A Survey of Signal Processing Problems and Tools in Holographic Three-Dimensional Television
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paper_title: A Large Scale Interactive Holographic Display

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Our work focuses on the development of interactive multi-user holographic displays that allow freely moving naked eye participants to share a three dimensional scene with fully continuous, observer independent, parallax. Our approach is based on a scalable design that exploits a specially arranged array of projectors and a holographic screen. The feasibility of such an approach has already been demonstrated with a working hardware and software 7.4M pixel prototype driven at 10-15Hz by two DVI streams. In this short contribution, we illustrate our progress, presenting a 50M pixel display prototype driven by a dedicated cluster hosting multiple consumer level graphic cards.
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paper_title: Interactive three-dimensional holographic displays: seeing the future in depth

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Computer graphics is confined chiefly to flat images. Images may look three-dimensional (3D), and sometimes create the illusion of 3D when displayed, for example, on a stereoscopic display [16, 13, 12]. Nevertheless, when viewing an image on most display systems, the human visual system (HVS) sees a flat plane of pixels. Volumetric displays can create a 3D computer graphics image, but fail to provide many visual depth cues (e.g. shading texture gradients) and cannot provide the powerful depth cue of overlap (occlusion). Discrete parallax displays (such as lenticular displays) promise to create 3D images with all of the depth cues, but are limited by achievable resolution. Only a real-time electronic holographic ("holovideo") display [11, 6, 8, 7, 9, 21, 22, 20, 2] can create a truly 3D computer graphics image with all of the depth cues (motion parallax, ocular accommodation, occlusion, etc.) and resolution sufficient to provide extreme realism [13]. Holovideo displays promise to enhance numerous applications in the creation and manipulation of information, including telepresence, education, medical imaging, interactive design and scientific visualization.The technology of electronic interactive three-dimensional holographic displays is in its first decade. Though fancied in popular science fiction, only recently have researchers created the first real holovideo systems by confronting the two basic requirements of electronic holography: computational speed and high-bandwidth modulation of visible light. This article describes the approaches used to address these problems, as well as emerging technologies and techniques that provide firm footing for the development of practical holovideo.
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paper_title: Holography, 1948-1971

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Lecture delivered by Dennis Gabor on the occasion of his receiving the 1971 Nobel Prize for Physics, Stockholm, Sweden, December 13 1971. Copyright (c) The Nobel Foundation 1972.
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paper_title: Speckle reduction in coherent information processing

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Speckle reduction techniques have been developed as one of the most active research fields in coherent optical information processing, together with the investigations of speckle statistics and coherence theory. The principles of speckle reduction are classified to five categories: 1) control of spatial coherence, 2) control of temporal coherence, 3) spatial sampling, 4) spatial averaging, and 5) digital image processing. This paper surveys the research works in the field of speckle reduction techniques involving four categories 1)-4) which have been conducted in the past 30 years.
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paper_title: Real-Time Display Of 3-D Computed Holograms By Scanning The Image Of An Acousto-Optic Modulator

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The invention of holography has sparked hopes for a three-dimensional electronic imaging systems analogous to television. Unfortunately, the extraordinary spatial detail of ordinary holographic recordings requires unattainable bandwidth and display resolution for three-dimensional moving imagery, effectively preventing their commercial development. However, the essential bandwidth of holographic images can be reduced enough to permit their transmission through fiber optic or coaxial cable, and the required resolution or space-bandwidth product of the display can be obtained by raster scanning the image of a commercially available acousto-optic modulator. No film recording or other photographic intermediate step is necessary as the projected modulator image is viewed directly. The design and construction of a working demonstration of the principles involved is also presented along with a discussion of engineering considerations in the system design. Finally, the theoretical and practical limitations of the system are addressed in the context of extending the system to real-time transmission of moving holograms synthesized from views of real and computer-generated three-dimensional scenes.
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paper_title: Color images with the MIT holographic video display

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The MIT holographic video display can be converted to color by illuminating the 3 acoustic channels of the acousto-optic modulator (AOM) with laser light corresponding to the red, green, and blue parts of the visible spectrum. The wavelengths selected are 633 nm (red), 532 nm (green), and 442 nm (blue). Since the AOM is operated in the Bragg regime, each wavelength is diffracted over a different angular range, resulting in a final image in which the three color primaries do not overlap. This situation can be corrected by shifting the diffracted spatial frequencies with an holographic optical element (HOE). This HOE consisting of a single grating is placed right after the AOM in the optical setup. Calculation of the required spatial frequency for the HOE must take into account the optical activity of the TeO2 crystal used in the AOM. The HOE introduces distortions in the final image, but these are so small as to be visually negligible. The final images are of a good quality and exhibit excellent color registration. The horizontal view zone, however, diminishes for the shorter wavelengths.
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paper_title: Dynamic holographic 3-D image projection

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The display of dynamic holographic images is possible by computing the hologram of objects in a three-dimensional scene and then transcribing the two-dimensional digital hologram onto a digital micromirror system illuminated with coherent light. Proof-of-principle instruments that reconstruct real and virtual images are described. The underlying process, its characteristics, limitations and utility are discussed.
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paper_title: Total internal reflection holographic gratings recorded in polymer-dispersed liquid crystals

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We report diffraction gratings recording in polymer dispersed liquid crystals (PDLC) material with a beam suffering total internal reflection from the sample air surface. The real time diffraction efficiency kinetics as well as the polarization, voltage behavior and angular selectivity of the gratings are studied at different reconstruction geometries. An increase of three times of the diffraction efficiency in case of applied voltage is observed for one of these geometries in opposite to typical PDLC switchable gratings. Such dependence appears in most of the geometries and is connected to the specific formation of two gratings by the Stetson scheme.
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paper_title: Hologram reconstruction using a digital micromirror device

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The aim of this work is an advancement of digital holographic interferometry by using a digital micromirror device (DMD) for real-time display of interferometric fringes. The theory describing the formation of the diffracted image produced by a coherently illuminated DMD is pre- sented. Digital holograms of simple structures are calculated assuming plane and spherical reference waves, respectively. These synthetic ho- lograms are used to control the micromirrors of the DMD, which now acts as a reflective hologram. Experimental holographic reconstructions of synthetic holograms prove the feasibility of DMDs in principle for this aim. Further experimental results show the reconstructed interferogram of fields recorded by digital holography and superposed numerically. This opens a new range of applications in digital holographic interferom- etry. © 2001 Society of Photo-Optical Instrumentation Engineers.
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paper_title: Optimization of hologram computation for real-time display

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Several methods of increasing the speed and simplicity of the computation of off-axis transmission holograms are presented, with applications to the real-time display of holographic images. A bipolar intensity approach enables a linear summation of interference fringes, a factor of two speed increase, and the elimination of image noise caused by object self- interference. An order of magnitude speed increase is obtained through the use of precomputed look-up tables containing a large array of elemental interference patterns corresponding to point source contributions from each of the possible locations in image space. Results achieved using a data-parallel supercomputer to compute horizontal-parallax- only holographic patterns containing 6 megasamples indicate that an image comprised of 10,000 points with arbitrary brightness (grayscale) can be computed in under one second.
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paper_title: Moire fringe reduction by optical filters in integral three-dimensional imaging on a color flat-panel display.

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We propose a method to reduce the color moire fringes that are attributable to the structure of a color flat-panel display in integral three-dimensional imaging. The method uses two types of optical low-pass filter, diffuser and defocus. The effectiveness of the method was confirmed in an experiment. We describe a way to design these filters with moire's residual energy and video signal energy as indices and demonstrate the validity of the model, which combines two filters to reduce moire fringes.
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paper_title: Three-dimensional integral imaging with large depth of focus by use of real and virtual image fields.

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We present an integral imaging method to enhance the depth of a three-dimensional image by displaying it throughout real and virtual image fields. When the product of depth and resolution square of the displayed three-dimensional image is used as a figure of merit in integral imaging systems, our method can maximize this merit especially when three-dimensional images with large depth of focus are displayed. The feasibility of our method is experimentally demonstrated by generation of elemental images by a computer.
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paper_title: Shannon number and information capacity of three-dimensional integral imaging

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Integral imaging systems performance has been previously investigated with regard to different parameters such as lateral resolution, field of view, and depth of view. Those parameters are linked to one another, and, since the information capacity of an integral imaging system is finite, there are always trade-offs among them. We use the Shannon number and information capacity limit as figures of merit of integral imaging systems. The Shannon number and information capacity provide compact assessments of the system and are useful for analysis and design. The limitations on the Shannon number and the information capacity of an integral imaging system are determined by the recording and display media.
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paper_title: Large depth-of-focus time-multiplexed three-dimensional integral imaging by use of lenslets with nonuniform focal lengths and aperture sizes

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Conventional integral-imaging systems utilize lenslet arrays with fixed focal lengths and aperture sizes. We propose a time-multiplexed integral-imaging method that enhances both the depth of focus and the resolution of a three-dimensional image by displaying it with an array of lenslets with different focal lengths and aperture sizes. The nonuniform lenslet parameters (focal lengths and aperture sizes) for our method are calculated. Our theoretical analysis indicates that significant improvements in both depth of focus and resolution can be obtained with the proposed technique.
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paper_title: Calculation of holograms from elemental images captured by integral photography.

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We describe a method in which holograms can be produced by calculation from images captured by integral photography (IP). We present a basic algorithm obtained by simulating IP reconstruction, in which conditions are set so as not to cause aliasing in the holograms after the calculations. To reduce the calculation load, we also propose a way to limit the range of calculation considering the distribution of light and a way to shift the optical field on the exit plane of microlenses in a lens array. Finally, by optical experiments, we confirm that three-dimensional images can be reconstructed from holograms calculated from an integral photograph of a real object captured with an IP camera.
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paper_title: 3D image sensing and reconstruction with time-division multiplexed computational integral imaging (CII)

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We present a method to compute high-resolution three-dimensional (3-D) images based on integral imaging. In order to overcome optical limitation on the spatial resolution of reconstructed 3-D image, we suggest performing time-division multiplexing by which a sequence of II's is captured with a slightly different location of the lenslet array. The II image sequence obtained by the detector array is processed digitally with super-resolution reconstruction algorithms to obtain a reconstructed image, appropriate to a viewing direction, which has a spatial resolution beyond the optical limitation.
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paper_title: Integral imaging with multiple image planes using a uniaxial crystal plate.

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Integral imaging has been attracting much attention recently for its several advantages such as full parallax, continuous view-points, and real-time full-color operation. However, the thickness of the displayed three-dimensional image is limited to relatively small value due to the degradation of the image resolution. In this paper, we propose a method to provide observers with enhanced perception of the depth without severe resolution degradation by the use of the birefringence of a uniaxial crystal plate. The proposed integral imaging system can display images integrated around three central depth planes by dynamically altering the polarization and controlling both elemental images and dynamic slit array mask accordingly. We explain the principle of the proposed method and verify it experimentally.
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paper_title: Multidimensional optical sensor and imaging system.

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We describe a multidimensional optical sensor and imaging system (MOSIS). Using a time-multiplexing, polarimetric, and multispectral imaging system, we are able to reconstruct a fully integrated multidimensional scene. Image fusion is used to integrate the multidimensional images. The fused image contains more information than the single two-dimensional and three-dimensional (3D) images. The multidimensional imaging system utilizes polarimetric imaging, multispectral imaging, 3D integral imaging with time and space multiplexing, and 3D image-fusion techniques to reconstruct the multidimensionally integrated scene. Optical experiments and computer simulations are presented.
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paper_title: Compression of ray information in three-dimensional integral imaging

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We present a method to compress ray information in three- dimensional 3-D integral imaging II using the Karhunen-Loeve trans- form KLT. Elemental images in II are highly correlated because they are picked-up by numerous lenslets, thus, the KLT can compress an integral image more effectively than 2-D ordinary images. In the hybrid coding scheme, the KLT is distinctly applied to disjoint subsets which are partitioned by a vector quantization VQ. In the optical experiments, we show that 3-D image quality degradation is negligible if approximately 10% of eigenvectors with the largest eigenvalues are used. We also evaluate the quality of uncompressed images by signal-to-noise ratio SNR and peak-to-peak signal-to-noise ratio PSNR. The presented coding scheme is shown to be better than JPEG Joint Photographic Experts Group for the lower bit rates in the II compression. Optical and numerical experiments are presented. © 2005 Society of Photo-Optical Instru-
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paper_title: Viewing-angle-enhanced integral imaging using lens switching

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The integral photography which is also called integral imaging (II) is an attractive autostereoscopic display method for its many advantages such as continuous viewpoints and no need for any viewing aids. In spite of many advantages of II, the narrow viewing angle has been a bottleneck of it. In this paper, we propose a method to enhance the viewing angle of II by opening and shutting the elemental lenses sequentially. We prove our idea by using a mask that has patterns of on/off. It has vertical or horizontal apertures in an array form, whose interval matches that of the lenses in the II. Both theoretical discussion and experimental result are presented.© (2002) COPYRIGHT SPIE--The International Society for Optical Engineering. Downloading of the abstract is permitted for personal use only.
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paper_title: Real-time pickup method for three-dimensional image based on integral photography

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We have studied integral photography (IP) that creates three-dimensional autostereoscopic images. In particularly we studied the possibility of a new method using a television camera to directly shoot numerous real images produced by a lens array. Unlike the conventional IP method in which film is placed immediately behind a lens array, this method employs a television camera, enabling us to shoot moving pictures. Of a number of factors affecting the process of image pickup, we examined some optical factors, and compared them with those obtained by the conventional IP method. The results show that, with this new direct pickup method using a television camera, we can obtain an IP image like those obtained using the conventional IP method. Further, we conducted an experiment using an HDTV camera, confirming the production of an autostereoscopic image using a display device that combines a liquid crystal panel and pinholes.
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paper_title: Multifacet structure of observed reconstructed integral images.

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Three-dimensional images generated by an integral imaging system suffer from degradations in the form of grid of multiple facets. This multifacet structure breaks the continuity of the observed image and therefore reduces its visual quality. We perform an analysis of this effect and present the guidelines in the design of lenslet imaging parameters for optimization of viewing conditions with respect to the multifacet degradation. We consider the optimization of the system in terms of field of view, observer position and pupil function, lenslet parameters, and type of reconstruction. Numerical tests are presented to verify the theoretical analysis.
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paper_title: Three-dimensional visualization of partially occluded objects using integral imaging

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A three-dimensional (3-D) scene may contain foreground objects which partially occlude background objects. In this paper, we present experiments on reconstruction of the unobstructed view of the background objects from a partially occluded 3-D scene using an integral imaging system. We have reconstructed images of the scene at various distances of interest through computational and optical integral imaging reconstruction in order to obtain images of the occluding foreground objects and partially occluded background objects.
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paper_title: Improvement of viewing angle in integral imaging by use of moving lenslet arrays with low fill factor.

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Lenslet arrays with a low fill factor can improve the viewing angle in integral imaging. However, the viewing resolution is degraded by low fill-factor lenslets because the spatial sampling rate of the ray information is reduced. We show that both the viewing resolution and the viewing angle of integral imaging can be improved by adopting a moving array-lenslet technique.
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paper_title: Very large-scale integral imaging (VLSII) for 3-D display

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Very large-scale integral imaging (VLSII) systems utilizing multiple micro-optics arrays and display devices can be used for wide viewing angle 3-D displays. We analyze the distortion of 3-D integral images when micro-optics arrays are placed on a spherical surface and are used in direct pickup integral imaging. We show that 3-D orthoscopic real images with non-negligible depth always suffer from image distor- tion, while orthoscopic virtual images do not. Also, nonplanar (curved) display panels and image sensors are not currently available. We pro- pose a solution to remedy this problem by placing a spherical lens in front of a planar lenslet array. © 2005 Society of Photo-Optical Instrumentation Engineers. (DOI: 10.1117/1.1824907) tion on VLSII using spherical surfaces to determine what kind of integral images are formed and how direct camera pickup can be used to obtain elemental images. We attempt to provide analytical answers to some of these questions. Also, nonplanar ~curved! display panels and image sensors are not currently available. We propose a solution to remedy this problem by modeling the VLSII with lenslet arrays arranged on spherical surfaces as an II system with planar lenslet arrays, on which thin lenses with proper focal lengths are attached. Our analysis illustrates that 3-D orthoscopic real images with large depth always suffer from image distortion, while orthoscopic virtual im- ages do not. Preliminary experiments are presented. 2 Analysis
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paper_title: Three-dimensional projection integral imaging using micro-convex-mirror arrays.

