proposed for the mathematical process of reconstruction of a three-dimensional object from its transmission shadowgraphs; it uses convolutions with functions defined in the real space of the object, without using Fourier transforms. The object is rotated about an axis at right angles to the direction of a parallel beam of radiation, and sections of it normal to the axis are reconstructed from data obtained by scanning the corresponding linear strips in the shadowgraphs at different angular settings. Since the formulae in the convolution method involve only summations over one variable at a time, while a two-dimensional reconstruction with the Fourier transform technique requires double summations, the convolution method is much faster (typically by a factor of 30); the relative increase in speed is larger where greater resolution is required. Tests of the convolution method with computer-simulated shadowgraphs show that it is also more accurate than the Fourier transform method. It has good potentialities for application in electron microscopy and x-radiography. A new method of reconstructing helical structures by this technique is also suggested.
---
paper_title: Evaluation of rotating slant-hole SPECT mammography using Monte-Carlo simulation methods

paper_content:

Multi-segment slant-hole (MSSH) collimators provide much higher detection efficiency of breast lesions at a single view than conventional low-energy high-resolution (LEHR) parallel-hole collimators. The purpose of this study is to evaluate rotating MSSH (RMSSH) SPECT mammography using Monte Carlo Simulation methods. We modified the SIMIND Monte Carlo code to generate RMSSH projection data including the effects of attenuation, collimator-detector response and scatter from a realistic NCAT phantom with an attached breast and a lesion placed at locations at the center and at the near and far walls of the breast. The projection data were reconstructed using analytic and iterative image reconstruction methods. For comparison, we examine an LEHR collimator with a resolution of 1 cm at 15.2 cm, which is the same as that of the MSSH at the center of the common volume-of-view. Also, we simulated the inclusion of a lead shield that covers the body to keep photons emitted from the other organs from reaching the MSSH collimator. Our results indicated that RMSSH SPECT provides greater breast lesion contrast than that of a planar image taken using a conventional LEHR collimator at all lesion locations. Depending on the lesion location, the spatial resolution of the RMSSH SPECT images may be comparable or slightly poorer than that of the planar scintimammographic images obtained using a LEHR collimator. Shielding of photon emissions from the other organs substantially improves the RMSSH SPECT image quality. We conclude that the Monte Carlo simulation methods are useful in evaluating the application of RMSSH SPECT to breast imaging as compared to conventional planar scintimammography and that RMSSH SPECT may enhance lesion detectability.
---
paper_title: The Exponential Radon Transform

paper_content:

The exponential Radon transform, a generalization of the Radon transform, is defined and is studied as a mapping of function spaces. An inversion formula is derived. The exponential Radon transform is represented in terms of Fourier transforms of its domain and range, and this leads to a characterization of the range of the transform.The exponential Radon transform is a model for some reconstruction problems in radionuclide emission computer tomography.
---
paper_title: An inversion formula for the attenuated X-ray transformation

paper_content:

The problem of inversion of the attenuated X-ray transformation is solved by an explicit formula. Several subsequent results are also given.
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paper_title: A unified analysis of exact methods of inverting the 2-D exponential radon transform, with implications for noise control in SPECT

paper_content:

Exact methods of inverting the two-dimensional (2-D) exponential Radon transform have been proposed by Bellini et al. (1979) and by Inouye et al. (1989), both of whom worked in the spatial-frequency domain to estimate the 2-D Fourier transform of the unattenuated sinogram; by Hawkins et al. (1988), who worked with circularly harmonic Bessel transforms; and by Tretiak and Metz (1980), who followed filtering of appropriately-modified projections by exponentially-weighted backprojection. With perfect sampling, all four of these methods are exact in the absence of projection-data noise, but empirical studies have shown that they propagate noise differently, and no underlying theoretical relationship among the methods has been evident. Here, an analysis of the 2-D Fourier transform of the modified sinogram reveals that all previously-proposed linear methods can be interpreted as special cases of a broad class of methods, and that each method in the class can be implemented, in principle, by any one of four distinct techniques. Moreover, the analysis suggests a new member of the class that Is predicted to have noise properties better than those of previously-proposed members.
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paper_title: Inversion of the attenuated Radon transform

paper_content:

We derive an exact inversion formula for the attenuated Radon transform. The formula is closely related to Novikov's inversion formula, but our derivation is completely different. We also give an implementation of the inversion formula very similar to the filtered backprojection algorithm of x-ray tomography and present numerical results.
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paper_title: A new SPECT reconstruction algorithm based on the Novikov explicit inversion formula

paper_content:

We present a new reconstruction algorithm for single-photon emission computed tomography. The algorithm is based on the Novikov explicit inversion formula for the attenuated Radon transform with non-uniform attenuation. Our reconstruction technique can be viewed as a generalization of both the filtered backprojection algorithm and the Tretiak-Metz algorithm. We test the performance of the present algorithm in a variety of numerical experiments. Our numerical examples show that the algorithm is capable of accurate image reconstruction even in the case of strongly non-uniform attenuation coefficient, similar to that occurring in a human thorax.
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paper_title: Compensation of tissue absorption in emission tomography

paper_content:

This paper presents a technique for overcoming the problem of tissue absorption in emission tomography. Given a set of equispaced projections in the interval (0, 2π), it is possible to derive an exact formula for recovering the spectrum of the image. The formula is obtained by solving a linear partial-differential equation that links the derivatives of the spectrum with respect to absorption and spatial frequencies. Examples of the application of the technique to synthetic and real data are given.
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paper_title: On the range characterization for the two-dimensional attenuated x-ray transformation

paper_content:

The problem of the range characterization for the two-dimensional attenuated x-ray transformation is solved (on the most essential level) by explicit integral relations.
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paper_title: Image reconstruction algorithm for a rotating slat collimator

paper_content:

A slat collimator in single photon emission computed tomography consists of a set of parallel slats. As the collimator spins, the detector measures a one-dimensional projection data set. A complete data set can be obtained by rotating the detector/collimator assembly around the object (patient) while the collimator spins continuously. The measured projection data are assumed to be weighted planar integrals of the object. This paper describes the development of an approximate three-dimensional image reconstruction algorithm for a rotating/spinning slat collimator. This algorithm is in filtered backprojection form. Computer simulations were performed to verify the effectiveness of the algorithm.
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paper_title: On single-photon emission computed tomography imaging based on an exact formula for the nonuniform attenuation correction

paper_content:

We develop an exact formula approach to nonuniform attenuation correction in single-photon emission computed tomography (SPECT). We discuss several reconstruction examples obtained by this method from real SPECT data.
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paper_title: Transmission scanning in emission tomography

paper_content:

Attenuation correction in single-photon (SPET) and positron emission (PET) tomography is now accepted as a vital component for the production of artefact-free, quantitative data. The most accurate attenuation correction methods are based on measured transmission scans acquired before, during, or after the emission scan. Alternative methods use segmented images, assumed attenuation coefficients or consistency criteria to compensate for photon attenuation in reconstructed images. This review examines the methods of acquiring transmission scans in both SPET and PET and the manner in which these data are used. While attenuation correction gives an exact correction in PET, as opposed to an approximate one in SPET, the magnitude of the correction factors required in PET is far greater than in SPET. Transmission scans also have a number of other potential applications in emission tomography apart from attenuation correction, such as scatter correction, inter-study spatial co-registration and alignment, and motion detection and correction. The ability to acquire high-quality transmission data in a practical clinical protocol is now an essential part of the practice of nuclear medicine.
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paper_title: A Combined PET/CT Scanner for Clinical Oncology

paper_content:

The availability of accurately aligned, whole-body anatomical (CT) and functional (PET) images could have a significant impact on diagnosing and staging malignant disease and on identifying and localizing metastases. Computer algorithms to align CT and PET images acquired on different scanners are generally successful for the brain, whereas image alignment in other regions of the body is more problematic. Methods: A combined PET/CT tomograph with the unique capability of acquiring accurately aligned functional and anatomical images for any part of the human body has been designed and built. The PET/CT scanner was developed as a combination of a Siemens Somatom AR.SP spiral CT and a partial-ring, rotating ECAT ART PET scanner. All components are mounted on a common rotational support within a single gantry. The PET and CT components can be operated either separately, or in combined mode. In combined mode, the CT images are used to correct the PET data for scatter and attenuation. Fully quantitative whole-body images are obtained for an axial extent of 100 cm in an imaging time of less than 1 h. When operated in PET mode alone, transmission scans are acquired with dual 137 Cs sources. Results: The scanner is fully operational and the combined device has been operated successfully in a clinical environment. Over 110 patients have been imaged, covering a range of different cancers, including lung, esophageal, head and neck, melanoma, lymphoma, pancreas, and renal cell. The aligned PET and CT images are used both for diagnosing and staging disease and for evaluating response to therapy. We report the first performance measurements from the scanner and present some illustrative clinical studies acquired in cancer patients. Conclusion: A combined PET and CT scanner is a practical and effective approach to acquiring co-registered anatomical and functional images in a single scanning session.
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paper_title: Noise propagation from attenuation correction into PET reconstruction

paper_content:

It is useful to have the ability to analyze the propagation of errors from transmission data into the resulting attenuation-corrected emission reconstruction PET. The authors develop theoretical expressions for the mean and covariance of the emission reconstruction when the only noise source is that in the transmission data. There are several ways to impose an attenuation correction onto the PET reconstructions, but here the authors theoretically analyze two cases: (1) A linear (on log data) transmission reconstruction is reprojected to get an ACF (Attenuation Correction Factor) estimate which is then used in a linear emission estimate. (2) PET reconstruction is a nonlinear estimate based on maximizing a regularized likelihood objective, and attenuation is modeled directly in the objective function. A validation study is presented for the mean and covariance expressions for case (1).
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paper_title: Determination of the Attenuation Map in Emission Tomography

paper_content:

Reliable attenuation correction methods for quantitative emission CT (ECT) require accurate delineation of the body contour and often necessitate knowledge of internal anatomic structure. Two broad classes of methods have been used to calculate the attenuation map: transmissionless and transmission-based attenuation correction techniques. Whereas calculated attenuation correction belonging to the first class of methods is appropriate for brain studies, more adequate methods must be performed in clinical applications, where the attenuation coefficient distribution is not known a priori, and for areas of inhomogeneous attenuation such as the chest. Measured attenuation correction overcomes this problem and uses different approaches to determine this map, including transmission scanning, segmented magnetic resonance images, or appropriately scaled CT scans acquired either independently on separate or simultaneously on multimodality imaging systems. Combination of data acquired from different imagers suffers from the usual problems of working with multimodality images-namely, accurate coregistration from the different modalities and assignment of attenuation coefficients. A current trend in ECT is to use transmission scanning to reconstruct the attenuation map. Combined ECT/CT imaging is an interesting approach; however, it considerably complicates both the scanner design and the data acquisition and processing protocols. Moreover, the cost of such systems may be prohibitive for small nuclear medicine departments. A dramatic simplification could be made if the attenuation map could be obtained directly from the emission projections, without the use of a transmission scan. This is being investigated either using a statistical model of emission data or applying the consistency conditions that allow one to identify the operator of the problem and, thus, to reconstruct the attenuation map. This article presents the physical and methodologic basis of attenuation correction and summarizes recent developments in algorithms used to compute the attenuation map in ECT. Other potential applications are also discussed.
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paper_title: Image fusion using an integrated, dual-head coincidence camera with X-ray tube-based attenuation maps.

paper_content:

UNLABELLED ::: The purpose of this study was to characterize a dual-head gamma camera capable of FDG imaging using coincidence detection and equipped with an integrated x-ray transmission system for attenuation correction, anatomic mapping, and image fusion. ::: ::: ::: METHODS ::: Radiation dose (425 mrads skin dose) and tissue contrast (0.7% deviation from expected values) were assessed for the x-ray system. Registration of transmission and emission scans was validated using a hot sphere phantom and was verified in selected patient studies. ::: ::: ::: RESULTS ::: Fusion of anatomic maps and FDG images allowed precise anatomic localization of lesions identified using dual-head coincidence imaging. ::: ::: ::: CONCLUSION ::: The combined approach of x-ray attenuation, anatomic mapping, and image fusion with scintigraphic studies provides a new diagnostic tool for nuclear medicine and fertile ground for future research.
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paper_title: Attenuation correction of SPECT using X-ray CT on an emission-transmission CT system: Myocardial perfusion assessment

paper_content:

We present experimental results for attenuation correction of emission tomography through use of an attenuation map derived from X-ray CT data. Both the X-ray and gamma-ray data were obtained on a prototype emission-transmission imaging system ("X-SPECT") which incorporates a third-generation X-ray CT scanner and a single-headed SPECT scanner. The two commercial imaging devices are juxtaposed such that the CT table can move the patient directly into the SPECT scanner after the X-ray scan. This enables systematic registration of SPECT and X-ray CT images. To calibrate attenuation map creation, a set of CT scans for a specific phantom geometry and X-ray technique were acquired, with varying attenuating materials in each scan. A piecewise linear fit was applied to the CT numbers and known attenuation coefficients at 140 keV, and the result is used to create attenuation maps at 140 keV from given CT images. This technique has been applied to small pigs in myocardial perfusion studies. Correcting for attenuation increased the image-derived myocardial radionuclide concentration values from 16% of the values observed for excised myocardium to 35% (ignoring background). Adding an additional correction for partial volume effects increases the image-derived value to 93% of the expected value (ignoring background). Precision, however, remains limited.
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paper_title: A comparison of segmentation and emission subtraction for singles transmission in PET

paper_content:

Single photon transmission scans, using /sup 137/Cs (662 keV gamma rays), are performed post-injection for clinical whole body PET studies. Two methods of processing the transmission data are investigated. The first method segments the transmission image into lung and soft tissue volumes. Average attenuation coefficients for 511 keV are applied and this transmission image is forward projected for attenuation correction. In the second method short emission contamination (EC) scans are performed after the singles transmission scans. These are singles transmission scans without a /sup 137/Cs transmission source. These EC scans are then subtracted from the transmission scans to remove contamination of the 662 keV energy window by 511 keV emission gamma rays. From ten patients studied by this method, the EC subtracted transmission images have average soft tissue attenuation coefficients of 0.074/cm, compared to the expected value of 0.086/cm for 662 keV gamma rays. Simple scaling of these transmission images to 511 keV values (0.095/cm) results in quantitative attenuation correction. A comparison of the two methods is made for these ten patient studies. Both methods result in accurate attenuation correction. The segmentation method performs low noise attenuation correction but is more limited to torso studies with the limbs out of the FOV. The EC subtraction method is more versatile and is applicable to the entire body but results in some more noise in the fully corrected images.
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paper_title: Postinjection single photon transmission tomography with ordered-subset algorithms for whole-body PET imaging

paper_content:

We have evaluated the penalized ordered-subset transmission reconstruction (OSTR) algorithm for postinjection single photon transmission scanning. The OSTR algorithm of Erdogan and Fessler (1999) uses a more accurate model for transmission tomography than ordered-subsets expectation-maximization (OSEM) when OSEM is applied to the logarithm of the transmission data. The OSTR algorithm is directly applicable to postinjection transmission scanning with a single photon source, as emission contamination from the patient mimics the effect, in the original derivation of OSTR, of random coincidence contamination in a positron source transmission scan. Multiple noise realizations of simulated postinjection transmission data were reconstructed using OSTR, filtered backprojection (FBP), and OSEM algorithms. Due to the nonspecific task performance, or multiple uses, of the transmission image, multiple figures of merit were evaluated, including image noise, contrast, uniformity, and root mean square (rms) error. We show that: 1) the use of a three-dimensional (3-D) regularizing image roughness penalty with OSTR improves the tradeoffs in noise, contrast, and rms error relative to the use of a two-dimensional penalty; 2) OSTR with a 3-D penalty has improved tradeoffs in noise, contrast, and rms error relative to FBP or OSEM; and 3) the use of image standard deviation from a single realization to estimate the true noise can be misleading in the case of OSEM. We conclude that using OSTR with a 3-D penalty potentially allows for shorter postinjection transmission scans in single photon transmission tomography in positron emission tomography (PET) relative to FBP or OSEM reconstructed images with the same noise properties. This combination of singles+OSTR is particularly suitable for whole-body PET oncology imaging.
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paper_title: Gamma camera-mounted anatomical X-ray tomography: technology, system characteristics and first images

paper_content:

Scintigraphic diagnosis, based on functional image interpretation, becomes more accurate and meaningful when supported by corresponding anatomical data. In order to produce anatomical images that are inherently registered with images of emission computerised tomography acquired with a gamma camera, an X-ray transmission system was mounted on the slip-ring gantry of a GEMS Millennium VG gamma camera. The X-ray imaging system is composed of an X-ray tube and a set of detectors located on opposite sides of the gantry rotor that moves around the patient along with the nuclear detectors. A cross-sectional anatomical transmission map is acquired as the system rotates around the patient in a manner similar to a third-generation computerised tomography (CT) system. Following transmission, single-photon emission tomography (SPET) or positron emission tomography (PET) coincidence detection images are acquired and the resultant emission images are thus inherently registered to the anatomical maps. Attenuation correction of the emission images is performed with the same anatomical maps to generate transmission maps. Phantom experiments of system performance and examples of first SPET and coincidence detection patient images are presented. Despite limitations of the system when compared with a state of the art CT scanner, the transmission anatomical maps allow for precise anatomical localisation and for attenuation correction of the emission images.
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paper_title: Image reconstruction algorithm for a rotating slat collimator

paper_content:

A slat collimator in single photon emission computed tomography consists of a set of parallel slats. As the collimator spins, the detector measures a one-dimensional projection data set. A complete data set can be obtained by rotating the detector/collimator assembly around the object (patient) while the collimator spins continuously. The measured projection data are assumed to be weighted planar integrals of the object. This paper describes the development of an approximate three-dimensional image reconstruction algorithm for a rotating/spinning slat collimator. This algorithm is in filtered backprojection form. Computer simulations were performed to verify the effectiveness of the algorithm.
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paper_title: Quantitative Procedures in 3D PET

paper_content:

Positron tomography has always had a strong quantitative emphasis. The reconstructed volumes which result from 2D acquisition provide radioactivity concentrations accurate to within a few percent under well-controlled, calibrated conditions. This is mostly due to the low acceptance of scattered coincidences and the accuracy of the corrections for random coincidences, dead time, and attenuation. In 3D PET though, many of these factors are altered by the acquisition geometry and the procedures needed for 3D reconstruction. This chapter discusses the approaches taken at present to applying the various corrections needed for quantitative 3D PET.
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paper_title: Analytic determination of the pinhole collimator's point-spread function and RMS resolution with penetration

paper_content:

Pinhole collimators are widely used to image small organs and animals. The pinhole response function (PRF) of knife-edge pinhole collimators has been estimated previously using geometric constructions without considering penetration and using "roll-off" models that employ an exponential model for the flux. An exact expression for the PRF on the imaging plane that includes the effect of aperture penetration is derived in this paper by calculating the flux for photons passing through the aperture and those passing through the attenuating material. The PRF is then used to estimate the angular-dependent root-mean-square resolution in the directions parallel and perpendicular to the tilt of the point source. The formulas are then compared with experimental data.
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paper_title: A filtered backprojection algorithm for axial head motion correction in fan-beam SPECT.

paper_content:

The authors present an approximate, but practical, three-dimensional filtered backprojection (FBP) reconstruction algorithm in fan-beam SPECT to correct for axial motion (both translation and rotation). A one-dimensional filter kernel was applied to the projections. It is assumed that the object is rigid and that its axial motion can be characterized by three components: one-dimensional translation and yaw and pitch rotations. It is further assumed that the motions that have occurred during the SPECT acquisition have been determined separately. The determined angular-view-dependent translation/rotation parameters were incorporated into the proposed FBP algorithm to correct for multiple axial head motions. The proposed axial head motion correction algorithm was evaluated using simulated three-dimensional Hoffman brain phantom data. Projections both with axial translation and with axial rotation, and with their combinations were generated. Images of a Hoffman brain phantom reconstructed using the proposed FBP algorithm and the conventional FBP algorithm were compared. Artefacts were observed in images without motion correction, but the artefacts were greatly reduced using the proposed reconstruction algorithm.
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paper_title: Performance evaluation of a pinhole SPECT system for myocardial perfusion imaging of mice.

paper_content:

The increasing use of transgenic mice as models of human physiology and disease has motivated the development of dedicated in vivo imaging systems for anatomic and functional characterization of mice as an adjunct to or a replacement for established ex vivo techniques. We have developed a pinhole single photon emission computed tomography (SPECT) system for high resolution imaging of mice with cardiovascular imaging as the primary application. In this work, we characterize the system performance through phantom studies. The spatial resolution and sensitivity were measured from images of a line source and point source, respectively, and were reported for a range of object-to-pinhole distances and pinhole diameters. Tomographic images of a uniform cylindrical phantom, Defrise phantom, and grid phantom were used to characterize the image uniformity and spatial linearity. The uniform phantom image did not contain any ring or reconstruction artifacts, but blurring in the axial direction was evident in the Defrise phantom images. The grid phantom images demonstrated excellent spatial linearity. A novel phantom modeling perfusion of the left ventricle of a mouse was designed and built with perfusion defects of varying sizes to evaluate the system performance for myocardial perfusion imaging of mice. The defect volumes were measured from the pinhole SPECT images and correlated to the actual defect volumes calculated according to geometric formulas. Linear regression analysis produced a correlation coefficient of r = 0.995 (p < 0.001), demonstrating the feasibility for measurement of perfusion defect size in mice using pinhole SPECT. We have performed phantom studies to characterize the spatial resolution, sensitivity, image uniformity, and spatial linearity of the pinhole SPECT system. Measurement of the perfusion defect size is a valuable phenotypic assessment and will be useful for hypothesis testing in murine models of cardiovascular disease.
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paper_title: Positron-emission tomography

paper_content:

We review positron-emission tomography (PET), which has inherent advantages that avoid the shortcomings of other nuclear medicine imaging methods. PET image reconstruction methods with origins in signal and image processing are discussed, including the potential problems of these methods. A summary of statistical image reconstruction methods, which can yield improved image quality, is also presented.
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paper_title: Image reconstruction algorithm for a rotating slat collimator

paper_content:

A slat collimator in single photon emission computed tomography consists of a set of parallel slats. As the collimator spins, the detector measures a one-dimensional projection data set. A complete data set can be obtained by rotating the detector/collimator assembly around the object (patient) while the collimator spins continuously. The measured projection data are assumed to be weighted planar integrals of the object. This paper describes the development of an approximate three-dimensional image reconstruction algorithm for a rotating/spinning slat collimator. This algorithm is in filtered backprojection form. Computer simulations were performed to verify the effectiveness of the algorithm.
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paper_title: Fast reconstruction of 3-D PET data with accurate statistical modeling

paper_content:

This paper presents the results of combining high sensitivity 3D PET whole-body acquisition followed by fast 2D iterative reconstruction methods based on accurate statistical models. This combination is made possible by Fourier rebinning (FORE), which accurately converts a 3D data set to a set of 2D sinograms. The combination of volume imaging with statistical reconstruction allows improvement of noise-bias trade-offs when image quality is dominated by measurement statistics. The rebinning of the acquired data into a 2D data set reduces the computation time of the reconstruction. For both penalized weighted least squares (PWLS) and ordered-subset EM (OSEM) reconstruction methods, the usefulness of a realistic model of the expected measurement statistics is shown when the data are pre-corrected for attenuation and random and scattered coincidences, as required for the FORE rebinning algorithm. The results presented are based on 3D simulations of whole body scans that include the major statistical effects of PET acquisition and data correction procedures. As the PWLS method requires knowledge of the variance of the projection data, a simple model for the effect of FORE rebinning on data variance is developed.
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paper_title: Novel Properties Of The Fourier Decomposition Of The Sinogram

paper_content:

The double Fourier decomposition of the sinogram is obtained by first taking the Fourier transform of each parallel-ray projection and then calculating the coefficients of a Fourier series with respect to angle for each frequency component of the transformed projections. The values of these coefficients may be plotted on a two-dimensional map whose coordinates are spatial frequency w (continuous) and angular harmonic number n (discrete). For |w| large, the Fourier coefficients on the line n=kw of slope k through the origin of the coefficient space are found to depend strongly on the contributions to the projection data that, for each view, come from a certain distance to the detector plane, where the distance is a linear function of k. The values of these coefficients depend only weakly on contributions from other distances from the detector. The theoretical basis of this property is presented in this paper and a potential application to emission computerized tomography is discussed.
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paper_title: Exact rebinning methods for three-dimensional PET

paper_content:

The high computational cost of data processing in volume PET imaging is still hindering the routine application of this successful technique, especially in the case of dynamic studies. This paper describes two new algorithms based on an exact rebinning equation, which can be applied to accelerate the processing of three-dimensional (3-D) PET data. The first algorithm, FOREPROJ, is a fast-forward projection algorithm that allows calculation of the 3-D attenuation correction factors (ACF's) directly from a two-dimensional (2-D) transmission scan, without first reconstructing the attenuation map and then performing a 3-D forward projection. The use of FOREPROJ speeds up the estimation of the 3-D ACF's by more than a factor five. The second algorithm, FOREX, is a rebinning algorithm that is also more than five times faster, compared to the standard reprojection algorithm (3DRP) and does not suffer from the image distortions generated by the even faster approximate Fourier rebinning (FORE) method at large axial apertures. However, FOREX is probably not required by most existing scanners, as the axial apertures are not large enough to show improvements over FORE with clinical data. Both algorithms have been implemented and applied to data simulated for a scanner with a large axial aperture (30 degrees), and also to data acquired with the ECAT HR and the ECAT HR+ scanners. Results demonstrate the excellent accuracy achieved by these algorithms and the important speedup when the sinogram sizes are powers of two.
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paper_title: LROC analysis of detector-response compensation in SPECT

paper_content:

Localization ROC (LROC) observer studies examined whether detector response compensation (DRC) in ordered-subset, expectation-maximization (OSEM) reconstructions helps in the detection and localization of hot tumors. Simulated gallium (Ga-67) images of the thoracic region were used in the study. The projection data modeled the acquisition of attenuated 93- and 185-keV photons with a medium-energy parallel-hole collimator, but scatter was not modeled. Images were reconstructed with five strategies: 1) OSEM with no DRC; 2) OSEM preceded by restoration filtering; 3) OSEM with iterative DRC; 4) OSEM with an ideal DRC; and 5) filtered backprojection (FBP) with no DRC. All strategies included attenuation correction. There were four LROC studies conducted. In a study using a single tumor activity, the ideal DRC offered the best performance, followed by iterative DRC, restoration filtering, OSEM with no DRC, and FBP. Statistical significance at the 5% level was found between all pairs of strategies except for restoration filtering and OSEM with no DRC. A similar ranking was found for a more realistic study using multiple tumor activities. Additional studies considered the effects of OSEM iteration number and tumor activity on the detection improvement that iterative DRC offered with respect to OSEM with no DRC.
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paper_title: Implementation of an accelerated iterative algorithm for cone-beam SPECT

paper_content:

Describes the implementation of an accelerated iterative reconstruction algorithm for cone-beam projections using a single circular orbit in single-photon-emission computed tomography (SPECT). This algorithm is a modified maximum-likelihood-expectation-maximization (ML-EM) algorithm and several approaches have been used to accelerate the reconstruction process. These approaches include: (i) the use of ordered subsets; (ii) the use of active areas and volumes; and (iii) the storing in memory of the transition vector for a given ray (during the forward projection step). This algorithm, which compensates for collimator geometric sensitivity variation as a function of position and makes uniform attenuation corrections has been evaluated using experimentally acquired phantom data. The results demonstrate a two-orders-of-magnitude decrease of the computational time of this algorithm over the conventional ML-EM algorithm with similar convergence properties.
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paper_title: Evaluating Image Reconstruction Methods for Tumor Detection in 3-Dimensional Whole-Body PET Oncology Imaging

paper_content:

UNLABELLED ::: We compare 3 image reconstruction algorithms for use in 3-dimensional (3D) whole-body PET oncology imaging. We have previously shown that combining Fourier rebinning (FORE) with 2-dimensional (2D) statistical image reconstruction via the ordered-subsets expectation-maximization (OSEM) and attenuation-weighted OSEM (AWOSEM) algorithms demonstrates improvements in image signal-to-noise ratios compared with the commonly used analytic 3D reprojection (3DRP) or FORE+FBP (2D filtered backprojection) reconstruction methods. To assess the impact of these reconstruction methods on detecting and localizing small lesions, we performed a human observer study comparing the different reconstruction methods. The observer study used the same volumetric visualization software tool that is used in clinical practice, instead of a planar viewing mode as is generally used with the standard receiver operating characteristic (ROC) methodology. This change in the human evaluation strategy disallowed the use of a ROC analysis, so instead we compared the fraction of actual targets found and reported (fraction-found) and also investigated the use of an alternative free-response operating characteristic (AFROC) analysis. ::: ::: ::: METHODS ::: We used a non-Monte Carlo technique to generate 50 statistically accurate realizations of 3D whole-body PET data based on an extended mathematic cardiac torso (MCAT) phantom and with noise levels typical of clinical scans performed on a PET scanner. To each realization, we added 7 randomly located 1-cm-diameter lesions (targets) whose contrasts were varied to sample the range of detectability. These targets were inserted in 3 organs of interest: lungs, liver, and soft tissues. The images were reconstructed with 3 reconstruction strategies (FORE+OSEM, FORE+AWOSEM, and FORE+FBP). Five human observers reported (localized and rated) 7 targets within each volume image. An observer's performance accuracy with each algorithm was measured, as a function of the lesion contrast and organ type, by the fraction of those targets reported and by the area below the AFROC curve. This AFROC curve plots the fraction of reported targets at each rating threshold against the fraction of cases with (> or =1) similarly rated false reports. ::: ::: ::: RESULTS ::: Images reconstructed with FORE+AWOSEM yielded the best overall target detection as compared with FORE+FBP and FORE+OSEM, although these differences in detectability were region specific. The FORE+FBP and FORE+AWOSEM algorithms had similar performances for liver targets. The FORE+OSEM algorithm performed significantly worse at target detection, especially in the liver. We speculate that this is the result of using an incorrect statistical model for OSEM and that the incorporation of attenuation weighting in AWOSEM largely compensates for this model inaccuracy. These results were consistent for both the fraction of actual targets found and the AFROC analysis. ::: ::: ::: CONCLUSION ::: We demonstrated the efficacy of performing observer detection studies using the same visualization tools as those used in clinical PET oncology imaging. These studies demonstrated that the FORE+AWOSEM algorithm led to the best overall detection and localization performance for 1-cm-diameter targets compared with the FORE+OSEM and FORE+FBP algorithms.
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paper_title: A factorization method for the 3D X-ray transform

paper_content:

We consider the inversion of the three-dimensional (3D) X-ray transform with a limited data set containing the line integrals which have two intersections with the lateral surface of a cylindrical detector. The usual solution to this problem is based on 3D filtered-backprojection, but this method is slow. This paper presents a new algorithm which factors the 3D reconstruction problem into a set of independent 2D radon transforms for a stack of parallel slices. Each slice is then reconstructed using standard 2D filtered-backprojection. The algorithm is based on the application of the stationary-phase approximation to the 2D Fourier transform of the data, and is an extension to three dimensions of the frequency-distance relation derived by Edholm et al.(1986) for the 2D radon transform. Error estimates are also obtained.
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paper_title: A filtered backprojection algorithm for axial head motion correction in fan-beam SPECT.

paper_content:

The authors present an approximate, but practical, three-dimensional filtered backprojection (FBP) reconstruction algorithm in fan-beam SPECT to correct for axial motion (both translation and rotation). A one-dimensional filter kernel was applied to the projections. It is assumed that the object is rigid and that its axial motion can be characterized by three components: one-dimensional translation and yaw and pitch rotations. It is further assumed that the motions that have occurred during the SPECT acquisition have been determined separately. The determined angular-view-dependent translation/rotation parameters were incorporated into the proposed FBP algorithm to correct for multiple axial head motions. The proposed axial head motion correction algorithm was evaluated using simulated three-dimensional Hoffman brain phantom data. Projections both with axial translation and with axial rotation, and with their combinations were generated. Images of a Hoffman brain phantom reconstructed using the proposed FBP algorithm and the conventional FBP algorithm were compared. Artefacts were observed in images without motion correction, but the artefacts were greatly reduced using the proposed reconstruction algorithm.
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paper_title: A fast rebinning algorithm for 3D positron emission tomography using John's equation

paper_content:

Volume imaging in positron emission tomography (PET) requires the inversion of the three-dimensional (3D) x-ray transform. The usual solution to this problem is based on 3D filtered-backprojection (FBP), but is slow. Alternative methods have been proposed which factor the 3D data into independent 2D data sets corresponding to the 2D Radon transforms of a stack of parallel slices. Each slice is then reconstructed using 2D FBP. These so-called rebinning methods are numerically efficient but are approximate. In this paper a new exact rebinning method is derived by exploiting the fact that the 3D x-ray transform of a function is the solution to the second-order partial differential equation first studied by John. The method is proposed for two sampling schemes, one corresponding to a pair of infinite plane detectors and another one corresponding to a cylindrical multi-ring PET scanner. The new FORE-J algorithm has been implemented for this latter geometry and was compared with the approximate Fourier rebinning algorithm FORE and with another exact rebinning algorithm, FOREX. Results with simulated data demonstrate a significant improvement in accuracy compared to FORE, while the reconstruction time is doubled. Compared to FOREX, the FORE-J algorithm is slightly less accurate but more than three times faster.
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paper_title: Fourier correction for spatially variant collimator blurring in SPECT

paper_content:

In single-photon emission computed tomography (SPECT), projection data are acquired by rotating the photon detector around a patient, either in a circular orbit or in a noncircular orbit. The projection data of the desired spatial distribution of emission activity is blurred by the point-response function of the collimator that is used to define the range of directions of gamma-ray photons reaching the detector. The point-response function of the collimator is not spatially stationary, but depends on the distance from the collimator to the point. Conventional methods for deblurring collimator projection data are based on approximating the actual distance-dependent point-response function by a spatially invariant blurring function, so that deconvolution methods can be applied independently to the data at each angle of view. A method is described here for distance-dependent preprocessing of SPECT projection data prior to image reconstruction. Based on the special distance-dependent characteristics of the Fourier coefficients of the sinogram, a spatially variant inverse filter can be developed to process the projection data in all views simultaneously. The algorithm is first derived from Fourier analysis of the projection data from the circular orbit geometry. For circular orbit projection data, experimental results from both simulated data and real phantom data indicate the potential of this method. It is shown that the spatial filtering method can be extended to the projection data from the noncircular orbit geometry. Experiments on simulated projection data from an elliptical orbit demonstrate correction of the spatially variant blurring and distortion in the reconstructed image caused by the noncircular orbit geometry. >
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paper_title: Reducing the effect of non-stationary resolution on activity quantitation with the frequency distance relationship in SPECT

paper_content:

The determination of quantitative parameters such as the maximum and total counts in a volume are influenced by the partial volume effect. The magnitude of this effect varies with the non-stationary spatial resolution inherent in SPECT imaging compared to the size and shape of the object, and the relative concentration of the object to its background. The objective of this investigation was to determine if the FDR (Frequency Distance Relationship) restoration filtering can reduce the impact of distance dependent spatial resolution on the quantitation of activity. An analytical projector which incorporates attenuation and distance dependent blurring was used to simulate small hot spherical objects in an cylindrical attenuator as imaged with a LEUHR collimator. FDR restoration filtering regularized using different Gaussian and parametric Wiener filters, was employed after attenuation correction with Bellini's method. Projections were also processed using Bellini's attenuation correction method followed by filtered backprojection and 3D Butterworth filtering with different cut-off frequencies. CCR's (Center Count Ratios) and TCR's (Total Count Ratios) were determined as the observed counts over true counts. Results show that after FDR restoration the CCR and TCR become approximately position invariant. However, when regularizing the FDR inverse filter with a Gaussian function, CCR's become highly unstable as the standard deviation (/spl sigma/) decreased below that at the center of rotation. The use of the Gaussian and parametric Wiener filters to regularize FDR filtering introduce noise in the CCR's but improve recovery of TCR's over that of the center of rotation.
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paper_title: A combined half-cone beam and parallel hole collimation system for SPECT brain imaging

paper_content:

The authors' research group has recently examined new types of collimator designs with the goal of improving sensitivity and lesion detection for head imaging. One of these collimator designs is a half-cone beam collimator. However, the sensitivity is reduced as objects are removed from the focal line and it does not satisfy Tuy's sufficiency condition. Parallel hole collimation does not have this problem with axial blurring, however, values for sensitivity are not as large as for half-cone collimators. A potential configuration is to use one parallel hole collimator in conjunction with two half-cone beam collimators on a triple camera single photon emission computed tomography (SPECT) system. This might produce results that represent the best of both collimation systems. The authors acquired projection data with a Defrise disk phantom filled with Tc-99m. A half-cone beam collimator with a focal length of 50 cm was placed on one head of a triple camera SPECT system. A low energy super high resolution (LESR) parallel hole collimator was placed on a second head. Different projection data sets were combined to model acquisition in the three headed gamma camera with three half-cone beam collimators, two half-cone beam and one parallel beam collimators, one half-cone beam and two parallel beam collimators and three parallel beam collimators. Image reconstruction used a modified maximum likelihood maximization-expectation (ML-EM) algorithm. For the reconstruction with three half-cone beam collimators, the authors observe axial blurring. This is largely reduced with two half-cone beam and one parallel beam collimators. Graphs of the image profiles demonstrate that the blurring along the axial direction is decreased with the addition of parallel hole collimators.
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paper_title: Iterative reconstruction with attenuation compensation from cone-beam projections acquired via nonplanar orbits

paper_content:

Single photon emission computed tomography (SPECT) imaging with cone-beam collimators provides improved sensitivity and spatial resolution for imaging small objects with large field-of-view detectors. It is known that Tuy's (1983) cone-beam data sufficiency condition must be met in order to obtain artifact-free reconstructions. Even though Tuy's condition was derived for attenuation-free situation, the authors hypothesize that an artifact-free reconstruction can be obtained even if the cone-beam data are attenuated, provided the imaging orbit satisfies Tuy's condition and the exact attenuation map. In the authors' studies, emission data are acquired using nonplanar orbit such as circle-and-line orbit to acquire sufficient data for cone-beam tomographic reconstruction. An iterative conjugate gradient reconstruction algorithm is used to reconstruct projection data with a pre-acquired attenuation map. Quantitative accuracy of the attenuation corrected emission reconstruction is significantly improved. The authors observe that the cone-beam iterative algorithms are very sensitive to the projection and backprojection models, because each voxel is sampled anisotropically. Sampling artifacts (i.e., aliasing artifacts) sometimes are very severe for non-planar orbit data acquisition geometries. Better voxel models and sampling geometries are required in further investigations.
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paper_title: Use of recovery coefficients as a test of system linearity of response in positron emission tomography (PET).

paper_content:

Quantitative determination of activity concentration has always been regarded as a goal of PET scanner design. Therefore, a linear response with respect to activity concentration is essential. To prove system linearity of response a test is proposed based on the determination of recovery coefficients (RC) for different object geometries. If all degrading effects, especially scatter and attenuation, are corrected to a sufficient degree of accuracy, the system behaves linearly resulting in a theoretical relationship between hot (HSRC) and cold spot recovery coefficients (CSRC). Therefore this relationship, applied to appropriate phantom data containing hot or cold spheres, combined with the constraints on asymptotic behaviour of RC can be used as a test of system linearity of response. After applying the system's standard corrections RC were determined using both 2D and 3D data acquisition modes. For both 2D and 3D data, HSRC directly measured showed an excellent agreement with the HSRC calculated from the CSRC. The analysis was extended to radionuclides of different positron ranges, to the effect of data smoothing, and to that of iterative reconstruction. The agreement between the RC determined under varying border conditions demonstrates the effectiveness of the corrections used in constituting system linearity. Therefore, the method proposed is used as a test of system linearity of response.
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paper_title: Spatial Resolution Properties of Penalized-Likelihood Image Reconstruction: Space-Invariant Tomographs

paper_content:

This paper examines the spatial resolution properties of penalized-likelihood image reconstruction methods by analyzing the local impulse response. The analysis shows that standard regularization penalties induce space-variant local impulse response functions, even for space-invariant tomographic systems. Paradoxically, for emission image reconstruction, the local resolution is generally poorest in high-count regions. We show that the linearized local impulse response induced by quadratic roughness penalties depends on the object only through its projections. This analysis leads naturally to a modified regularization penalty that yields reconstructed images with nearly uniform resolution. The modified penalty also provides a very practical method for choosing the regularization parameter to obtain a specified resolution in images reconstructed by penalized-likelihood methods.
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paper_title: Noise properties of the EM algorithm: II. Monte Carlo simulations.