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Three-dimensional projection integral imaging using micro-convex-mirror arrays is presented. In this scheme, a pseudoscopic to orthoscopic image conversion process is not required even if elemental images of direct camera pickup are used. In addition, a wide viewing angle without image flipping can be achieved even if optical barriers are not used. The feasibility of the proposed scheme was experimentally demonstrated, in which the viewing angle of more than 60 degrees was obtained. To the best of our knowledge, this is the largest viewing angle demonstrated by a single integral imaging system.
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paper_title: Integral imaging with improved depth of field by use of amplitude-modulated microlens arrays

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One of the main challenges in three-dimensional integral imaging is its limited depth of field. Such a limitation is imposed by diffraction, among other factors. The easiest way to improve the depth of field is by reducing the numerical aperture of the microlenses. However, such an improvement is obtained at the expense of an important deterioration in the spatial resolution. We propose a technique, which is novel in the context of integral imaging, for improving the depth of field with no deterioration of the spatial resolution. The technique, based on amplitude modulation of the array of phase elements, can substantially improve the figure of merit of the product of depth of the focus and the squared resolution.
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paper_title: New Characteristic Equation of Three-Dimensional Integral Imaging System and its Applications

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A three-dimensional integral imaging system is analyzed using the viewing parameters, namely, the image resolution, the viewing angle and the image depth. A new characteristic equation of this integral imaging system, which shows the relationship among the viewing parameters, is presented. This characteristic equation can be adopted as a design criterion for the integral imaging system. This can also be used as a figure of merit for the quantitative analysis of the integral imaging system.
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paper_title: Multiple-viewing-zone integral imaging using a dynamic barrier array for three-dimensional displays

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In spite of many advantages of integral imaging, the viewing zone in which an observer can see three-dimensional images is limited within a narrow range. Here, we propose a novel method to increase the number of viewing zones by using a dynamic barrier array. We prove our idea by fabricating and locating the dynamic barrier array between a lens array and a display panel. By tilting the barrier array, it is possible to distribute images for each viewing zone. Thus, the number of viewing zones can be increased with an increment of the states of the barrier array tilt.
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paper_title: Stereoscopic and autostereoscopic display systems

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In this article, the requirements for 3D television are outlined, and the suitability of different display types are considered. A brief description of our own approach is offered, which suggests that the first generation of 3D television will be a two-image system comprising a pair of images directed to the appropriate viewers' eyes. This will be achieved by means of steering optics controlled by a head-position tracker.
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paper_title: Three-dimensional image processing in the future of immersive media

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This survey paper discusses the three-dimensional image processing challenges posed by present and future immersive telecommunications, especially immersive video conferencing and television. We introduce the concepts of presence, immersion, and co-presence and discuss their relation to virtual collaborative environments in the context of communications. Several examples are used to illustrate the current state of the art. We highlight the crucial need of real-time, highly realistic video with adaptive viewpoint for future immersive communications and identify calibration, multiple-view analysis, tracking, and view synthesis as the fundamental image-processing modules addressing such a need. For each topic, we sketch the basic problem and representative solutions from the image processing literature.
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paper_title: Introduction to Fourier Optics

paper_content:

Analysis of two-dimensional signals and systems foundation of scalar diffraction theory Fresnel and Fraunhofer diffraction wave-optics analysis of coherent optical systems frequency analysis of optical imaging systems wavefront modulation analog optical information processing holography. Appendices: delta function and Fourier transform theorems introduction to paraxial geometrical optics polarization and Jones matrices.
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paper_title: Free-space beam propagation between arbitrarily oriented planes based on full diffraction theory: a fast Fourier transform approach

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A fast and accurate numerical method for free-space beam propagation between arbitrarily oriented planes is developed. The only approximation made in the development of the method was that the vector nature of light was ignored. The method is based on evaluating the Rayleigh–Sommerfeld diffraction integral by use of the fast Fourier transform with a special transformation to handle tilts and offsets of planes. The fundamental aspects of a software package based on the developed method are presented. A numerical example realized with the software package is presented to establish the validity of the method.
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paper_title: Simulation of scalar optical diffraction between arbitrarily oriented planes

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Scalar optical diffraction between arbitrarily oriented planes for monochromatic waves is analyzed and a simulator based on a discrete model is developed. The model is based directly on the Rayleigh-Sommerfeld diffraction integral; there is no need for Fresnel and Fraunhofer approximations. Furthermore, the model permits to use of the FFT algorithm. The simulator results are satisfactory.
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paper_title: Engineering Optics

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Engineering Optics is a book for students who want to apply their knowledge of optics to engineering problems, as well as for engineering students who want to acquire the basic principles of optics. It covers such important topics as optical signal processing, holography, tomography, holographic radars, fiber optical communication, electro- and acousto-optic devices, and integrated optics (including optical bistability). As a basis for understanding these topics, the first few chapters give easy-to-follow explanations of diffraction theory, Fourier transforms, and geometrical optics. Practical examples, such as the video disk, the Fresnel zone plate, and many more, appear throughout the text, together with numerous solved exercises. There is an entirely new section in this updated edition on 3-D imaging.
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paper_title: Digital Holography for Metrologic Applications

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In digital holography the holograms are recorded by CCD-arrays. The reconstruction of the complex wavefields is performed numerically in the computer. This enormously improves metrologic applications like deformation measurement, contour measurement, vibration analysis, refractive index determination, particle analysis, or microscopy.
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paper_title: Fundamentals of Photonics

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Preface to the Second Edition. Preface to the First Edition. 1 Ray Optics. 2 Wave Optics. 3 Beam Optics. 4 Fourier Optics. 5 Electromagnetic Optics. 6 Ploarization Optics. 7 Photonic-Crystal Optics. 8 Guided-Wave Optics. 9 Fiber Optics. 10 Resonator Optics. 11 Statistical Optics. 12 Photon Optics. 13 Photon and Atoms. 14 Laser Amplifiers. 15 Lasers. 16 Semiconductor Optics. 17 Semiconductor Photon Sources. 18 Semiconductor Photon Detectors. 19 Acousto-Optics. 20 Electro-Optics. 21 Nonlinear Optics. 22 Ultrafast Optics. 23 Optical Interconnects and Switches. 24 Optical Fiber Communications. A Fourier Transform. B Linear Systems. C Modes of Linear Systems. Symbols and Units. Authors. Index.
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paper_title: Introduction to Fourier Optics

paper_content:

Analysis of two-dimensional signals and systems foundation of scalar diffraction theory Fresnel and Fraunhofer diffraction wave-optics analysis of coherent optical systems frequency analysis of optical imaging systems wavefront modulation analog optical information processing holography. Appendices: delta function and Fourier transform theorems introduction to paraxial geometrical optics polarization and Jones matrices.
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paper_title: Signal processing issues in diffraction and holographic 3DTV

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Image capture and image display will most likely be decoupled in future 3DTV systems. Due to the need to convert abstract representations of 3D images to display driver signals, and to explicitly consider optical diffraction and propagation effects, it is expected that signal processing issues will be of fundamental importance in 3DTV systems. Since diffraction between two parallel planes can be represented as a 2D linear shift-invariant system, various signal processing techniques naturally play an important role. Diffraction between tilted planes can also be modeled as a relatively simple system, leading to efficient discrete computations. Two fundamental problems are digital computation of the optical field arising from a 3D object, and finding the driver signals for a given optical display device which will then generate a desired optical field in space. The discretization of optical signals leads to several interesting issues; for example, it is possible to violate the Nyquist rate while sampling, but still achieve full reconstruction. The fractional Fourier transform is another signal processing tool which finds applications in optical wave propagation.
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paper_title: Fast algorithms for free-space diffraction patterns calculation

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Here we present a fast algorithm for Fresnel integral calculation. Some fast algorithms using the fast Fourier transform are analysed and their performance has been checked. These methods are of easy implementation, but are only valid for a specific range of distances. Fast algorithms based on the Fractional Fourier transform allow accurate evaluation of the Fresnel integral from object to Fraunhofer domain in a single step.
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paper_title: Compression of Digital Holograms for Three-Dimensional Object Reconstruction and Recognition

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We present the results of applying lossless and lossy data compression to a three-dimensional object reconstruction and recognition technique based on phase-shift digital holography. We find that the best lossless (Lempel-Ziv, Lempel-Ziv-Welch, Huffman, Burrows-Wheeler) compression rates can be expected when the digital hologram is stored in an intermediate coding of separate data streams for real and imaginary components. The lossy techniques are based on subsampling, quantization, and discrete Fourier transformation. For various degrees of speckle reduction, we quantify the number of Fourier coefficients that can be removed from the hologram domain, and the lowest level of quantization achievable, without incurring significant loss in correlation performance or significant error in the reconstructed object domain.
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paper_title: Wave-front propagation by Gaussian superposition

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Abstract A linear superposition of Gaussians is proposed for representing and propagating a plane or defocused scalar wave-front with an arbitrary amplitude distribution in a linear, homogeneous and isotropic medium. A comparison is made against a wave-front that is propagated using the discrete Fresnel diffraction integral. At distances of propagation where the quadratic phase in the discrete Fresnel integral oscillates slowly, both techniques give similar results. At distances where the quadratic phase undergoes a large number of oscillations, the number of samples when using the discrete Fresnel integral must be increased to fit within Nyquist criteria. In comparison, the wave-front expressed as a linear Gaussian superposition can be calculated precisely, without any approximation, making unnecessary to increase the number of samples. Then, shorter processing time can be attained without losing the resolution in computing the propagated field. Thus, the Gaussian superposition technique becomes especially efficient when complex multistage optical systems are analyzed.
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paper_title: Computation of holographic patterns between tilted planes

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ABSTRACT Computation of the diffraction pattern that gives the desired reconstruction of an object upon proper illuminationis an important process in computer generated holography. A fast computational method, based on the plane wave decomposition of 3D field in free-space, is presented to find the desired diffraction pattern. The computationalburden includes two FFT algorithms in addition to a shuffling of the frequency components that needs aninterpolation in the frequency domain. The algorithm is based on the exact diffraction formulation; there isno need for Fresnel or Fraunhofer approximations. The developed model is utilized to calculate the scalaroptical diffraction between tilted planes for monochromatic light. The performance of the presented algorithmis satisfactory for tilt angles up to 600. Keywords: Digital Holography, Diffraction, Plane Wave Decomposition 1. INTRODUCTION The holography is based on duplication of information-carrying optical waves which come from a 3D scene inthe absence of the original source. Holography can be easily understood by the diffraction principles of optical
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paper_title: Light propagation in the Fresnel region. New numerical approach

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Abstract The paper presents the comparison of light propagation simulation methods in the Fresnel region. The theoretical approach is based on the Kirchhof scalar theory of diffraction. In the case of the complex amplitude calculated by using the spectrum of the plane waves a new approach is included. Advantages and disadvantages of the presented methods are compared and discussed. The simulated results are verified in an experimental optical setup.
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paper_title: Wave propagation with rotating intensity distributions

paper_content:

Wave fields containing invariant features have recently stimulated the interest of the scientific community. Typical examples of such fields are Gaussian modes, Bessel beams @1#, and wave fronts containing phase dislocations @2#. Bessel beams are solutions of the wave equation that propagate with invariant intensity. Phase dislocations are discontinuities of the phase in a wave front such that the circulation of the phase around its axis is an integral multiple of 2p. Thus, they determine lines of zero intensity in space. Experimental evidences of optical dislocations can be found, for example, in Refs. @3‐5#. It was noted in Refs. @4,6# that, under certain circumstances, an array of dislocations nested in a Gaussian beam rotates by p/2 rad from the waist to the far field, expanding with the host beam. The objective of this paper is to investigate general solutions having rotating intensity distributions around and along the propagation axis. We start by demonstrating that these solutions are easily obtained in terms of the superposition of Gauss-Laguerre ~GL! modes. The rotation rate along the propagation is then derived and the set of all possible solutions presenting a specific total rotation angle is characterized. Finally, we analyze the limit of the rotation rate and present experimental results for optical beams. Let a scalar wave be represented by the function
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paper_title: Monochromatic electromagnetic wavelets and the Huygens principle

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For the first time, to our knowledge, optical diffraction is shown to be a wavelet transform with the electromagnetic wavelets. We show that the optical wavelets proposed by Onural [Opt. Lett.18 , 846 (1993)] are the Huygens wavelets under a Fresnel approximation, and the electromagnetic wavelets proposed by Kaiser [A Friendly Guide to Wavelets (Birkhauser, Boston, Mass., 1994)] reduce to Hyugens wavelets in the case of a monochromatic field.
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paper_title: Efficient computation of quadratic-phase integrals in optics

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We present a fast NlogN time algorithm for computing quadratic-phase integrals. This three-parameter class of integrals models propagation in free space in the Fresnel approximation, passage through thin lenses, and propagation in quadratic graded-index media as well as any combination of any number of these and is therefore of importance in optics. By carefully managing the sampling rate, one need not choose N much larger than the space-bandwidth product of the signals, despite the highly oscillatory integral kernel. The only deviation from exactness arises from the approximation of a continuous Fourier transform with the discrete Fourier transform. Thus the algorithm computes quadratic-phase integrals with a performance similar to that of the fast-Fourier-transform algorithm in computing the Fourier transform, in terms of both speed and accuracy.
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paper_title: Diffraction field computation from arbitrarily distributed data points in space

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Computation of the diffraction field from a given set of arbitrarily distributed data points in space is an important signal processing problem arising in digital holographic 3D displays. The field arising from such distributed data points has to be solved simultaneously by considering all mutual couplings to get correct results. In our approach, the discrete form of the plane wave decomposition is used to calculate the diffraction field. Two approaches, based on matrix inversion and on projections on to convex sets (POCS), are studied. Both approaches are able to obtain the desired field when the number of given data points is larger than the number of data points on a transverse cross-section of the space. The POCS-based algorithm outperforms the matrix-inversion-based algorithm when the number of known data points is large.
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paper_title: Fast numerical calculation of Fresnel patterns in convergent systems

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In this communication, the authors study diffraction patterns calculation under convergent illumination. We analyze known fast algorithms using the Fourier transform and the fractional Fourier transform (FRT). We show that interference between convergent phase factor and the kernel of the different numerical Fresnel transformation may impose serious restrictions on the range of distances, where diffraction patterns can be accurately calculated. We review free space diffraction patterns calculation algorithms. In that case, the FRT provides a unified algorithm for the calculation of amplitude patterns in all the range of distances. On the contrary, we show below that, when convergent illumination is used, none of the algorithms is capable of providing a unified framework of calculation for the entire region of interest from object to focus.
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paper_title: Fourier diffraction through a tilted slit

paper_content:

Describes a simplified treatment of the diffraction from a single slit, in a particular case of oblique incidence. Using the methods of Fourier optics, one can see that the angular spectrum consists of two factors: the first is the same as that obtained by normal incidence and the second represents a law of 'reflection' or 'refraction' for the diffracted wave. This property substantially constitutes the 'law of edge diffraction' and from it, as a limiting case, one can deduce the laws of geometrical optics.
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paper_title: Calculation of the Rayleigh–Sommerfeld diffraction integral by exact integration of the fast oscillating factor

paper_content:

We describe a numerical method that can be used to calculate the propagation of light in a medium of constant (possibly complex) index of refraction n. The method integrates the Rayleigh–Sommerfeld diffraction integral numerically. After an appropriate change of integration variables, the integrand of the diffraction integral is split into a slowly varying and an (often fast) oscillating quadratic factor. The slowly varying factor is approximated by a spline fit, and the resulting Fresnel integrals are subsequently integrated exactly. Although the method is not as fast as methods involving a fast Fourier transform, such as plane-wave propagation or Fresnel approximation, it is accurate over a greater range than these methods.
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paper_title: DIGITAL DECODING OF IN-LINE HOLOGRAMS

paper_content:

Digitally sampled in-line holograms may be linearly filtered to reconstruct a representation of the original object distribution, thereby decoding the information contained in the hologram. The decoding process is performed by digital computation rather than optically. Substitution of digital for optical decoding has several advantages, including selective suppression of the twin-image artifact, elimination of the far-field requirement, and automation of the data reduction and analysis process. The proposed filter is a truncated series expansion of the inverse of that operator that maps object opacity function to hologram intensity. The first term of the expansion is shown to be equivalent to conventional (optical) reconstruction, with successive terms increasingly sup-pressing the twin image. The algorithm is computationally efficient, requiring only a single fast Fourier transform pair.
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paper_title: Computer-generated holograms of 3-D objects composed of tilted planar segments

paper_content:

The Fresnel diffraction of tilted planar objects is determined. It can be calculated by a Fourier transformation, a coordinate transformation, and a subsequent multiplication by a quadratic phase. Images of 3-D objects are composed by superimposing several planar segments. The resultant complex amplitude is coded and stored in a computer-generated hologram. Optical reconstructions of planar segments nearly parallel to the optical axis are demonstrated.
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paper_title: Generalized propagation-invariant wave fields

paper_content:

Necessary and sufficient conditions are presented that determine a generalized class of propagation-invariant wave fields. The existence of wave fields with transverse distributions that are periodically reproduced with different azimuthal orientations is demonstrated. These fields are conveniently described in the longitudinal-azimuthal frequency representation. An interesting subclass is characterized by aperiodic rotated self-images, in the sense that they never return to their original orientation along the propagation. Other subclasses include the conventional self-imaging wave fields, the so-called nondiffracting beams, and the rotating wave fields.
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paper_title: Aliasing error in digital holography.

paper_content:

The use of the discrete Fourier transform in digital holography introduces aliasing error in the reconstructed image. Spectrum shaping to reduce dynamic range may also result in a serious increase in aliasing error. The effect of aliasing in digital holography is analyzed. It is proposed that the bandwidth be constrained when shaping the image spectrum. Experimental results show the approach to be quite effective.
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paper_title: Dark beams with a constant notch

paper_content:

Dark beams are wave fields carrying information in dark regions. We experimentally demonstrate dark beams that maintain a constant notch shape and size along a predetermined domain in free space. The energy is concentrated around the dark region with negligible sidelobes. These beams are generated with low-information-content phase-only diffractive elements in an on-axis configuration.
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paper_title: Synthesis of three-dimensional light fields and applications

paper_content:

The possibility of synthesizing light fields satisfying given requirements within a three-dimensional (3-D) space domain was proposed and demonstrated during recent years. In this paper we present fundamental physical properties characterizing 3-D fields and propose analytical and numerical procedures to synthesize them. These methods solve the proper wave equation under 3-D constraints. Since the manipulation of light is a basic task, this is an effective approach to deal with classical as well as novel problems in optics and photonics technologies. We discuss exciting new potential areas of application and extensions of this concept.
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paper_title: GENERATION OF PSEUDO-NONDIFFRACTING BEAMS WITH USE OF DIFFRACTIVE PHASE ELEMENTS DESIGNED BY THE CONJUGATE-GRADIENT METHOD

paper_content:

The design of diffractive phase elements (DPE’s) for generating pseudo-nondiffracting beams (PNDB’s) is carried out by employing the conjugate-gradient method. By selecting a trapezoid profile as the preset axial intensity distribution to guide the design and by using an error function with a weighting factor to evaluate the performance of the DPE, we develop a model design and obtain a satisfactory diffractive pattern of a single-segment PNDB. We demonstrate the effectiveness of the conjugate-gradient method in the design of DPE’s that produce multiple-segment PNDB’s in which two consecutive segments are separated by a dark region along the optical axis. We also evaluate the degree of nonuniformity of the axial intensity over a given axial region. The design results match the requirements quite well.
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paper_title: Fast calculation method for optical diffraction on tilted planes by use of the angular spectrum of plane waves

paper_content:

A novel method for simulating field propagation is presented. The method, based on the angular spectrum of plane waves and coordinate rotation in the Fourier domain, removes geometric limitations posed by conventional propagation calculation and enables us to calculate complex amplitudes of diffracted waves on a plane not parallel to the aperture. This method can be implemented by using the fast Fourier transformation twice and a spectrum interpolation. It features computation time that is comparable with that of standard calculation methods for diffraction or propagation between parallel planes. To demonstrate the method, numerical results as well as a general formulation are reported for a single-axis rotation.
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paper_title: Snake Beam: A Paraxial Arbitrary Focal Line

paper_content:

The creation of paraxial arbitrary focal lines by a Fourier computer-generated hologram is demonstrated. The desired focal line is represented by a series of connected straight line segments, each of which is implemented by a radial harmonic function located on a different radial portion of the entire hologram. Each subhologram is multiplied by appropriate linear and quadratic phase functions and is shifted by some distance from the center. The two phase factors determine the location of each line segment, while the in-plane shift determines the tilt angle of the segment.
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paper_title: Frequency analysis of light diffraction between rotated planes.

paper_content:

The relation for the angular spectra of rotated planes is evaluated, starting from the knowledge of the mono-chromatic scalar field on a given plane. Diffracted light on a tilted plane can then be calculated in the frequency domain by fast-Fourier-transform algorithms. Unlike the Fresnel and Fraunhofer approaches, this analysis does not require approximations; as a consequence, it permits any positions in space for the planes under investigation. Digital images are generated, which show the effects of rotation.
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paper_title: Generalizing, optimizing, and inventing numerical algorithms for the fractional Fourier, Fresnel, and linear canonical transforms

paper_content:

By use of matrix-based techniques it is shown how the space–bandwidth product (SBP) of a signal, as indicated by the location of the signal energy in the Wigner distribution function, can be tracked through any quadratic-phase optical system whose operation is described by the linear canonical transform. Then, applying the regular uniform sampling criteria imposed by the SBP and linking the criteria explicitly to a decomposition of the optical matrix of the system, it is shown how numerical algorithms (employing interpolation and decimation), which exhibit both invertibility and additivity, can be implemented. Algorithms appearing in the literature for a variety of transforms (Fresnel, fractional Fourier) are shown to be special cases of our general approach. The method is shown to allow the existing algorithms to be optimized and is also shown to permit the invention of many new algorithms.
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paper_title: Electromagnetic Degrees of Freedom of an Optical System

paper_content:

We present a rigorous electromagnetic formalism for defining, evaluating, and optimizing the degrees of freedom of an optical system. The analysis is valid for the delivery of information with electromagnetic waves under arbitrary boundary conditions communicating between domains in three-dimensional space. We show that, although in principle there is an infinity of degrees of freedom, the effective number is finite owing to the presence of noise. This is in agreement with the restricted classical theories that showed this property for specific optical systems and within the scalar and paraxial approximations. We further show that the best transmitting and receiving functions are the solutions of well-defined eigenvalue equations. The present approach is useful for understanding and designing modern optical systems for which the previous approaches are not applicable, as well as for application in inverse and synthesis problems.
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paper_title: WAVE FIELDS IN THREE DIMENSIONS : ANALYSIS AND SYNTHESIS

paper_content:

Distributions of wave fields in three-dimensional domains are analyzed, synthesized, and generated experimentally. Fundamental limits are discussed and sampling conditions are derived for their generation, with use of a single diffractive element. A general design procedure, based on optimization algorithms, is developed and implemented. Experimental results show that special three-dimensional light distributions can be achieved with low-information-content elements in on-axis configurations.
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paper_title: Computer generated holography using parallel commodity graphics hardware

paper_content:

This paper presents a novel method for using programmable graphics hardware to generate fringe patterns for SLM-based holographic displays. The algorithm is designed to take the programming constraints imposed by the graphics hardware pipeline model into consideration, and scales linearly with the number of object points. In contrast to previous methods we do not have to use the Fresnel approximation. The technique can also be used on several graphics processors in parallel for further optimization. We achieve real-time frame rates for objects consisting of a few hundred points at a resolution of 960x600 pixels and over 10 frames per second for 1000 points.
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paper_title: Optical Systems and Processes

paper_content:

Review of Electromagnetic Wave Propagation Linear Systems Theory Wavefront Transformations Basic Optical Systems Non-Ideal Optical Systems Statistical Aspects of Light Interference and Interferometers Polarization Spatial Light Modulation Holography Advanced Operator Algebra Optical Information Processing Appendix A - Summary of Operator Relations Appendix B - Bibliography Appendix C - Problems and Solutions.
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paper_title: Fresnel propagation and diffraction and paraxial wave equation

paper_content:

Examines the diffraction and the free propagation of an optical scalar field by means of the Fresnel approximation. In a unified way, deduces its properties from the differential equation of paraxial waves. By means of the variable separation method, one can obtain directly significant field distributions like: pseudoplane waves, pseudospherical waves and Gaussian beams. The following topics are also treated: the Fresnel diffraction integral, inverse propagation, the Fresnel transformations, some invariant properties for affinity transformations, and formulas which permit an evaluation, in every point of the field, of the distribution of the error connected to the Fresnel approximation.
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paper_title: Diffraction from a wavelet point of view

paper_content:

The system impulse response representing the Fresnel diffraction is shown to form a wavelet family of functions. The scale parameter of the wavelet family represents the depth (distance). This observation relates the diffraction-holography-related studies and the wavelet theory. The results may be used in various optical applications such as designing robust volume optical elements for optical signal processing and finding new formulations for optical inverse problems. The results also extend the wavelet concept to the nonbandpass family of functions with the implication of new applications in signal processing. The presented wavelet structure, for example, is a tool for space–depth analysis.
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paper_title: Computation considerations and fast algorithms for calculating the diffraction integral

paper_content:

Abstract One of the most basic optical ‘components’ is free-space propagation. A common approximation used when calculating the resultant field distribution after propagation is the Fresnel integral. This integral can be evaluated in two ways: directly or by using the angular spectrum. In this paper, we estimate the regions in which each mode of evaluation is preferable according to computing efficiency and accuracy considerations. A fast numerical algorithm is introduced for each region. The result is relevant also for the evaluation of the Rayleigh-Sommerfeld diffraction formula.
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paper_title: Fast-Fourier-transform based numerical integration method for the Rayleigh-Sommerfeld diffraction formula

paper_content:

The numerical calculation of the Rayleigh-Sommerfeld diffraction integral is investigated. The implementation of a fast-Fourier-transform (FFT) based direct integration ::: (FFT-DI) method is presented, and Simpson's rule is used to improve the calculation accuracy. The sampling interval, the size of the computation window, and their influence on numerical accuracy and on computational complexity are discussed for the FFT-DI and the FFT-based angular spectrum (FFT-AS) methods. The performance of the FFT-DI method is verified by numerical simulation and compared with that of the FFT-AS method.
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paper_title: Free-space beam propagation between arbitrarily oriented planes based on full diffraction theory: a fast Fourier transform approach

paper_content:

A fast and accurate numerical method for free-space beam propagation between arbitrarily oriented planes is developed. The only approximation made in the development of the method was that the vector nature of light was ignored. The method is based on evaluating the Rayleigh–Sommerfeld diffraction integral by use of the fast Fourier transform with a special transformation to handle tilts and offsets of planes. The fundamental aspects of a software package based on the developed method are presented. A numerical example realized with the software package is presented to establish the validity of the method.
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paper_title: Scattering and diffraction in physical optics

paper_content:

This book presents a comprehensive tutorial on propagation, diffraction and scattering problems from the basic principles of physical optics. Beginning with the fundamental differential and integral equations for wavefields, the text presents an exhaustive discussion on the extinction theorem as a non-local boundary condition; this has been extensively employed for the rigorous solution of scattering and diffraction problems. There is also an in-depth presentation of the topic of scattering from rough surfaces, in particular the phenomenon of enhanced backscattering, as well as a detailed development of the angular spectrum representation of fields leading to questions on non-diffraction beams. Of key interest in near field optical microscopy and nanooptics, the S-matrix theory based on the angular spectrum for propagating components and the recently discovered properties of the S-matrix for evanescent components of wavefields are considered. In addition, the book deals with the healing effect of phase conjugation on waves, and focuses on some applications concerning the relationship with time reversal. Readers will also find discussions on image recovery from partial information data (phase problems and super-resolution problems), as well as a chapter on the fundamentals of near field optical microscopy techniques, including the hot topic of propagation in negative index media.
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paper_title: Family of scaling chirp functions, diffraction, and holography

paper_content:

It is observed that diffraction is a convolution operation with a chirp kernel whose argument is scaled. Family of functions obtained from a prototype by shifting and argument scaling form the essential ground for wavelet framework. Therefore, a connection between diffraction and wavelet transform is developed. However, wavelet transform is essentially prescribed for time-frequency and/or multiresolution analysis which is irrelevant in our case. Instead, the proposed framework is useful in various location-depth type of analysis in imaging. The linear transform when the analyzing functions are the chirps is called the scaling chirp transform. The scaled chirp functions do not satisfy the commonly used admissibility condition for wavelets. However, it is formally shown that these neither band nor time limited signals can be used as wavelet functions and the inversion is still possible. Diffraction and in-line holography are revisited within the scaling chirp transform context. It is formally proven that a volume in-line hologram gives perfect reconstruction. The developed framework for wave propagation based phenomena has the potential of advancing both signal processing and optical applications. >
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paper_title: Large objects reconstructed from computer-generated holograms

paper_content:

In digital holography the first step is the determination of the complex amplitude in the hologram. Large objects can be synthesized from tilted planar segments. In the calculation step, configurations of inclined object and inclined hologram plane are considered. The Fresnel transformation is generalized, and its implementation is discussed. Optical reconstructions show the feasibility.
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paper_title: Computer-generated holograms of tilted planes by a spatial frequency approach

paper_content:

Computer-generated holograms of plane surfaces tilted and shifted with respect to the hologram plane are considered. The analysis is made in the spatial frequency domain by using the translation and rotation transformations of angular spectra. The frequency approach permits the use of the fast-Fourier-transform algorithm, which decreases the computation time and makes it possible to consider any position of the planes in space. Various configurations of tilted and shifted planes have been investigated, and computer-generated holograms of off-axis planes have been obtained. Computer and optical reconstructions, both of which show the feasibility of the proposed approach, have been carried out.
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paper_title: Fresnelets: new multiresolution wavelet bases for digital holography

paper_content:

We propose a construction of new wavelet-like bases that are well suited for the reconstruction and processing of optically generated Fresnel holograms recorded on CCD-arrays. The starting point is a wavelet basis of L2 to which we apply a unitary Fresnel transform. The transformed basis functions are shift-invariant on a level-by-level basis but their multiresolution properties are governed by the special form that the dilation operator takes in the Fresnel domain. We derive a Heisenberg-like uncertainty relation that relates the localization of Fresnelets with that of their associated wavelet basis. According to this criterion, the optimal functions for digital hologram processing turn out to be Gabor functions, bringing together two separate aspects of the holography inventor's work. We give the explicit expression of orthogonal and semi-orthogonal Fresnelet bases corresponding to polynomial spline wavelets. This special choice of Fresnelets is motivated by their near-optimal localization properties and their approximation characteristics. We then present an efficient multiresolution Fresnel transform algorithm, the Fresnelet transform. This algorithm allows for the reconstruction (backpropagation) of complex scalar waves at several user-defined, wavelength-independent resolutions. Furthermore, when reconstructing numerical holograms, the subband decomposition of the Fresnelet transform naturally separates the image to reconstruct from the unwanted zero-order and twin image terms. This greatly facilitates their suppression. We show results of experiments carried out on both synthetic (simulated) data sets as well as on digitally acquired holograms.
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paper_title: Optimization of hologram computation for real-time display

paper_content:

Several methods of increasing the speed and simplicity of the computation of off-axis transmission holograms are presented, with applications to the real-time display of holographic images. A bipolar intensity approach enables a linear summation of interference fringes, a factor of two speed increase, and the elimination of image noise caused by object self- interference. An order of magnitude speed increase is obtained through the use of precomputed look-up tables containing a large array of elemental interference patterns corresponding to point source contributions from each of the possible locations in image space. Results achieved using a data-parallel supercomputer to compute horizontal-parallax- only holographic patterns containing 6 megasamples indicate that an image comprised of 10,000 points with arbitrary brightness (grayscale) can be computed in under one second.
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paper_title: One-Dimensional Beam Shaping

paper_content:

A general method is presented of optimizing one-dimensional holograms to realize an arbitrary diffracted field distribution over the propagation volume. A computationally simpler approach is developed for the case in which only the transverse distribution in one plane and the longitudinal distribution on one axis are of interest. Scaling effects in the optimized beams are studied and compared with those of canonical beams.
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paper_title: Autofocus for digital Fresnel holograms by use of a Fresnelet-sparsity criterion

paper_content:

We propose a robust autofocus method for reconstructing digital Fresnel holograms. The numerical reconstruction involves simulating the propagation of a complex wave front to the appropriate distance. Since the latter value is difficult to determine manually, it is desirable to rely on an automatic procedure for finding the optimal distance to achieve high-quality reconstructions. Our algorithm maximizes a sharpness metric related to the sparsity of the signal's expansion in distance-dependent waveletlike Fresnelet bases. We show results from simulations and experimental situations that confirm its applicability.
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paper_title: Diffraction-specific fringe computation for electro-holography

paper_content:

Diffraction-specific fringe computation is a novel system for the generation of holographic fringe patterns for real-time display. This thesis describes the development, implementation, and analysis of diffraction-specific computation, an approach that considers the reconstruction process rather than the interference process in optical holography. The primary goal is to increase the speed of holographic computation for real-time three-dimensional electro-holographic (holovideo) displays. Diffraction-specific fringe computation is based on the discretization of space and spatial frequency in the fringe pattern. Two holographic fringe encoding techniques are developed from diffraction-specific fringe computation and applied to make most efficient use of hologram channel capacity. A "hogel-vector encoding" technique is based on undersampling the fringe spectra. A "fringelet encoding" technique is designed to increase the speed and simplicity of decoding. The analysis of diffraction-specific computation focuses on the trade-offs between compression ratio, image fidelity, and image depth. The decreased image resolution (increased point spread) that is introduced into holographic images due to encoding is imperceptible to the human visual system under certain conditions. A compression ratio of 16 is achieved (using either encoding method) with an acceptably small loss in image resolution. Total computation time is reduced by a factor of over 100 to less than 7.0 seconds per 36-MB holographic fringe using the fringelet encoding method. Diffraction-specific computation more efficiently matches the information content of holographic fringes to the human visual system. Diffraction-specific holographic encoding allows for "visual-bandwidth holography," i.e., holographic imaging that requires a bandwidth commensurate with the usable visual information contained in an image. Diffraction-specific holographic encoding enables the integration of holographic information with other digital media, and is therefore vital to applications of holovideo in the areas of visualization, entertainment, and information, including education, telepresence, medical imaging, interactive design, and scientific visualization. Copies available exclusively from MIT Libraries, Rm. 14-0551, Cambridge, MA 02139-4307. Ph. 617-253-5668; Fax 617-253-1690.
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paper_title: Synthesis of nondiffracting beams in free space.

paper_content:

A novel method of designing the longitudinal intensity profile of a diffracted beam in free space is proposed. By use of this method illumination of an arbitrarily synthesized aperture can yield a desired intensity distribution, including a constant (nondiffracting) curve, along a chosen limited range.
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paper_title: Simulation of scalar optical diffraction between arbitrarily oriented planes

paper_content:

Scalar optical diffraction between arbitrarily oriented planes for monochromatic waves is analyzed and a simulator based on a discrete model is developed. The model is based directly on the Rayleigh-Sommerfeld diffraction integral; there is no need for Fresnel and Fraunhofer approximations. Furthermore, the model permits to use of the FFT algorithm. The simulator results are satisfactory.
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paper_title: Encoding amplitude information onto phase-only filters

paper_content:

We report a new, to our knowledge, technique for encoding amplitude information onto a phase-only filter with a single liquid-crystal spatial light modulator. In our approach we spatially modulate the phase that is encoded onto the filter and, consequently, spatially modify the diffraction efficiency of the filter. Light that is not diffracted into the first order is sent into the zero order, effectively allowing for amplitude modulation of either the first-order or the zero-order diffracted light. This technique has several applications in both optical pattern recognition and image processing, including amplitude modulation and inverse filters. Experimental results are included for the new technique.
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paper_title: Reconstruction algorithm for high-numerical-aperture holograms with diffraction-limited resolution

paper_content:

A fast algorithm is proposed for the reconstruction of digital holograms that are recorded at high numerical aperture. The method directly evaluates the Rayleigh-Sommerfeld diffraction integral by use of a fast convolution algorithm. A shift parameter that accounts for the coordinate system's transverse displacement of the object plane and the hologram plane is introduced in a discrete representation of the diffraction kernel. Combination of the samplings reconstructed with different shift values yields diffraction-limited resolution over the full field of view. The algorithm is suitable for various applications such as holographic microscopy and metrology. Simulation and experimental results are presented.
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paper_title: Improvement of accuracy in digital holography by use of multiple holograms.

paper_content:

Speckle pattern decorrelation reduces the accuracy of interferometric shape and deformation measurements. We introduce a technique for the reduction of speckle noise in digital holography. The method is not based on classical filtering techniques such as median filters. Instead it utilizes the shift theorem of the Fourier transform. For this method several holograms of the same object under test are recorded. The reconstruction leads to a set of object wave fields with different speckle patterns. A proper averaging procedure, taking into account the properties of the wrapped phases, leads to an improvement of the accuracy in the resulting phase difference. The theory of the applied method is described and our first results for technical components with an improvement of accuracy up to 1/57 of the wavelength are presented.
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paper_title: Optimally tuned spatial light modulators for digital holography

paper_content:

Digital holography and comparative digital holography are applications that require computer-addressable modulators for the optical reconstruction of digital holograms. The quality of the reconstructed holograms depends on the modulator's properties. Therefore a characterization of the modulators is required. We show the result of a modulator characterization and the modulator's influence on the quality of the reconstructed hologram. We then compare qualitatively and quantitatively the optical reconstruction of phase and amplitude holograms by considering their level of detail and their diffraction efficiency.
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paper_title: Computer-generated holograms of 3-D objects composed of tilted planar segments

paper_content:

The Fresnel diffraction of tilted planar objects is determined. It can be calculated by a Fourier transformation, a coordinate transformation, and a subsequent multiplication by a quadratic phase. Images of 3-D objects are composed by superimposing several planar segments. The resultant complex amplitude is coded and stored in a computer-generated hologram. Optical reconstructions of planar segments nearly parallel to the optical axis are demonstrated.
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paper_title: Real color fractional Fourier transform holograms

paper_content:

Real color fractional Fourier transform holography is proposed based on fractional Fourier transform holography. The method of fabrication, the principle of reconstruction and the design of system parameters are discussed in detail. The experiments prove real color fractional Fourier transform holograms possess the better function of anti-counterfeit than common fractional Fourier transform holograms.
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paper_title: Pixel resolution control in numerical reconstruction of digital holography

paper_content:

A new method for resolution control in numerical reconstruction of digital holography is proposed. The wave field on a tilted or vertical plane can be reconstructed without being subject to the minimum object-to-hologram distance requirement, and the pixel resolution can be easily controlled by adjusting the position of a transitional plane. The proposed method solves the problem of pixel resolution control for small object-to-hologram distances and is especially useful for multicolor, multiwavelength digital holography and metrological applications. Experimental results are presented to verify the idea.
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paper_title: Investigation of M2 factor influence for paraxial computer generated hologram reconstruction using a statistical method

paper_content:

Abstract In this paper, we study the influence of the M 2 quality factor of an incident beam on the reconstruction performance of a computer generated hologram (CGH). We use a statistical method to analyze the evolution of different quality criteria such as diffraction efficiency, root mean square error, illumination uniformity or correlation coefficient calculated in the numerical reconstruction versus the increasing M 2 quality factor. The simulation results show us that this factor must always be taken into account in the CGH design when the M 2 value is bigger than 2.
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paper_title: Recurrence formulas for fast creation of synthetic three-dimensional holograms

paper_content:

A method for accelerating the synthesis of computer-generated ::: three-dimensional (3-D) holograms, based on conventional ray ::: tracing, is proposed. In ray tracing, computers expend almost all ::: of their resources in calculating the 3-D distances between each one of ::: the point sources composing an object and a sampling point on the ::: hologram. We present recurrence formulas that precisely compute the ::: distances and reduce the computation time for synthesizing holograms to ::: one half to one quarter, depending on the processor type. We ::: demonstrate that a full-parallax hologram with an area of 4800 × ::: 4800 pixels, synthesized for a 3-D object containing 966 point sources ::: of light, is computed within 17 min and is optically ::: reconstructed.
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paper_title: Color computer-generated holograms from projection images.

paper_content:

A novel method for the procurement of full-color three-dimensional (3-D) images of real objects has been developed. This method is based on extracting information from 3-D Fourier spectra, which are calculated from several projection images recorded using a white light source. 3-D Fourier spectra for three colors were obtained separately for projection images recorded with a color-CCD camera. Three computer-generated holograms (CGHs) were then synthesized from those Fourier spectra. The resulting numerically and optically reconstructed full-color images are presented.
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paper_title: Design of Binary Computer Generated Holograms by Simulated Annealing: Observation of Metastable States

paper_content:

Abstract The simulated annealing algorithm allows optimal solutions to the binary hologram design problems to be located. When compared with sub-optimal techniques such as direct binary search it requires a significantly longer optimization time. We compare simulated annealing and direct binary search for the design of fanout holograms. Over a large range of design parameters no significant improvement is obtained by the use of simulated annealing. When diffraction efficiencies are maximized however metastable states are observed and simulated annealing results in a 6·6% increase in diffraction efficiency and a tenfold reduction in reconstruction error.
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paper_title: Computer generated holograms: an historical review

paper_content:

Since Gabor's first hologram, the meanings of that term have grown with increased use of the invention. Computers expanded applications further, and this paper surveys the methods and techniques of computer generated holograms.
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paper_title: Fast calculation method for cylindrical computer-generated holograms.

paper_content:

We propose a fast calculation method for diffraction to nonplanar surfaces using the fast-Fourier transform (FFT) algorithm. In his method, the diffracted wavefront on a cylindrical surface is expressed as a convolution between the point response function and the spatial distribution of objects wherein the convolution is calculated using FFT. The principle of the fast calculation and the simulation results are presented.
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paper_title: Complex spatial filtering with binary masks.

paper_content:

Usually a hologram is produced by means of an interference experiment. Here, however, we let a computer- guided plotter draw the hologram. The plot, which has to be minified and recorded on film, contains no grey, only binary transmittance values. Our binary holograms yield reconstructed images of a quality equal to that of images obtained from usual holograms of comparable dimensions. When a Fourier hologram is inserted into the Fraunhofer plane of a coherent image forming system, it acts as a special type of a spatial filter, a so-called optical matched filter. Our binary matched filter is suitable for optical character recognition, the same as the usual optical matched filter introduced by Vander Lugt.
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paper_title: Hologram reconstruction using a digital micromirror device

paper_content:

The aim of this work is an advancement of digital holographic interferometry by using a digital micromirror device (DMD) for real-time display of interferometric fringes. The theory describing the formation of the diffracted image produced by a coherently illuminated DMD is pre- sented. Digital holograms of simple structures are calculated assuming plane and spherical reference waves, respectively. These synthetic ho- lograms are used to control the micromirrors of the DMD, which now acts as a reflective hologram. Experimental holographic reconstructions of synthetic holograms prove the feasibility of DMDs in principle for this aim. Further experimental results show the reconstructed interferogram of fields recorded by digital holography and superposed numerically. This opens a new range of applications in digital holographic interferom- etry. © 2001 Society of Photo-Optical Instrumentation Engineers.
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paper_title: Accurate encoding of arbitrary complex fields with amplitude-only liquid crystal spatial light modulators.

paper_content:

We show that computer generated holograms, implemented with amplitude-only liquid crystal spatial light modulators, allow the synthesis of fully complex fields with high accuracy. Our main discussion considers modified amplitude holograms whose transmittance is obtained by adding an appropriate bias function to the real cosine computer hologram of the encoded signal. We first propose a bias function, given by a soft envelope of the signal modulus, which is appropriate for perfect amplitude modulators. We also consider a second bias term, given by a constant function, which results appropriate for modulators whose amplitude transmittance is coupled with a linear phase modulation. The influence of the finite pixel size of the spatial light modulator is compensated by digital pre-filtering of the encoded complex signal. The performance of the discussed amplitude CGHs is illustrated by means of numerical simulations and the experimental synthesis of high order Bessel beams.
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paper_title: Computer-Generated Holograms of Images Reconstructed on Curved Surfaces

paper_content:

When one illuminates a computer-generated hologram by a plane wave, ::: the obtained two-dimensional image is usually displayed on a planar ::: plane. Other possibilities for reconstructing images on arbitrary ::: curved surfaces are discussed herein. As an example, the ::: reconstruction of an image on a virtual spherical surface is ::: demonstrated.
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paper_title: Synthesis of digital holograms by direct binary search.

paper_content:

A new approach to the design of computer-generated holograms makes optimal use of the available device resolution. An iterative search algorithm minimizes an error criterion by directly manipulating the binary hologram and observing the effect on the desired reconstruction. Several measures of error and efficiency useful in assessing the optimality of digital holograms are defined. Methods for designing digital holograms that are based on projections and error diffusion are presented as established techniques for comparison to direct binary search.
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paper_title: Image reconstruction only by phase data in phase-shifting digital holography.

paper_content:

We describe data compression in phase-shifting digital holography. We demonstrate by experimentation that an image of a diffusely reflecting object can be reconstructed only by phase data of the derived complex amplitude. It is shown that reduction of the bit depth of the phase data does not seriously damage the image even down to 1 bit. We observe enhancement of halo in the image with low bit depths. This tendency is verified quantitatively by a one-dimensional simulation. Our procedure for smoothing the images that result from the data-compression methods is shown to be effective.
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paper_title: Some mathematical properties of the uniformly sampled quadratic phase function and associated issues in digital Fresnel diffraction simulations

paper_content:

The quadratic phase function is fundamental in describing and computing wave-propagation-related phenomena under the Fresnel approximation; it is also frequently used in many signal processing algorithms. This function has interesting properties and Fourier transform relations. For example, the Fourier transform of the sampled chirp is also a sampled chirp for some sampling rates. These properties are essential in interpreting the aliasing and its effects as a consequence of sampling of the quadratic phase function, and lead to interesting and efficient algorithms to simulate Fresnel diffraction. For example, it is possible to construct discrete Fourier transform (DFT)-based algorithms to compute exact continuous Fresnel diffraction patterns of continuous, not necessarily bandlimited, periodic masks at some specific distances.
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paper_title: Complex-valued Fresnel-transform sampling.

paper_content:

VanderLugt [Appl. Opt. 29, 3352 (1990)] presented sampling rates for the amplitude of Fresnel diffraction patterns. These apply to any plane in a coherent optical system. Although these sampling rates represent the amplitude of diffraction patterns accurately, they are not adequate for the retention of complete information in complex-valued Fresnel diffraction patterns. I show this by considering the ability to reconstruct the original input image through backward diffraction of the forward diffraction pattern of such an image. I then extend the VanderLugt sampling techniques such that reliable sampling of the phase of these Fresnel diffraction patterns can also be achieved. The analysis is restricted to lensless optical systems. The new sampling rates are tested with numerical computations of Fresnel diffraction patterns and rigorous scalar diffraction patterns in both forward and backward directions.
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paper_title: Direct coarse sampling of electronic holograms.

paper_content:

We describe a series of experiments in order to test a new sampling criterion for wavefront reconstruction from a carrier-frequency signal as obtained in electronic holography. Both a Fourier-transform configuration and a Fresnel-zone form of a digital holographic microscope are described. Direct coarse sampling of holograms with no phase-shifting elements can be used to obtain wavefront reconstruction, with the number of samples required reduced by the ratio f0/(2Bs), where f0 is the carrier frequency and 2Bs is the signal bandwidth.
---
paper_title: Sampling-50 years after Shannon

paper_content:

This paper presents an account of the current state of sampling, 50 years after Shannon's formulation of the sampling theorem. The emphasis is on regular sampling, where the grid is uniform. This topic has benefitted from a strong research revival during the past few years, thanks in part to the mathematical connections that were made with wavelet theory. To introduce the reader to the modern, Hilbert-space formulation, we reinterpret Shannon's sampling procedure as an orthogonal projection onto the subspace of band-limited functions. We then extend the standard sampling paradigm for a presentation of functions in the more general class of "shift-in-variant" function spaces, including splines and wavelets. Practically, this allows for simpler-and possibly more realistic-interpolation models, which can be used in conjunction with a much wider class of (anti-aliasing) prefilters that are not necessarily ideal low-pass. We summarize and discuss the results available for the determination of the approximation error and of the sampling rate when the input of the system is essentially arbitrary; e.g., nonbandlimited. We also review variations of sampling that can be understood from the same unifying perspective. These include wavelets, multiwavelets, Papoulis generalized sampling, finite elements, and frames. Irregular sampling and radial basis functions are briefly mentioned.
---
paper_title: A measurement method based on a modified version of the chirplet transform for instantaneous frequency estimation

paper_content:

A measurement method for instantaneous frequency estimation is presented in this paper. The method is based on the use of the chirplet transform, a linear time-frequency representation (TFR) allowing additional modifications of each cell on the time-frequency plane with respect to other TFRs. In particular, a modified version of this transform is proposed here; a bending effect can be further imposed on the cells. Thanks both to this feature and a suitable measurement procedure, properly set up by the authors, the method assures a satisfying accuracy in reconstructing the instantaneous frequency trajectory of monocomponent signals as well as a good resolving capability in the analysis of multicomponent signals whose instantaneous frequency trajectories are strongly nonlinear and very close to one another. Theoretical details concerning the chirplet transform and its modified version are first given. Then, the proposed measurement procedure for choosing the optimal values of the parameters of the transform according to the local features of the analyzed signal is described. At the end, the results of several experimental tests conducted both on monocomponent and multicomponent signals are presented; advantages over other solutions are also highlighted.
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paper_title: Time-frequency perspectives: the 'chirplet' transform

paper_content:

The authors have developed an expansion they call the chirplet transform. It has been successfully applied to a wide variety of signal processing applications, including radar and image processing. There has been a recent debate as to the relative merits of an affine-in-time (wavelet) transform and the classical short-time Fourier transform (STFT) for the analysis of nonstationary phenomena. Chirplet filters embody both the wavelet and STFT as special cases by decoupling the filter bandwidths and center frequencies. Chirplets, by their embodiment of affine geometry in the time-frequency (TF) plane, may also include shears in time and frequency (chirps) and even time-bandwidth product variation (noise bursts) if desired. The most general chirplets may be derived from one or more basic ('mother') chirplets by the transformations or perspective geometry in the TF plane. >
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paper_title: Holographic particle image velocimetry: signal recovery from under-sampled CCD data

paper_content:

Holographic particle image velocimetry (HPIV) has now been demonstrated by several research groups as a method to make three-component velocity measurements from a three-dimensional fluid flow field. More recently digital HPIV has become a hot topic with the promise of near-real-time measurements without the often cumbersome optics and wet processing associated with traditional holographic methods. It is clear, however, that CCD cameras have a limited number of pixels and are not capable of resolving more than a small fraction of the interference pattern that is recorded by a typical particulate hologram. In this paper, we consider under-sampling of the interference pattern to reduce the information content and to allow recordings to be made on a CCD sensor. We describe the basic concept of model fitting to under-sampled data and demonstrate signal recovery through computer simulation. A three-dimensional analysis shows that in general, periodic sampling strategies can result in multiple particle images in the reconstruction. It is shown, however, that the significance of these peaks is reduced in the case of high numerical aperture (NA) reconstruction and can be virtually eliminated by dithering the position of sampling apertures.
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paper_title: Optical transforms in digital holography

paper_content:

Digital holography is a symbiosis of physical optics and digital computers. In numerical reconstruction of optical ::: holograms, optical wavefront is sampled, and obtained numerical data are transformed in computers for evaluating ::: physical properties of objects that produced that wave front. In fabricating computer-generated holograms, numerical ::: data produced by a computer are converted into a physical hologram or an optical element intended for forming real ::: optical beams. Therefore mutual correspondence between optical transformations and their computer representations is ::: of fundamental importance for digital holography. In the paper, this problem is addressed and different computer ::: representations of basic optical transforms such as convolution, Fourier and Fresnel integral transforms are briefly ::: reviewed.
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paper_title: Diffraction from a wavelet point of view

paper_content:

The system impulse response representing the Fresnel diffraction is shown to form a wavelet family of functions. The scale parameter of the wavelet family represents the depth (distance). This observation relates the diffraction-holography-related studies and the wavelet theory. The results may be used in various optical applications such as designing robust volume optical elements for optical signal processing and finding new formulations for optical inverse problems. The results also extend the wavelet concept to the nonbandpass family of functions with the implication of new applications in signal processing. The presented wavelet structure, for example, is a tool for space–depth analysis.
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paper_title: The Chirplet Transform: Physical Considerations

paper_content:

We consider a multidimensional parameter space formed by inner products of a parameterizable family of chirp functions with a signal under analysis. We propose the use of quadratic chirp functions (which we will call q-chirps for short), giving rise to a parameter space that includes both the time-frequency plane and the time-scale plane as 2-D subspaces. The parameter space contains a "time-frequency-scale volume" and thus encompasses both the short-time Fourier transform (as a slice along the time and frequency axes) and the wavelet transform (as a slice along the time and scale axes). In addition to time, frequency, and scale, there are two other coordinate axes within this transform space: shear in time (obtained through convolution with a q-chirp) and shear in frequency (obtained through multiplication by a q-chirp). Signals in this multidimensional space can be obtained by a new transform, which we call the "q-chirplet transform" or simply the "chirplet transform". The proposed chirplets are generalizations of wavelets related to each other by 2-D affine coordinate transformations (translations, dilations, rotations, and shears) in the time-frequency plane, as opposed to wavelets, which are related to each other by 1-D affine coordinate transformations (translations and dilations) in the time domain only.
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paper_title: Family of scaling chirp functions, diffraction, and holography

paper_content:

It is observed that diffraction is a convolution operation with a chirp kernel whose argument is scaled. Family of functions obtained from a prototype by shifting and argument scaling form the essential ground for wavelet framework. Therefore, a connection between diffraction and wavelet transform is developed. However, wavelet transform is essentially prescribed for time-frequency and/or multiresolution analysis which is irrelevant in our case. Instead, the proposed framework is useful in various location-depth type of analysis in imaging. The linear transform when the analyzing functions are the chirps is called the scaling chirp transform. The scaled chirp functions do not satisfy the commonly used admissibility condition for wavelets. However, it is formally shown that these neither band nor time limited signals can be used as wavelet functions and the inversion is still possible. Diffraction and in-line holography are revisited within the scaling chirp transform context. It is formally proven that a volume in-line hologram gives perfect reconstruction. The developed framework for wave propagation based phenomena has the potential of advancing both signal processing and optical applications. >
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paper_title: Fresnelets: new multiresolution wavelet bases for digital holography

paper_content:

We propose a construction of new wavelet-like bases that are well suited for the reconstruction and processing of optically generated Fresnel holograms recorded on CCD-arrays. The starting point is a wavelet basis of L2 to which we apply a unitary Fresnel transform. The transformed basis functions are shift-invariant on a level-by-level basis but their multiresolution properties are governed by the special form that the dilation operator takes in the Fresnel domain. We derive a Heisenberg-like uncertainty relation that relates the localization of Fresnelets with that of their associated wavelet basis. According to this criterion, the optimal functions for digital hologram processing turn out to be Gabor functions, bringing together two separate aspects of the holography inventor's work. We give the explicit expression of orthogonal and semi-orthogonal Fresnelet bases corresponding to polynomial spline wavelets. This special choice of Fresnelets is motivated by their near-optimal localization properties and their approximation characteristics. We then present an efficient multiresolution Fresnel transform algorithm, the Fresnelet transform. This algorithm allows for the reconstruction (backpropagation) of complex scalar waves at several user-defined, wavelength-independent resolutions. Furthermore, when reconstructing numerical holograms, the subband decomposition of the Fresnelet transform naturally separates the image to reconstruct from the unwanted zero-order and twin image terms. This greatly facilitates their suppression. We show results of experiments carried out on both synthetic (simulated) data sets as well as on digitally acquired holograms.
---
paper_title: A fast refinement for adaptive Gaussian chirplet decomposition

paper_content:

The chirp function is one of the most fundamental functions in nature. Many natural events, for example, most signals encountered in seismology and the signals in radar systems, can be modeled as the superposition of short-lived chirp functions. Hence, the chirp-based signal representation, such as the Gaussian chirplet decomposition, has been an active research area in the field of signal processing. A main challenge of the Gaussian chirplet decomposition is that Gaussian chirplets do not form an orthogonal basis. A promising solution is to employ adaptive type signal decomposition schemes, such as the matching pursuit. The general underlying theory of the matching pursuit method has been well accepted, but the numerical implementation, in terms of computational speed and accuracy, of the adaptive Gaussian chirplet decomposition remains an open research topic. We present a fast refinement algorithm to search for optimal Gaussian chirplets. With a coarse dictionary, the resulting adaptive Gaussian chirplet decomposition is not only fast but is also more accurate than other known adaptive schemes. The effectiveness of the algorithm introduced is demonstrated by numerical simulations.
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paper_title: Diffraction-specific fringe computation for electro-holography

paper_content:

Diffraction-specific fringe computation is a novel system for the generation of holographic fringe patterns for real-time display. This thesis describes the development, implementation, and analysis of diffraction-specific computation, an approach that considers the reconstruction process rather than the interference process in optical holography. The primary goal is to increase the speed of holographic computation for real-time three-dimensional electro-holographic (holovideo) displays. Diffraction-specific fringe computation is based on the discretization of space and spatial frequency in the fringe pattern. Two holographic fringe encoding techniques are developed from diffraction-specific fringe computation and applied to make most efficient use of hologram channel capacity. A "hogel-vector encoding" technique is based on undersampling the fringe spectra. A "fringelet encoding" technique is designed to increase the speed and simplicity of decoding. The analysis of diffraction-specific computation focuses on the trade-offs between compression ratio, image fidelity, and image depth. The decreased image resolution (increased point spread) that is introduced into holographic images due to encoding is imperceptible to the human visual system under certain conditions. A compression ratio of 16 is achieved (using either encoding method) with an acceptably small loss in image resolution. Total computation time is reduced by a factor of over 100 to less than 7.0 seconds per 36-MB holographic fringe using the fringelet encoding method. Diffraction-specific computation more efficiently matches the information content of holographic fringes to the human visual system. Diffraction-specific holographic encoding allows for "visual-bandwidth holography," i.e., holographic imaging that requires a bandwidth commensurate with the usable visual information contained in an image. Diffraction-specific holographic encoding enables the integration of holographic information with other digital media, and is therefore vital to applications of holovideo in the areas of visualization, entertainment, and information, including education, telepresence, medical imaging, interactive design, and scientific visualization. Copies available exclusively from MIT Libraries, Rm. 14-0551, Cambridge, MA 02139-4307. Ph. 617-253-5668; Fax 617-253-1690.
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paper_title: Fast matching pursuit with a multiscale dictionary of gaussian chirps

paper_content:

We introduce a modified matching pursuit algorithm, called fast ridge pursuit, to approximate N-dimensional signals with M Gaussian chirps at a computational cost O(MN) instead of the expected O(MN/sup 2/logN). At each iteration of the pursuit, the best Gabor atom is first selected, and then, its scale and chirp rate are locally optimized so as to get a "good" chirp atom, i.e., one for which the correlation with the residual is locally maximized. A ridge theorem of the Gaussian chirp dictionary is proved, from which an estimate of the locally optimal scale and chirp is built. The procedure is restricted to a sub-dictionary of local maxima of the Gaussian Gabor dictionary to accelerate the pursuit further. The efficiency and speed of the method is demonstrated on a sound signal.
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paper_title: Frequency analysis of digital holography

paper_content:

A coherent optical imaging system consisting of recording a digital hologram by a CCD array and numerical reconstruction of the complex wave field in a computer is subjected to a frequency analysis. This analysis recognizes Fresnel and lensless Fourier transform holography; collimated and divergent reference waves; as well as the real image, the virtual image, and the dc term. The influences of finite sampling and the fill factor of the CCD pixels are examined. The impulse response of the system is a shifted Fraunhofer diffraction pattern of the aperture defined by the CCD. A fill factor below unity leads to a contrast decrease, which is quantitatively described in the modulation transfer function.
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paper_title: Fresnel transform and sampling theorem

paper_content:

Abstract A sampling theorem for functions whose Fresnel transform has finite support is established. Its possible applications to optics are outlined.
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paper_title: Adaptive chirplet based signal approximation

paper_content:

The chirp function is one of the most fundamental functions in nature. Many natural events can be roughly approximated by a group of chirp functions. In this paper, we present a practical adaptive chirplet based signal approximation algorithm. Unlike the other chirplet decompositions known so far, the elementary chirplet functions employed in this algorithm are adaptive. Therefore, the resulting approximation could better match the underlying signal and uses fewer coefficients. The effectiveness of the algorithm is demonstrated by numerical simulations.
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paper_title: Optimum sampling of Fresnel transforms.

paper_content:

The opportunity to process signals in domains other than the time or frequency domains arises naturally in coherently illuminated optical systems that produce Fourier transforms. It is well known that N samples are sufficient to represent the information content in the object, image, and Fourier planes. We extend these results to show that we can accurately represent the intensity signal in any Fresnel plane of a coherently illuminated optical system with exactly N samples, provided that we use a specified nonuniform sampling technique.
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paper_title: Compression of Digital Holograms for Three-Dimensional Object Reconstruction and Recognition

paper_content:

We present the results of applying lossless and lossy data compression to a three-dimensional object reconstruction and recognition technique based on phase-shift digital holography. We find that the best lossless (Lempel-Ziv, Lempel-Ziv-Welch, Huffman, Burrows-Wheeler) compression rates can be expected when the digital hologram is stored in an intermediate coding of separate data streams for real and imaginary components. The lossy techniques are based on subsampling, quantization, and discrete Fourier transformation. For various degrees of speckle reduction, we quantify the number of Fourier coefficients that can be removed from the hologram domain, and the lowest level of quantization achievable, without incurring significant loss in correlation performance or significant error in the reconstructed object domain.
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paper_title: Effects of quantization in phase-shifting digital holography.

paper_content:

We discuss quantization effects of hologram recording on the quality of reconstructed images in phase-shifting digital holography. We vary bit depths of phase-shifted holograms in both numerical simulation and experiments and then derived the complex amplitude, which is subjected to Fresnel transformation for the image reconstruction. The influence of bit-depth limitation in quantization has been demonstrated in a numerical simulation for spot-array patterns with linearly varying intensities and a continuous intensity object. The objects are provided with uniform and random phase modulation. In experiments, digital holograms are originally recorded at 8 bits and the bit depths are changed to deliver holograms at bit depths of 1 to 8 bits for the image reconstruction. The quality of the reconstructed images has been evaluated for the different quantization levels.
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paper_title: Optimum quantization in digital holography.

paper_content:

Finite plotter resolution introduces quantization into the representation of digital holograms. By utilizing the statistical properties of the Fourier transform of random phase images, optimum quantization schemes are derived and tabulated for the representation of these transforms. These schemes are applied to Lohmann and Lee type holograms where it is found that measured quantization errors agree with theoretical predictions. Methods for predicting optimum quantization schemes and associated mean squared errors are simplified to table-lookup procedures.
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paper_title: A companding approach for nonuniform quantization of digital holograms of three-dimensional objects

paper_content:

We apply two novel nonuniform quantization techniques to digital holograms of three-dimensional real-world objects. Our companding approach, combines the efficiency of uniform quantization with the improved performance of nonuniform quantization. We show that the performance of companding techniques can be comparable with k-means clustering and a competitive neural network, while only requiring a single-pass processing step. The quantized holographic pixels are coded using lossless techniques for the calculation of compression ratio.
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paper_title: The Shannon sampling theorem&#8212;Its various extensions and applications: A tutorial review

paper_content:

It has been almost thirty years since Shannon introduced the sampling theorem to communications theory. In this review paper we will attempt to present the various contributions made for the sampling theorems with the necessary mathematical details to make it self-contained. We will begin by a clear statement of Shannon's sampling theorem followed by its applied interpretation for time-invariant systems. Then we will review its origin as Whittaker's interpolation series. The extensions will include sampling for functions of more than one variable, random processes, nonuniform sampling, nonband-limited functions, implicit sampling, generalized functions (distributions), sampling with the function and its derivatives as suggested by Shannon in his original paper, and sampling for general integral transforms. Also the conditions on the functions to be sampled will be summarized. The error analysis of the various sampling expansions, including specific error bounds for the truncation, aliasing, jitter and parts of various other errors will be discussed and summarized. This paper will be concluded by searching the different recent applications of the sampling theorems in other fields, besides communications theory. These include optics, crystallography, time-varying systems, boundary value problems, spline approximation, special functions, and the Fourier and other discrete transforms.
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paper_title: Quadrature Formulae And Asymptotic Error Expansions For Wavelet Approximations Of Smooth Functions

paper_content:

This paper deals with typical problems that arise when using wavelets in numerical analysis applications. The first part involves the construction of quadrature formulae for the calculation of inner products of smooth functions and scaling functions. Several types of quadratures are discussed and compared for different classes of wavelets. Since their construction using monomials is ill-conditioned, also a modified, well-conditioned construction using Chebyshev polynomials is presented. The second part of the paper deals with pointwise asymptotic error expansions of wavelet approximations of smooth functions. They are used to derive asymptotic interpolating properties of the wavelet approximation and to construct a convergence acceleration algorithm. This is illustrated with numerical examples.
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paper_title: Irregular sampling and frames

paper_content:

Abstract . A theory of irregular sampling is developed for the class of real sampling sequences for which there are L 2 -convergent sampling formulas. The sampling sequences are effectively characterized, and the formulas are accompanied by methods of computing coefficients. These sampling formulas depend on the theory of coherent state (Gabor) frames and an analysis of the inverse frame operator. The results include regular sampling theory and the irregular sampling theory of Paley-Wiener, Levinson, Beutler, and Yao-Thomas. The chapter also presents a new aliasing technique, perspective on stability and uniqueness, and references to recent contributions by others.
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paper_title: Quantitative Fourier Analysis of Approximation Techniques: Part I - Interpolators and Projectors

paper_content:

We present a general Fourier-based method that provides an accurate prediction of the approximation error as a function of the sampling step T. Our formalism applies to an extended class of convolution-based signal approximation techniques, which includes interpolation, generalized sampling with prefiltering, and the projectors encountered in wavelet theory. We claim that we can predict the L/sup 2/-approximation error by integrating the spectrum of the function to approximate-not necessarily bandlimited-against a frequency kernel E(/spl omega/) that characterizes the approximation operator. This prediction is easier yet more precise than was previously available. Our approach has the remarkable property of providing a global error estimate that is the average of the true approximation error over all possible shifts of the input function. Our error prediction is exact for stationary processes, as well as for bandlimited signals. We apply this method to the comparison of standard interpolation and approximation techniques. Our method has interesting implications for approximation theory. In particular, we use our results to obtain some new asymptotic expansions of the error as T/spl rarr/0, as well as to derive improved upper bounds of the kind found in the Strang-Fix (1971) theory. We finally show how we can design quasi-interpolators that are near optimal in the least-squares sense.
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paper_title: Sampling procedures in function spaces and asymptotic equivalence with shannon's sampling theory

paper_content:

We view Shannon's sampling procedure as a problem of approximation in the space S = {s: s (x) = (c * sinc)(x)c e l 2}. We show that under suitable conditions on a generating function λ e L 2, the approximation problem onto the space V = {v:v(x) = (c * λ)(x)c e l 2} produces a sampling procedure similar to the classical one. It consists of an optimal prefiltering, a pure jitter-stable sampling, and a postfiltering for the reconstruction. We describe equivalent signal representations using generic, dual, cardinal, and orthogonal basis functions and give the expression of the corresponding filters. We then consider sequences λn, where λn denotes the n-fold convolution of λ. They provide a sequence of increasingly regular sampling schemes as the value of n increases. We show that the cardinal and orthogonal pre- and postfilters associated with these sequences asymptotically converge to the ideal lowpass filter of Shannon. The theory is illustrated using several eamples.
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paper_title: An overview of wavelet based multiresolution analyses

paper_content:

In this paper an overview of wavelet based multiresolution analyses is presented. First, the continuous wavelet transform in its simplest form is discussed. Then, the definition of a multiresolution analysis is given and how wavelets fit into it is shown. The authors take a closer look at orthogonal, biorthogonal and semiorthogonal wavelets. The fast wavelet transform, wavelets on an interval, multidimensional wavelets and wavelet packets are discussed. Several examples of wavelet families are introduced and compared. Finally, the essentials of two major applications are outlined: data compression and compression of linear operators.
---
paper_title: Sampling-50 years after Shannon

paper_content:

This paper presents an account of the current state of sampling, 50 years after Shannon's formulation of the sampling theorem. The emphasis is on regular sampling, where the grid is uniform. This topic has benefitted from a strong research revival during the past few years, thanks in part to the mathematical connections that were made with wavelet theory. To introduce the reader to the modern, Hilbert-space formulation, we reinterpret Shannon's sampling procedure as an orthogonal projection onto the subspace of band-limited functions. We then extend the standard sampling paradigm for a presentation of functions in the more general class of "shift-in-variant" function spaces, including splines and wavelets. Practically, this allows for simpler-and possibly more realistic-interpolation models, which can be used in conjunction with a much wider class of (anti-aliasing) prefilters that are not necessarily ideal low-pass. We summarize and discuss the results available for the determination of the approximation error and of the sampling rate when the input of the system is essentially arbitrary; e.g., nonbandlimited. We also review variations of sampling that can be understood from the same unifying perspective. These include wavelets, multiwavelets, Papoulis generalized sampling, finite elements, and frames. Irregular sampling and radial basis functions are briefly mentioned.
---
paper_title: A sampling theorem associated with quasi-Fourier transform

paper_content:

We propose, in this correspondence, a unified generalized bandlimited sampling theorem associated with quasi-Fourier transform of the form (F/sub /spl phi//f)(/spl xi/)=/spl int//sub D/f(x) exp(-i/spl epsiv//spl phi/(x)) dx, where the phase function satisfies /spl phi/'(x)>0 in its domain. Our result contains classical Shannon's sampling theorem and the Sundaram et al. (1997) scale-limited sampling theorem as a special case. New results such as the chirp-limited sampling theorem can be established.
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paper_title: The exact value of the Paley - Wiener constant

paper_content:

A method for the manufacture of matches which comprises applying to a match splint, before the formation thereon of the match head, a surface coating of an aqueous dispersion comprising (i) a low melting point solid fuel e.g. paraffin wax, (ii) a finely divided particulate or fibrous filler and (iii) either a water-immiscible volatile solvent for the fuel or a water-soluble or water-dispersible adhesive, or a mixture of the two; and thereafter drying the coated splint to deposit thereon adjacent the end thereof, on which the head is to be formed, a surface coating comprising a mixture of said fuel and said filler, and, if present, said adhesive.
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paper_title: Approximation Error for Quasi-Interpolators and ( Multi-) Wavelet Expansions

paper_content:

We investigate the approximation properties of general polynomial preserving operators that approximate a function into some scaled subspace of L 2 via an appropriate sequence of inner products. In particular, we consider integer shift-invariant approximations such as those provided by splines and wavelets, as well as finite elements and multi-wavelets which use multiple generators. We estimate the approximation error as a function of the scale parameter T when the function to approximate is sufficiently regular. We then present a generalized sampling theorem, a result that is rich enough to provide tight bounds as well as asymptotic expansions of the approximation error as a function of the sampling step T. Another more theoretical consequence is the proof of a conjecture by Strang and Fix, which states the equivalence between the order of a multi-wavelet space and the order of a particular subspace generated by a single function. Finally, we consider refinable generating functions and use the two-scale relation to obtain explicit formulae for the coefficients of the asymptotic development of the error. The leading constants are easily computable and can be the basis for the comparison of the approximation power of wavelet and multi-wavelet expansions of a given order L. © 1999 Academic Press
---
paper_title: Fresnelets: new multiresolution wavelet bases for digital holography

paper_content:

We propose a construction of new wavelet-like bases that are well suited for the reconstruction and processing of optically generated Fresnel holograms recorded on CCD-arrays. The starting point is a wavelet basis of L2 to which we apply a unitary Fresnel transform. The transformed basis functions are shift-invariant on a level-by-level basis but their multiresolution properties are governed by the special form that the dilation operator takes in the Fresnel domain. We derive a Heisenberg-like uncertainty relation that relates the localization of Fresnelets with that of their associated wavelet basis. According to this criterion, the optimal functions for digital hologram processing turn out to be Gabor functions, bringing together two separate aspects of the holography inventor's work. We give the explicit expression of orthogonal and semi-orthogonal Fresnelet bases corresponding to polynomial spline wavelets. This special choice of Fresnelets is motivated by their near-optimal localization properties and their approximation characteristics. We then present an efficient multiresolution Fresnel transform algorithm, the Fresnelet transform. This algorithm allows for the reconstruction (backpropagation) of complex scalar waves at several user-defined, wavelength-independent resolutions. Furthermore, when reconstructing numerical holograms, the subband decomposition of the Fresnelet transform naturally separates the image to reconstruct from the unwanted zero-order and twin image terms. This greatly facilitates their suppression. We show results of experiments carried out on both synthetic (simulated) data sets as well as on digitally acquired holograms.
---
paper_title: A sampling theorem for wavelet subspaces

paper_content:

The classical Shannon sampling theorem is extended to the subspaces used in the multiresolution analysis in wavelet theory. Under weak hypotheses, these subspaces are first shown to have a Riesz basis formed from the reproducing kernels. These in turn are used to construct the sampling sequences. Examples are given. >
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paper_title: Theory and Design of Local Interpolators

paper_content:

Abstract This paper shows that an error spectrum can be used to describe the performance of any convolutional interpolator used to shift an oversampled image. This spectrum is linear in the image power spectrum and in an error factor that depends only on the interpolator and the shift. The same form is shown to describe the interpolation of undersampled data, in an average sense. Simple formulas are derived for the error factor in either Fourier or real space, and standard interpolators are evaluated with them. Optimal interpolators are derived for various theoretical spectra: constant inband, Lorentzian, power law, and Gaussian. Practical methods of interpolator design are devised for use with image spectra that are known only partially or are not easily characterized analytically.
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paper_title: A Wavelet Tour of Signal Processing

paper_content:

Introduction to a Transient World. Fourier Kingdom. Discrete Revolution. Time Meets Frequency. Frames. Wavelet Zoom. Wavelet Bases. Wavelet Packet and Local Cosine Bases. An Approximation Tour. Estimations are Approximations. Transform Coding. Appendix A: Mathematical Complements. Appendix B: Software Toolboxes.
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paper_title: Atomic Decomposition by Basis Pursuit

paper_content:

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

paper_content:

Here we present a fast algorithm for Fresnel integral calculation. Some fast algorithms using the fast Fourier transform are analysed and their performance has been checked. These methods are of easy implementation, but are only valid for a specific range of distances. Fast algorithms based on the Fractional Fourier transform allow accurate evaluation of the Fresnel integral from object to Fraunhofer domain in a single step.
---
paper_title: Digital computation of the fractional Fourier transform

paper_content:

An algorithm for efficient and accurate computation of the fractional Fourier transform is given. For signals with time-bandwidth product N, the presented algorithm computes the fractional transform in O(NlogN) time. A definition for the discrete fractional Fourier transform that emerges from our analysis is also discussed.
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paper_title: Interpolating Between Periodicity and Discreteness Through the Fractional Fourier Transform

paper_content:

Periodicity and discreteness are Fourier duals in the same sense as operators such as coordinate multiplication and differentiation, and translation and phase shift. The fractional Fourier transform allows interpolation between such operators which gradually evolve from one member of the dual pair to the other as the fractional order goes from zero to one. Here, we similarly discuss the interpolation between the dual properties of periodicity and discreteness, showing how one evolves into the other as the order goes from zero to one. We also discuss the concepts of partial discreteness and partial periodicity and relate them to fractional discreteness and periodicity
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paper_title: General theory of linear, coherent optical data-processing systems

paper_content:

This paper concerns a general theory of linear optical data-processing systems. Well-known basic ideas are critically reviewed, fundamental constraints upon the system behavior are investigated, and a new operational notation is proposed. Every system is considered as a bidirectional processor and every two-dimensional signal is equivalently described in the space and frequency domain. Without referring to field-theoretical notation, the constraints of “reciprocity” and “losslessness” are introduced. In rough terms, reciprocity states equality of transmissions A → B and B → A, where A and B are points in the two reference planes, losslessness states equality of signal energies in the two planes. “Symmetry” has to be independently defined in the space and the frequency domain, and “shift invariance” and “spreadlessness” appear to be dual constraints. The important constraint of “time reversibility” is obtained when a system simultaneously satifies reciprocity and losslessness. In the second part, the general theory is applied to lenses, sections of free space, Fourier transformers, and magnifiers. With the aid of a suitable shorthand notation, the main results of Fourier optics can be easily derived.
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paper_title: On bandlimited signals with fractional Fourier transform

paper_content:

We study bandlimited signals with fractional Fourier transform (FRFT). We show that if a nonzero signal f is bandlimited with FRFT F/sub /spl alpha// for a certain real /spl alpha/, then it is not bandlimited with FRFT F/sub /spl beta// for any /spl beta/ with /spl beta//spl ne//spl plusmn//spl alpha/+n/spl pi/ for any integer n. This is a generalization of the fact that a nonzero signal can not be both timelimited and bandlimited. We also provide sampling theorems for bandlimited signals with FRFT that are similar to the Shannon sampling theorem.
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paper_title: A Linear Filtering Approach to the Computation of the Discrete Fourier Transform

paper_content:

This chapter contains sections titled: Introduction, An Algorithm Suggested by Chirp Filtering
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paper_title: Fractional-Fourier-transform calculation through the fast-Fourier-transform algorithm

paper_content:

A method for the calculation of the fractional Fourier transform (FRT) by means of the fast Fourier transform (FFT) algorithm is presented. The process involves mainly two FFT’s in cascade; thus the process has the same complexity as this algorithm. The method is valid for fractional orders varying from −1 to 1. Scaling factors for the FRT and Fresnel diffraction when calculated through the FFT are discussed.
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paper_title: Sparse Components of Images and Optimal Atomic Decompositions

paper_content:

Recently, Field, Lewicki, Olshausen, and Sejnowski have reported efforts to identify the ``Sparse Components'' of image data. Their empirical findings indicate that such components have elongated shapes and assume a wide range of positions, orientations, and scales.
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paper_title: Efficient computation of quadratic-phase integrals in optics

paper_content:

We present a fast NlogN time algorithm for computing quadratic-phase integrals. This three-parameter class of integrals models propagation in free space in the Fresnel approximation, passage through thin lenses, and propagation in quadratic graded-index media as well as any combination of any number of these and is therefore of importance in optics. By carefully managing the sampling rate, one need not choose N much larger than the space-bandwidth product of the signals, despite the highly oscillatory integral kernel. The only deviation from exactness arises from the approximation of a continuous Fourier transform with the discrete Fourier transform. Thus the algorithm computes quadratic-phase integrals with a performance similar to that of the fast-Fourier-transform algorithm in computing the Fourier transform, in terms of both speed and accuracy.
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paper_title: Improved discrete fractional Fourier transform

paper_content:

The fractional Fourier transform is a useful mathematical operation that generalizes the well-known continuous Fourier transform. Several discrete fractional Fourier transforms (DFRFT’s) have been developed, but their results do not match those of the continuous case. We propose a new DFRFT. This improved DFRFT provides transforms similar to those of the continuous fractional Fourier transform and also retains the rotation properties.
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paper_title: Calculation of the Rayleigh–Sommerfeld diffraction integral by exact integration of the fast oscillating factor

paper_content:

We describe a numerical method that can be used to calculate the propagation of light in a medium of constant (possibly complex) index of refraction n. The method integrates the Rayleigh–Sommerfeld diffraction integral numerically. After an appropriate change of integration variables, the integrand of the diffraction integral is split into a slowly varying and an (often fast) oscillating quadratic factor. The slowly varying factor is approximated by a spline fit, and the resulting Fresnel integrals are subsequently integrated exactly. Although the method is not as fast as methods involving a fast Fourier transform, such as plane-wave propagation or Fresnel approximation, it is accurate over a greater range than these methods.
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paper_title: Fresnel diffraction and the fractional-order Fourier transform

paper_content:

Fractional-order Fourier transforms are adapted to the mathematical expression of Fresnel diffraction, just as the standard Fourier transform is adapted to Fraunhofer diffraction. The continuity of fractional Fourier transforms with respect to their orders corresponds to the continuity of wave propagation, and their composition is in accordance with the Huygens principle.
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paper_title: Fractional discrete Fourier transforms

paper_content:

Direct calculation of fractional Fourier transforms from the expressions derived for their optical implementation is laborious. An extension of the discrete Fourier transform would have only O(N2) computational complexity. We define such a system, offer a general way to compute the fractional discrete Fourier transform matrix, and numerically validate the algorithm.
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paper_title: Brushlets: A Tool for Directional Image Analysis and Image Compression

paper_content:

Abstract We construct a new adaptive basis of functions which is reasonably well localized with only one peak in frequency. We develop a compression algorithm that exploits this basis to obtain the most economical representation of the image in terms of textured patterns with different orientations, frequencies, sizes, and positions. The technique directly works in the Fourier domain and has potential applications for compression of highly textured images, texture analysis, etc.
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paper_title: A measurement method based on a modified version of the chirplet transform for instantaneous frequency estimation

paper_content:

A measurement method for instantaneous frequency estimation is presented in this paper. The method is based on the use of the chirplet transform, a linear time-frequency representation (TFR) allowing additional modifications of each cell on the time-frequency plane with respect to other TFRs. In particular, a modified version of this transform is proposed here; a bending effect can be further imposed on the cells. Thanks both to this feature and a suitable measurement procedure, properly set up by the authors, the method assures a satisfying accuracy in reconstructing the instantaneous frequency trajectory of monocomponent signals as well as a good resolving capability in the analysis of multicomponent signals whose instantaneous frequency trajectories are strongly nonlinear and very close to one another. Theoretical details concerning the chirplet transform and its modified version are first given. Then, the proposed measurement procedure for choosing the optimal values of the parameters of the transform according to the local features of the analyzed signal is described. At the end, the results of several experimental tests conducted both on monocomponent and multicomponent signals are presented; advantages over other solutions are also highlighted.
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paper_title: An Efficient Approach to the Linear Least Squares Problem

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In this paper we present a partially orthogonal decomposition for a matrix A. Using this decomposition the linear least squares problem is reduced to solving two linear systems. The matrix of the first system is symmetric and positive definite, and the matrix of the second system is nonsingular upper triangular. We show that this approach can provide computational savings.
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paper_title: Projection-Slice Theorem as a Tool for Mathematical Representation of Diffraction

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Although the impulse (Dirac delta) function has been widely used as a tool in signal processing, its more complicated counterpart, the impulse function over higher dimensional manifolds in RopfN, did not get such a widespread utilization. Based on carefully made definitions of such functions, it is shown that many higher dimensional signal processing problems can be better formulated, yielding more insight and flexibility, using these tools. The well-known projection-slice theorem is revisited using these impulse functions. As a demonstration of the utility of the projection-slice formulation using impulse functions over hyperplanes, the scalar optical diffraction is reformulated in a more general context
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paper_title: Impulse functions over curves and surfaces and their applications to diffraction

paper_content:

Abstract An explicit preferred definition of impulse functions (Dirac delta functions) over lower-dimensional manifolds in R N is given in such a way to assure uniform concentration per geometric unit of the manifold. Some related properties are presented. An application related to diffraction is demonstrated.
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paper_title: Two-dimensional spectral analysis of cortical receptive field profiles

paper_content:

Abstract Most vision research embracing the spatial frequency paradigm has been conceptually and mathematically a one-dimensional analysis of two-dimensional mechanisms. Spatial vision models and the experiments sustaining them have generally treated spatial frequency as a one-dimensional variable, even though receptive fields and retinal images are two-dimensional and linear transform theory obliges any frequency analysis to preserve dimension. Four models of cortical receptive fields are introduced and studied here in 2D form, in order to illustrate the relationship between their excitatory/inhibitory spatial structure and their resulting 2D spectral properties. It emerges that only a very special analytic class of receptive fields possess independent tuning functions for spatial frequency and orientation; namely, those profiles whose two-dimensional Fourier Transforms are expressible as the separable product of a radial function and an angular function. Furthermore, only such receptive fields would have the same orientation tuning curve for single bars as for gratings. All classes lacking this property would describe cells responsive to different orientations for different spatial frequencies and vice versa; this is shown to be the case, for example, for the Hubel & Wiesel model of cortical orientation-tuned simple cells receiving inputs from an aligned row of center/surround LGN cells. When these results are considered in conjunction with psychophysical evidence for nonseparability of spatial frequency and orientation tuning properties within a “channel”, it becomes mandatory that future spatial vision research of the Fourier genre take on an explicitly two-dimensional character.
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paper_title: IV: Fractional Transformations in Optics

paper_content:

Publisher Summary This chapter discusses the fractional transformations in optics. The chapter presents the fundamentals of the fractional Fourier transformation (FRT). A complete signal description, simultaneously displaying space and frequency information, can be achieved by the space-frequency Wigner distribution function (WDF). An application of the FRT is related to signal multiplexing. Due to the ability of the FRT to rotate the Wigner chart, the Wigner distribution of a signal may be arranged in a more efficient manner. The chapter explores the fractionalization procedure of Zernike's phase contrast microscope. This method is a spatial filtering process similar to the optical Hilbert transformation. The chapter discusses the harmonic real transformations, and provides the applications of several fractional transformations. Some aspects of the fractionalization techniques are also discussed in the chapter. These applications are “filtering processes”. The chapter also describes FRT and some other optical transformations based on the framework of group theory.
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paper_title: Complex Wavelets for Shift Invariant Analysis and Filtering of Signals

paper_content:

Abstract This paper describes a form of discrete wavelet transform, which generates complex coefficients by using a dual tree of wavelet filters to obtain their real and imaginary parts. This introduces limited redundancy (2m:1 for m-dimensional signals) and allows the transform to provide approximate shift invariance and directionally selective filters (properties lacking in the traditional wavelet transform) while preserving the usual properties of perfect reconstruction and computational efficiency with good well-balanced frequency responses. Here we analyze why the new transform can be designed to be shift invariant and describe how to estimate the accuracy of this approximation and design suitable filters to achieve this. We discuss two different variants of the new transform, based on odd/even and quarter-sample shift (Q-shift) filters, respectively. We then describe briefly how the dual tree may be extended for images and other multi-dimensional signals, and finally summarize a range of applications of the transform that take advantage of its unique properties.
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paper_title: Relationships between the Radon–Wigner and fractional Fourier transforms

paper_content:

Two recently described transforms are shown to be related. The Radon–Wigner transform is the squared modulus of the fractional Fourier transform. This new theorem may serve to translate signal and image processing results between different signal representations. Some consequences regarding moments are presented, including a new fractional-Fourier-transform uncertainty relation. Implications for processing are suggested.
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paper_title: Convolution, filtering, and multiplexing in fractional Fourier domains and their relation to chirp and wavelet transforms

paper_content:

A concise introduction to the concept of fractional Fourier transforms is followed by a discussion of their relation to chirp and wavelet transforms. The notion of fractional Fourier domains is developed in conjunction with the Wigner distribution of a signal. Convolution, filtering, and multiplexing of signals in fractional domains are discussed, revealing that under certain conditions one can improve on the special cases of these operations in the conventional space and frequency domains. Because of the ease of performing the fractional Fourier transform optically, these operations are relevant for optical information processing.
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paper_title: The Wigner distribution function applied to optical signals and systems

paper_content:

In this paper the Wigner distribution function has been introduced for optical signals and systems. The Wigner distribution function of an optical signal appears to be in close resemblance to the ray concept in geometrical optics. This resemblance reaches even farther: although derived from Fourier optics, the Wigner distribution functions of some elementary optical systems can directly be interpreted in terms of geometrical optics.
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paper_title: Time-frequency localization operators: A geometric phase space approach

paper_content:

The author defines a set of operators which localize in both time and frequency. These operators are similar to but different from the low-pass time-limiting operator, the singular functions of which are the prolate spheroidal wave functions. The author's construction differs from the usual approach in that she treats the time-frequency plane as one geometric whole (phase space) rather than as two separate spaces. For disk-shaped or ellipse-shaped domains in time-frequency plane, the associated localization operators are remarkably simple. Their eigenfunctions are Hermite functions, and the corresponding eigenvalues are given by simple explicit formulas involving the incomplete gamma functions. >
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paper_title: The fractional fourier transform and the wigner distribution

paper_content:

The Wigner distribution and many other members of the Cohen class of generalized phase-space distributions of a signal all share certain translation properties and the property that their two marginal distributions of energy density along the time and along the frequency axes equal the signal power and the spectral energy density. A natural generalization of this last property is shown to be a certain relationship through the Radon transform between the distribution and the signal's fractional Fourier transform. It is shown that the Wigner distribution is now distinguished by being the only member of the Cohen class that has this generalized property as well as a generalized translation property. The inversion theorem for the Wigner distribution is then extended to yield the fractional Fourier transforms.
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paper_title: Misaligned first-order optics: canonical operator theory

paper_content:

Canonical operator theory of paraxial optics is generalized to address the case of misaligned optics. The formal group structure is extended from the aligned case in terms of Heisenberg–Weil and inhomogeneous canonical transforms and the associated 3 × 3 augmented ray matrices. Certain misalignment phase shifts that are often mistreated and ignored have been derived and incorporated into the theory.
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paper_title: Generalizing, optimizing, and inventing numerical algorithms for the fractional Fourier, Fresnel, and linear canonical transforms

paper_content:

By use of matrix-based techniques it is shown how the space–bandwidth product (SBP) of a signal, as indicated by the location of the signal energy in the Wigner distribution function, can be tracked through any quadratic-phase optical system whose operation is described by the linear canonical transform. Then, applying the regular uniform sampling criteria imposed by the SBP and linking the criteria explicitly to a decomposition of the optical matrix of the system, it is shown how numerical algorithms (employing interpolation and decimation), which exhibit both invertibility and additivity, can be implemented. Algorithms appearing in the literature for a variety of transforms (Fresnel, fractional Fourier) are shown to be special cases of our general approach. The method is shown to allow the existing algorithms to be optimized and is also shown to permit the invention of many new algorithms.
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paper_title: Application of the two-dimensional fractional-order Fourier transformation to particle field digital holography

paper_content:

We demonstrate that the fractional-order Fourier transformation is a suitable method to analyze the diffraction patterns of particle field holograms. This method permits reconstruction of in-line digital holograms beyond the Fraunhofer condition (d2/λz≃10). We show that the diameter of spherical particles is measured with good accuracy. Simulation and experimental results are presented.
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paper_title: The discrete fractional Fourier transform

paper_content:

We propose and consolidate a definition of the discrete fractional Fourier transform which generalizes the discrete Fourier transform (DFT) in the same sense that the continuous fractional Fourier transform (FRT) generalizes the continuous ordinary Fourier Transform. This definition is based on a particular set of eigenvectors of the DFT which constitutes the discrete counterpart of the set of Hermite-Gaussian functions. The fact that this definition satisfies all the desirable properties expected of the discrete FRT, supports our confidence that it will be accepted as the definitive definition of this transform.
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paper_title: Fourier optics described by operator algebra

paper_content:

Fresnel diffraction is described by replacing the Fresnel-Kirchhoff integral, the lens transfer factor, and other operations by operators. The resulting operator algebra leads to the description of Fourier optics in a simple and compact way, bypassing the cumbersome integral calculus. Aberration effects and Gaussian beam illumination are also treated as a simple extension of the present theory.
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paper_title: The Fractional Fourier Transform: With Applications in Optics and Signal Processing

paper_content:

Preface. Acknowledgments. Introduction. Signals, Systems, and Transformations. Wigner Distributions and Linear Canonical Transforms. The Fractional Fourier Transform. Time-Order and Space-Order Representations. The Discrete Fractional Fourier Transform. Optical Signals and Systems. Phase-Space Optics. The Fractional Fourier Transform in Optics. Applications of the Fractional Fourier Transform to Filtering, Estimation, and Signal Recovery. Applications of the Fractional Fourier Transform to Matched Filtering, Detection, and Pattern Recognition. Bibliography on the Fractional Fourier Transform. Other Cited Works. Credits. Index.
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paper_title: Joint Time-Frequency Analysis

paper_content:

It has been well understood that a given signal can be represented in an infinite number of different ways. Different signal representations can be used for different applications. For example, signals obtained from most engineering applications are usually functions of time. But when studying or designing the system, we often like to study signals and systems in the frequency domain. Although the frequency content of the majority of signals in the real world evolves over time, the classical power spectrum does not reveal such important information. In order to overcome this problem, many alternatives, such as the Gabor (1946) expansion, wavelets, and time-dependent spectra, have been developed and widely studied. In contrast to the classical time and frequency analysis, we name these new techniques joint time-frequency analysis. We introduce the basic concepts and well-tested algorithms for joint time-frequency analysis. Analogous to the classical Fourier analysis, we roughly partition this article into two parts: the linear (e.g., short-time Fourier transform, Gabor expansion) and the quadratic transforms (e.g., Wigner-Ville (1932, 1948) distribution). Finally, we introduce the so-called model-based (or parametric) time-frequency analysis method.
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paper_title: Adapted wavelet analysis from theory to software

paper_content:

This detail-oriented text is intended for engineers and applied mathematicians who must write computer programs to perform wavelet and related analysis on real data. It contains an overview of mathematical prerequisites and proceeds to describe hands-on programming techniques to implement special programs for signal analysis and other applications. From the table of contents: - Mathematical Preliminaries - Programming Techniques - The Discrete Fourier Transform - Local Trigonometric Transforms - Quadrature Filters - The Discrete Wavelet Transform - Wavelet Packets - The Best Basis Algorithm - Multidimensional Library Trees - Time-Frequency Analysis - Some Applications - Solutions to Some of the Exercises - List of Symbols - Quadrature Filter Coefficients
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paper_title: Mathematical Methods and Algorithms for Signal Processing

paper_content:

I. INTRODUCTION AND FOUNDATIONS. 1. Introduction and Foundations. II. VECTOR SPACES AND LINEAR ALGEBRA. 2. Signal Spaces. 3. Representation and Approximation in Vector Spaces. 4. Linear Operators and Matrix Inverses. 5. Some Important Matrix Factorizations. 6. Eigenvalues and Eigenvectors. 7. The Singular Value Decomposition. 8. Some Special Matrices and Their Applications. 9. Kronecker Products and the Vec Operator. III. DETECTION, ESTIMATION, AND OPTIMAL FILTERING. 10. Introduction to Detection and Estimation, and Mathematical Notation. 11. Detection Theory. 12. Estimation Theory. 13. The Kalman Filter. IV. ITERATIVE AND RECURSIVE METHODS IN SIGNAL PROCESSING. 14. Basic Concepts and Methods of Iterative Algorithms. 15. Iteration by Composition of Mappings. 16. Other Iterative Algorithms. 17. The EM Algorithm in Signal Processing. V. METHODS OF OPTIMIZATION. 18. Theory of Constrained Optimization. 19. Shortest-Path Algorithms and Dynamic Programming. 20. Linear Programming. APPENDIXES. A. Basic Concepts and Definitions. B. Completing the Square. C. Basic Matrix Concepts. D. Random Processes. E. Derivatives and Gradients. F. Conditional Expectations of Multinomial and Poisson r.v.s.
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paper_title: Primal-Dual Interior-Point Methods

paper_content:

Preface Notation 1. Introduction. Linear Programming Primal-Dual Methods The Central Path A Primal-Dual Framework Path-Following Methods Potential-Reduction Methods Infeasible Starting Points Superlinear Convergence Extensions Mehrotra's Predictor-Corrector Algorithm Linear Algebra Issues Karmarkar's Algorithm 2. Background. Linear Programming and Interior-Point Methods Standard Form Optimality Conditions, Duality, and Solution Sets The B {SYMBOL 200 \f "Symbol"} N Partition and Strict Complementarity A Strictly Interior Point Rank of the Matrix A Bases and Vertices Farkas's Lemma and a Proof of the Goldman-Tucker Result The Central Path Background. Primal Method Primal-Dual Methods. Development of the Fundamental Ideas Notes and References 3. Complexity Theory. Polynomial Versus Exponential, Worst Case vs Average Case Storing the Problem Data. Dimension and Size The Turing Machine and Rational Arithmetic Primal-Dual Methods and Rational Arithmetic Linear Programming and Rational Numbers Moving to a Solution from an Interior Point Complexity of Simplex, Ellipsoid, and Interior-Point Methods Polynomial and Strongly Polynomial Algorithms Beyond the Turing Machine Model More on the Real-Number Model and Algebraic Complexity A General Complexity Theorem for Path-Following Methods Notes and References 4. Potential-Reduction Methods. A Primal-Dual Potential-Reduction Algorithm Reducing Forces Convergence A Quadratic Estimate of \Phi _{\rho } along a Feasible Direction Bounding the Coefficients in The Quadratic Approximation An Estimate of the Reduction in \Phi _{\rho } and Polynomial Complexity What About Centrality? Choosing {SYMBOL 114 \f "Symbol"} and {SYMBOL 97 \f "Symbol"} Notes and References 5. Path-Following Algorithms. The Short-Step Path-Following Algorithm Technical Results The Predictor-Corrector Method A Long-Step Path-Following Algorithm Limit Points of the Iteration Sequence Proof of Lemma 5.3 Notes and References 6. Infeasible-Interior-Point Algorithms. The Algorithm Convergence of Algorithm IPF Technical Results I. Bounds on \nu _k \delimiter "026B30D (x^k,s^k) \delimiter "026B30D Technical Results II. Bounds on (D^k)^{-1} \Delta x^k and D^k \Delta s^k Technical Results III. A Uniform Lower Bound on {SYMBOL 97 \f "Symbol"}k Proofs of Theorems 6.1 and 6.2 Limit Points of the Iteration Sequence 7. Superlinear Convergence and Finite Termination. Affine-Scaling Steps An Estimate of ({SYMBOL 68 \f "Symbol"}x, {SYMBOL 68 \f "Symbol"} s). The Feasible Case An Estimate of ({SYMBOL 68 \f "Symbol"} x, {SYMBOL 68 \f "Symbol"} s). The Infeasible Case Algorithm PC Is Superlinear Nearly Quadratic Methods Convergence of Algorithm LPF+ Convergence of the Iteration Sequence {SYMBOL 206 \f "Symbol"}(A,b,c) and Finite Termination A Finite Termination Strategy Recovering an Optimal Basis More on {SYMBOL 206 \f "Symbol"} (A,b,c) Notes and References 8. Extensions. The Monotone LCP Mixed and Horizontal LCP Strict Complementarity and LCP Convex QP Convex Programming Monotone Nonlinear Complementarity and Variational Inequalities Semidefinite Programming Proof of Theorem 8.4. Notes and References 9. Detecting Infeasibility. Self-Duality The Simplified HSD Form The HSDl Form Identifying a Solution-Free Region Implementations of the HSD Formulations Notes and References 10. Practical Aspects of Primal-Dual Algorithms. Motivation for Mehrotra's Algorithm The Algorithm Superquadratic Convergence Second-Order Trajectory-Following Methods Higher-Order Methods Further Enhancements Notes and References 11. Implementations. Three Forms of the Step Equation The Cholesky Factorization Sparse Cholesky Factorization. Minimum-Degree Orderings Other Orderings Small Pivots in the Cholesky Factorization Dense Columns in A The Augmented System Formulat
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paper_title: Computation considerations and fast algorithms for calculating the diffraction integral

paper_content:

Abstract One of the most basic optical ‘components’ is free-space propagation. A common approximation used when calculating the resultant field distribution after propagation is the Fresnel integral. This integral can be evaluated in two ways: directly or by using the angular spectrum. In this paper, we estimate the regions in which each mode of evaluation is preferable according to computing efficiency and accuracy considerations. A fast numerical algorithm is introduced for each region. The result is relevant also for the evaluation of the Rayleigh-Sommerfeld diffraction formula.
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paper_title: Gabor’s Signal Expansion and Its Relation to Sampling of the Sliding-Window Spectrum

paper_content:

It is sometimes convenient to describe a time signal ϕ(t), say, not in the time domain, but in the frequency domain by means of its frequency spectrum, i.e., the Fourier transform \(\bar{\varphi}(\omega)\) of the function ϕ(t) which is defined by ::: ::: $$\bar{\varphi }(\omega ) = \int {\varphi (t){{e}^{{ - j\omega t}}}dt;}$$ ::: ::: a bar on top of a symbol will mean throughout that we are dealing with a function in the frequency domain. (Unless otherwise stated, all integrations and summations in this contribution extend from -∞ to +∞.) The frequency spectrum shows us the global distribution of the energy of the signal as a function of frequency. However, one is often more interested in the momentary or local distribution of the energy as a function of frequency.
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paper_title: An orthonormal class of exact and simple DFT eigenvectors with a high degree of symmetry

paper_content:

The paper presents a novel orthonormal class of eigenvectors of the discrete Fourier transform (DFT) whose order N is factored as N=rM/sup 2/. The DFT eigenvectors have the form e=E/spl alpha/, where /spl alpha/ are eigenvectors of some /spl lscr/ /spl times//spl lscr/ matrices, given by, or related to, the DFT matrix of order r, with /spl lscr/ = r, 2r, or 4r, and the matrix E expands /spl alpha/ to the full DFT size N=rM/sup 2/. In particular, when N is an arbitrarily large power of 2, r may be 1 or 2. The resulting eigenvectors are expressed exactly with simple exponential expressions, have a considerable number of elements constrained to 0, and show a high degree of symmetry. The derivation of such a class is based on a partition of the N-dimensional linear space into subspaces of very small dimension (r, 2r or 4r).
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paper_title: Discrete Fourier transforms when the number of data samples is prime

paper_content:

The discrete Fourier transform of a sequence of N points, where N is a prime number, is shown to be essentially a circular correlation. This can be recognized by rearranging the members of the sequence and the transform according to a rule involving a primitive root of N. This observation permits the discrete Fourier transform to be computed by means of a fast Fourier transform algorithm, with the associated increase in speed, even though N is prime.
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paper_title: Wigner distribution function and its application to first-order optics

paper_content:

The Wigner distribution function of optical signals and systems has been introduced. The concept of such functions is not restricted to deterministic signals, but can be applied to partially coherent light as well. Although derived from Fourier optics, the description of signals and systems by means of Wigner distribution functions can be interpreted directly in terms of geometrical optics: (i) for quadratic-phase signals (and, if complex rays are allowed to appear, for Gaussian signals, too), it leads immediately to the curvature matrix of the signal; (ii) for Luneburg’s first-order system, it directly yields the ray transformation matrix of the system; (iii) for the propagation of quadratic-phase signals through first-order systems, it results in the well-known bilinear transformation of the signal’s curvature matrix. The zeroth-, first-, and second-order moments of the Wigner distribution function have been interpreted in terms of the energy, the center of gravity, and the effective width of the signal, respectively. The propagation of these moments through first-order systems has been derived. Since a Gaussian signal is completely described by its three lowest-order moments, the propagation of such a signal through first-order systems is known as well.
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paper_title: Continuous and discrete fractional Fourier domain decomposition

paper_content:

We introduce the fractional Fourier domain decomposition for continuous and discrete signals and systems. A procedure called pruning, analogous to truncation of the singular-value decomposition, underlies a number of potential applications, among which we discuss the fast implementation of space-variant linear systems.
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paper_title: Linear Programming and Extensions

paper_content:

In real-world problems related to finance, business, and management, mathematicians and economists frequently encounter optimization problems. In this classic book, George Dantzig looks at a wealth of examples and develops linear programming methods for their solutions. He begins by introducing the basic theory of linear inequalities and describes the powerful simplex method used to solve them. Treatments of the price concept, the transportation problem, and matrix methods are also given, and key mathematical concepts such as the properties of convex sets and linear vector spaces are covered."The author of this book was the main force in establishing a new mathematical discipline, and he has contributed to its further development at every stage and from every angle. This volume ... is a treasure trove for those who work in this field--teachers, students, and users alike. Its encyclopaedic coverage, due in part to collaboration with other experts, makes it an absolute must."--S. Vajda, Zentralblatt fYr Mathematik und ihre Grenzgebiete
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paper_title: Fractional Fourier transform as a tool for analyzing beam propagation and spherical mirror resonators

paper_content:

The complex amplitude distributions on two spherical reference surfaces of given curvature and spacing are simply related by a fractional Fourier transform. The order of the fractional Fourier transform is proportional to the Gouy phase shift between the two surfaces. This result provides new insight into wave propagation and spherical mirror resonators as well as the possibility of exploiting the fractional Fourier transform as a mathematical tool in analyzing such systems.
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paper_title: A fast refinement for adaptive Gaussian chirplet decomposition

paper_content:

The chirp function is one of the most fundamental functions in nature. Many natural events, for example, most signals encountered in seismology and the signals in radar systems, can be modeled as the superposition of short-lived chirp functions. Hence, the chirp-based signal representation, such as the Gaussian chirplet decomposition, has been an active research area in the field of signal processing. A main challenge of the Gaussian chirplet decomposition is that Gaussian chirplets do not form an orthogonal basis. A promising solution is to employ adaptive type signal decomposition schemes, such as the matching pursuit. The general underlying theory of the matching pursuit method has been well accepted, but the numerical implementation, in terms of computational speed and accuracy, of the adaptive Gaussian chirplet decomposition remains an open research topic. We present a fast refinement algorithm to search for optimal Gaussian chirplets. With a coarse dictionary, the resulting adaptive Gaussian chirplet decomposition is not only fast but is also more accurate than other known adaptive schemes. The effectiveness of the algorithm introduced is demonstrated by numerical simulations.
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paper_title: Fast matching pursuit with a multiscale dictionary of gaussian chirps

paper_content:

We introduce a modified matching pursuit algorithm, called fast ridge pursuit, to approximate N-dimensional signals with M Gaussian chirps at a computational cost O(MN) instead of the expected O(MN/sup 2/logN). At each iteration of the pursuit, the best Gabor atom is first selected, and then, its scale and chirp rate are locally optimized so as to get a "good" chirp atom, i.e., one for which the correlation with the residual is locally maximized. A ridge theorem of the Gaussian chirp dictionary is proved, from which an estimate of the locally optimal scale and chirp is built. The procedure is restricted to a sub-dictionary of local maxima of the Gaussian Gabor dictionary to accelerate the pursuit further. The efficiency and speed of the method is demonstrated on a sound signal.
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paper_title: The fractional Fourier transform and time-frequency representations

paper_content:

The functional Fourier transform (FRFT), which is a generalization of the classical Fourier transform, was introduced a number of years ago in the mathematics literature but appears to have remained largely unknown to the signal processing community, to which it may, however, be potentially useful. The FRFT depends on a parameter /spl alpha/ and can be interpreted as a rotation by an angle /spl alpha/ in the time-frequency plane. An FRFT with /spl alpha/=/spl pi//2 corresponds to the classical Fourier transform, and an FRFT with /spl alpha/=0 corresponds to the identity operator. On the other hand, the angles of successively performed FRFTs simply add up, as do the angles of successive rotations. The FRFT of a signal can also be interpreted as a decomposition of the signal in terms of chirps. The authors briefly introduce the FRFT and a number of its properties and then present some new results: the interpretation as a rotation in the time-frequency plane, and the FRFT's relationships with time-frequency representations such as the Wigner distribution, the ambiguity function, the short-time Fourier transform and the spectrogram. These relationships have a very simple and natural form and support the FRFT's interpretation as a rotation operator. Examples of FRFTs of some simple signals are given. An example of the application of the FRFT is also given. >
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paper_title: Fourier transforms of fractional order and their optical interpretation

paper_content:

Fourier transforms of fractional order a are defined in a manner such that the common Fourier transform is a special case with order a = 1. An optical interpretation is provided in terms of quadratic graded index media and discussed from both wave and ray viewpoints. Fractional Fourier transforms can extend the range of spatial filtering operations.
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paper_title: z-transform DFT filters and FFT's

paper_content:

The paper shows how discrete Fourier transformation can be implemented as a filter bank in a way which reduces the number of filter coefficients. A particular implementation of such a filter bank is directly related to the normal complex FFT algorithm. The principle developed further leads to types of DFT filter banks which utilize a minimum of complex coefficients. These implementations lead to new forms of FFT's, among which is a \cos/\sin FFT for a real signal which only employs real coefficients. The new FFT algorithms use only half as many real multiplications as does the classical FFT.
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paper_title: An affine scaling methodology for best basis selection

paper_content:

A methodology is developed to derive algorithms for optimal basis selection by minimizing diversity measures proposed by Wickerhauser (1994) and Donoho (1994). These measures include the p-norm-like (l/sub (p/spl les/1)/) diversity measures and the Gaussian and Shannon entropies. The algorithm development methodology uses a factored representation for the gradient and involves successive relaxation of the Lagrangian necessary condition. This yields algorithms that are intimately related to the affine scaling transformation (AST) based methods commonly employed by the interior point approach to nonlinear optimization. The algorithms minimizing the (l/sub (p/spl les/1)/) diversity measures are equivalent to a previously developed class of algorithms called focal underdetermined system solver (FOCUSS). The general nature of the methodology provides a systematic approach for deriving this class of algorithms and a natural mechanism for extending them. It also facilitates a better understanding of the convergence behavior and a strengthening of the convergence results. The Gaussian entropy minimization algorithm is shown to be equivalent to a well-behaved p=0 norm-like optimization algorithm. Computer experiments demonstrate that the p-norm-like and the Gaussian entropy algorithms perform well, converging to sparse solutions. The Shannon entropy algorithm produces solutions that are concentrated but are shown to not converge to a fully sparse solution.
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paper_title: Deriving algorithms for computing sparse solutions to linear inverse problems

paper_content:

A novel methodology is employed to develop algorithms for computing sparse solutions to linear inverse problems, starting from suitably defined diversity measures whose minimization promotes sparsity. These measures include p-norm-like (/spl Lscr//sub (p/spl les/1)/) diversity measures, and the Gaussian and Shannon entropies. The algorithm development methodology uses a factored representation of the gradient, and involves successive relaxation of the Lagrangian necessary condition. The general nature of the methodology provides a systematic approach for deriving a class of algorithms called FOCUSS (FOCal Underdetermined System Solver), and a natural mechanism for extending them.
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paper_title: Gabor's expansion of a signal into Gaussian elementary signals

paper_content:

It is proved that Gabor's expansion of a signal into a discrete set of Gaussian elementary signals exists. An expansion into another discrete set of functions is defined, which functions are biorthonormal, to the Gaussian elementary signals. Hence, the expansion coefficients of the two expansions can be determined easily.
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paper_title: A Wavelet Tour of Signal Processing

paper_content:

Introduction to a Transient World. Fourier Kingdom. Discrete Revolution. Time Meets Frequency. Frames. Wavelet Zoom. Wavelet Bases. Wavelet Packet and Local Cosine Bases. An Approximation Tour. Estimations are Approximations. Transform Coding. Appendix A: Mathematical Complements. Appendix B: Software Toolboxes.
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Title: A Survey of Signal Processing Problems and Tools in Holographic Three-Dimensional Television

Section 1: INTRODUCTION
Description 1: Provide an introduction to the topic of true 3-D video displays, the role of optics, and the importance of signal processing in holographic 3DTV. 

Section 2: Holography and Holographic 3DTV Displays
Description 2: Discuss the principles of holography, the requirements for dynamic holographic displays, and potential technologies for holographic 3DTV displays.

Section 3: Integral Imaging
Description 3: Explain the technology of integral imaging, including recent advances and techniques for improving image resolution and depth of field.

Section 4: Stereoscopic 3DTV Displays
Description 4: Describe stereoscopic 3DTV displays, their reliance on multiview video, and related signal processing issues.

Section 5: BASICS OF DIFFRACTION
Description 5: Provide an overview of the fundamentals of optical wave diffraction and its relevance to the computation of 3-D optical fields in holography.

Section 6: FUNDAMENTAL PROBLEMS IN HOLOGRAPHIC 3DTV
Description 6: Identify and define the forward and inverse problems in holographic 3DTV from a signal processing perspective, including discretization issues.

Section 7: REVIEW OF TECHNIQUES AND ALGORITHMS FOR WAVE PROPAGATION, DIFFRACTION, AND HOLOGRAPHY
Description 7: Review available techniques and algorithms in the literature for solving wave propagation, diffraction, and holography problems.

Section 8: COMPUTER GENERATED HOLOGRAPHY
Description 8: Discuss the history, methods, and applications of computer-generated holography (CGH) in achieving high-quality 3D images.

Section 9: DISCRETIZATION AND QUANTIZATION ISSUES IN DIFFRACTION AND HOLOGRAPHY
Description 9: Examine the specific characteristics of diffraction-related signals and the challenges of their discretization and quantization.

Section 10: SIGNAL PROCESSING TOOLS FOR DIFFRACTION AND HOLOGRAPHY RELATED PROBLEMS
Description 10: Present signal processing tools and techniques that can advance the state-of-the-art in holographic 3DTV, including sampling, transformation theory, and space-frequency analysis.

Section 11: CONCLUSION
Description 11: Summarize the survey, highlighting the areas needing further research and how improvements can advance 3DTV and contribute to optics and signal processing theory.