paper_content:

In an earlier paper we derived a theoretical formulation for estimating the statistical properties of images reconstructed using the iterative ML-EM algorithm. To gain insight into this complex problem, two levels of approximation were considered in the theory. These techniques revealed the dependence of the variance and covariance of the reconstructed image noise on the source distribution, imaging system transfer function, and iteration number. In this paper a Monte Carlo approach was taken to study the noise properties of the ML-EM algorithm and to test the predictions of the theory. The study also served to evaluate the approximations used in the theory. Simulated data from phantoms were used in the Monte Carlo experiments. The ML-EM statistical properties were calculated from sample averages of a large number of images with different noise realizations. The agreement between the more exact form of the theoretical formulation and the Monte Carlo formulation was better than 10% in most cases examined, and for many situations the agreement was within the expected error of the Monte Carlo experiments. Results from the studies provide valuable information about the noise characteristics of ML-EM reconstructed images. Furthermore, the studies demonstrate the power of the theoretical and Monte Carlo approaches for investigating noise properties of statistical reconstruction algorithms.
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paper_title: Radiological Imaging The Theory Of Image Formation Detection And Processing

paper_content:

Thank you for reading radiological imaging the theory of image formation detection and processing. Maybe you have knowledge that, people have search numerous times for their favorite books like this radiological imaging the theory of image formation detection and processing, but end up in malicious downloads. Rather than reading a good book with a cup of coffee in the afternoon, instead they are facing with some malicious virus inside their laptop.
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paper_title: Noise Reduction in Oncology FDG PET Images by Iterative Reconstruction: A Quantitative Assessment

paper_content:

Tumor detection depends on the contrast between tumor activity and background activity and on the image noise in these 2 regions. The lower the image noise, the easier the tumor detection. Tumor activity contrast is determined by physiology. Noise, however, is affected by many factors, including the choice of reconstruction algorithm. Previous simulation and phantom measurements indicated that the ordered-subset expectation maximization (OSEM) algorithm may produce less noisy images than does the usual filtered backprojection (FBP) method, at equivalent resolution. To see if this prediction would hold in actual clinical situations, we quantified noise in clinical images reconstructed with both OSEM and FBP. Methods: Three patients (2 with colon cancer, 1 with breast cancer) were imaged with FDG PET using a gated replicate technique that permitted accurate measurement of noise at each pixel. Each static image was acquired as a gated image sequence, using a pulse generator with a 1-s period, yielding 40 replicate images over the 10- to 15-min imaging time. The images were or were not precorrected for attenuation and were reconstructed with both FBP and OSEM at comparable resolution. From these data, images of pixel mean, SD, and signal-to-noise ratio (S/N) could be produced, reflecting only noise caused by the statistical fluctuations in the emission process. Results: Noise did not vary greatly over each FBP image, even when image intensity varied greatly from one region to the next, causing S/N to be worse in low-activity regions than in high-activity regions. In contrast, OSEM had high noise in hot regions and low noise in cold regions. OSEM had a much better S/N than did FBP in cold regions of the image, such as the lungs (in the attenuation-corrected images), where improvements in S/N averaged 160%. Improvements with OSEM were less dramatic in hotter areas such as the liver (averaging 25% improvement in the attenuation-corrected images). In very hot tumors, FBP actually produced higher S/Ns than did OSEM. Conclusion: We conclude that OSEM reconstruction can significantly reduce image noise, especially in relatively low-count regions. OSEM reconstruction failed to improve S/N in very hot tumors, in which S/N may already be adequate for tumor detection.
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paper_title: Quantitative SPECT imaging: A review and recommendations by the focus committee of the society of nuclear medicine Computer and Instrumentation Council

paper_content:

This article is a review of the physics principles, instrumentation and reconstruction methods behind SPECT imaging. Particular attention is paid to the mechanisms that can significantly affect the accuracy of a SPECT image. We describe instrumentation advances and reconstruction methods used to correct images to improve image quality and produce quantitative images. The clinical importance of improved image quality and quantitation are also reviewed.
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paper_title: Statistical Approaches in Quantitative Positron Emission Tomography

paper_content:

Positron emission tomography is a medical imaging modality for producing 3D images of the spatial distribution of biochemical tracers within the human body. The images are reconstructed from data formed through detection of radiation resulting from the emission of positrons from radioisotopes tagged onto the tracer of interest. These measurements are approximate line integrals from which the image can be reconstructed using analytical inversion formulae. However these direct methods do not allow accurate modeling either of the detector system or of the inherent statistical fluctuations in the data. Here we review recent progress in developing statistical approaches to image estimation that can overcome these limitations. We describe the various components of the physical model and review different formulations of the inverse problem. The wide range of numerical procedures for solving these problems are then reviewed. Finally, we describe recent work aimed at quantifying the quality of the resulting images, both in terms of classical measures of estimator bias and variance, and also using measures that are of more direct clinical relevance.
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paper_title: Recovery correction for quantitation in emission tomography: a feasibility study

paper_content:

In emission tomography, the spread of regional tracer uptake to surrounding areas caused by limited spatial resolution of the tomograph must be taken into account when quantitating activity concentrations in vivo. Assuming linearity and stationarity, the relationship between imaged activity concentration and true activity concentration is only dependent on the geometric relationship between the limited spatial resolution of the tomograph in all three dimensions and the three-dimensional size and shape of the object. In particular it is independent of the type of object studied. This concept is characterized by the term "recovery coefficient". Recovery effects can be corrected for by recovery coefficients determined in a calibration measurement for lesions of simple geometrical shape. This method works on anatomical structures that can be approximated to simple geometrical objects. The aim of this study was to investigate whether recovery correction of appropriate structures is feasible in a clinical setting. Measurements were done on a positron emission tomography (PET) scanner in the 2D and 3D acquisition mode and on an analogue and digital single-photon emission tomography (SPET) system using commercially available software for image reconstruction and correction of absorption and scatter effects. The results of hot spot and cold spot phantom measurements were compared to validate the assumed conditions of linearity and stationarity. It can be concluded that a recovery correction is feasible for PET scanners down to lesions measuring about 1.5xFWHM in size, whereas with simple correction schemes, which are widely available, an object-independent recovery correction for SPET cannot be performed. This result can be attributed to imperfections in the commercially available methods for attenuation and scatter correction in SPET, which are only approximate.
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paper_title: Resolution and noise properties of MAP reconstruction for fully 3-D PET

paper_content:

Derives approximate analytical expressions for the local impulse response and covariance of images reconstructed from fully three-dimensional (3-D) positron emission tomography (PET) data using maximum a posteriori (MAP) estimation. These expressions explicitly account for the spatially variant detector response and sensitivity of a 3-D tomograph. The resulting spatially variant impulse response and covariance are computed using 3-D Fourier transforms. A truncated Gaussian distribution is used to account for the effect on the variance of the nonnegativity constraint used in MAP reconstruction. Using Monte Carlo simulations and phantom data from the microPET small animal scanner, the authors show that the approximations provide reasonably accurate estimates of contrast recovery and covariance of MAP reconstruction for priors with quadratic energy functions. They also describe how these analytical results can be used to achieve near-uniform contrast recovery throughout the reconstructed volume.
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paper_title: Regularization for uniform spatial resolution properties in penalized-likelihood image reconstruction

paper_content:

Traditional space-invariant regularization methods in tomographic image reconstruction using penalized-likelihood estimators produce images with nonuniform spatial resolution properties. The local point spread functions that quantify the smoothing properties of such estimators are space-variant, asymmetric, and object-dependent even for space-invariant imaging systems. We propose a new quadratic regularization scheme for tomographic imaging systems that yields increased spatial uniformity and is motivated by the least-squares fitting of a parameterized local impulse response to a desired global response. We have developed computationally efficient methods for PET systems with shift-invariant geometric responses. We demonstrate the increased spatial uniformity of this new method versus conventional quadratic regularization schemes in simulated PET thorax scans.
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paper_title: The convergence of object dependent resolution in maximum likelihood based tomographic image reconstruction.

paper_content:

Study of the maximum likelihood by EM algorithm (ML) with a reconstruction kernel equal to the intrinsic detector resolution and sieve regularization has demonstrated that any image improvements over filtered backprojection (FBP) are a function of image resolution. Comparing different reconstruction algorithms potentially requires measuring and matching the image resolution. Since there are no standard methods for describing the resolution of images from a nonlinear algorithm such as ML, we have defined measures of effective local Gaussian resolution (ELGR) and effective global Gaussian resolution (EGGR) and examined their behaviour in FBP images and in ML images using two different measurement techniques. For FBP these two resolution measures are equal and exhibit the standard convolution behaviour of linear systems. For ML, the FWHM of the ELGR monotonically increased with decreasing Gaussian object size due to slower convergence rates for smaller objects. For the simple simulated phantom used, this resolution dependence is independent of object position. With increasing object size, number of iterations and sieve size the object size dependence of the ELGR decreased. The FWHM of the EGGR converged after approximately 200 iterations, masking the fact that the ELGR for small objects was far from convergence. When FBP is compared to a nonlinear algorithm such as ML, it is recommended that at least the EGGR be matched; for ML this requires more than the number of iterations (e.g., < 100) that are typically run to minimize the mean square error or to satisfy a feasibility or similar stopping criterion. For many tasks, matching the EGGR of ML to FBP images may be insufficient and >> 200 iterations may be needed, particularly for small objects in the ML image because their ELGR has not yet converged.
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paper_title: Noise properties of the EM algorithm: I. Theory.

paper_content:

The expectation-maximization (EM) algorithm is an important tool for maximum-likelihood (ML) estimation and image reconstruction, especially in medical imaging. It is a non-linear iterative algorithm that attempts to find the ML estimate of the object that produced a data set. The convergence of the algorithm and other deterministic properties are well established, but relatively little is known about how noise in the data influences noise in the final reconstructed image. In this paper we present a detailed treatment of these statistical properties. The specific application we have in mind is image reconstruction in emission tomography, but the results are valid for any application of the EM algorithm in which the data set can be described by Poisson statistics. We show that the probability density function for the grey level at a pixel in the image is well approximated by a log-normal law. An expression is derived for the variance of the grey level and for pixel-to-pixel covariance. The variance increases rapidly with iteration number at first, but eventually saturates as the ML estimate is approached. Moreover, the variance at any iteration number has a factor proportional to the square of the mean image (though other factors may also depend on the mean image), so a map of the standard deviation resembles the object itself. Thus low-intensity regions of the image tend to have low noise. By contrast, linear reconstruction methods, such as filtered back-projection in tomography, show a much more global noise pattern, with high-intensity regions of the object contributing to noise at rather distant low-intensity regions. The theoretical results of this paper depend on two approximations, but in the second paper in this series we demonstrate through Monte Carlo simulation that the approximations are justified over a wide range of conditions in emission tomography. The theory can, therefore, be used as a basis for calculation of objective figures of merit for image quality.
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Title: Overview of methods for image reconstruction from projections in emission computed tomography
Section 1: INTRODUCTION
Description 1: Introduce the significance of emission computed tomography (ECT), its major subfields (PET and SPECT), and the multidisciplinary nature of the algorithms used for image reconstruction from projections in ECT.

Section 2: BRIEF OUTLINE OF DATA COLLECTION IN EMISSION TOMOGRAPHY
Description 2: Provide an overview of the data collection techniques in PET and SPECT, discussing the detection of gamma rays, the use of labeled molecules, and typical sources of errors in the data.

Section 3: PROBLEM FORMULATION FOR EMISSION TOMOGRAPHY
Description 3: Formulate the problem of image reconstruction in ECT, explaining the linear, spatially variant (LSV) model and the inclusion of components such as scatter and random coincidences in PET data.

Section 4: IMAGE RECONSTRUCTION APPROACH BASED ON D-C MODEL
Description 4: Describe the approach and theoretical foundations for solving the direct continuous (D-C) model of image reconstruction, including mathematical formulations and computational challenges.

Section 5: IMAGE RECONSTRUCTION APPROACH BASED ON D-D MODEL
Description 5: Present the discrete-discrete (D-D) model approach, including the representation of the function by basis functions, estimation techniques for model parameters, and iterative algorithms.

Section 6: IMAGE RECONSTRUCTION APPROACH BASED ON C-C MODEL
Description 6: Discuss the continuous-continuous (C-C) model approach, simplified integration models, Fourier transform techniques, and numeric evaluation of inversion formulas.

Section 7: INVERSION FOR C-C MODEL WITH ATTENUATION
Description 7: Explain the adjustments required in the C-C model to account for the attenuation of photons, and the challenges in deriving and applying inversion formulas for SPECT and PET with attenuation considerations.

Section 8: PROCESSING OF TRANSMISSION DATA TO OBTAIN MAP OF ATTENUATION COEFFICIENT
Description 8: Describe methods for estimating the spatial distribution of attenuation coefficients using transmission data and its importance in processing emission data.

Section 9: PREPROCESSING OF EMISSION DATA TO COMPENSATE FOR ATTENUATION
Description 9: Detail the preprocessing steps required to adjust PET data for attenuation and normalization factors before reconstruction algorithms are applied.

Section 10: FOURIER DEBLURRING OF DATA IN SPECT, AND FOURIER REBINNING OF DATA IN PET
Description 10: Introduce preprocessing techniques to address the collimator response in SPECT and describe Fourier rebinning procedures in PET to reduce computational load.

Section 11: DISCUSSION-LINEARITY VERSUS OPTIMALITY
Description 11: Contrast the linearity of C-C-based methods and the statistical optimality of non-linear approaches, discussing their noise properties, spatial resolution, and quantitative capabilities in ECT.

Section 12: CONCLUSION
Description 12: Summarize the key points discussed in the paper, the variety of image reconstruction methods in ECT, and the challenges posed by computational resources and the nature of different applications.