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445
An iterative thresholding algorithm for linear inverse problems with a sparsity constraint
, 2008
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Multiple Description Coding: Compression Meets the Network
, 2001
"... This article focuses on the compressed representations of the pictures ..."
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Cited by 433 (9 self)
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This article focuses on the compressed representations of the pictures
Image Representation Using 2D Gabor Wavelets
 IEEE Trans. Pattern Analysis and Machine Intelligence
, 1996
"... This paper extends to two dimensions the frame criterion developed by Daubechies for onedimensional wavelets, and it computes the frame bounds for the particular case of 2D Gabor wavelets. Completeness criteria for 2D Gabor image representations are important because of their increasing role in man ..."
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Cited by 375 (4 self)
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This paper extends to two dimensions the frame criterion developed by Daubechies for onedimensional wavelets, and it computes the frame bounds for the particular case of 2D Gabor wavelets. Completeness criteria for 2D Gabor image representations are important because of their increasing role in many computer vision applications and also in modeling biological vision, since recent neurophysiological evidence from the visual cortex of mammalian brains suggests that the filter response profiles of the main class of linearlyresponding cortical neurons (called simple cells) are best modeled as a family of selfsimilar 2D Gabor wavelets. We therefore derive the conditions under which a set of continuous 2D Gabor wavelets will provide a complete representation of any image, and we also find selfsimilar wavelet parameterizations which allow stable reconstruction by summation as though the wavelets formed an orthonormal basis. Approximating a "tight frame" generates redundancy which allows lowresolution neural responses to represent highresolution images, as we illustrate by image reconstructions with severely quantized 2D Gabor coefficients. Index TermsGabor wavelets, coarse coding, image representation, visual cortex, image reconstruction.
Sampling—50 years after Shannon
 Proceedings of the IEEE
, 2000
"... 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 benefited from a strong research revival during the past few years, thanks in part to the math ..."
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Cited by 337 (27 self)
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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 benefited 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, Hilbertspace formulation, we reinterpret Shannon’s sampling procedure as an orthogonal projection onto the subspace of bandlimited functions. We then extend the standard sampling paradigm for a representation of functions in the more general class of “shiftinvariant” functions 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 (antialiasing) prefilters that are not necessarily ideal lowpass. 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. Keywords—Bandlimited functions, Hilbert spaces, interpolation, least squares approximation, projection operators, sampling,
Nonuniform Sampling and Reconstruction in ShiftInvariant Spaces
, 2001
"... This article discusses modern techniques for nonuniform sampling and reconstruction of functions in shiftinvariant spaces. It is a survey as well as a research paper and provides a unified framework for uniform and nonuniform sampling and reconstruction in shiftinvariant spaces by bringing togeth ..."
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Cited by 216 (13 self)
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This article discusses modern techniques for nonuniform sampling and reconstruction of functions in shiftinvariant spaces. It is a survey as well as a research paper and provides a unified framework for uniform and nonuniform sampling and reconstruction in shiftinvariant spaces by bringing together wavelet theory, frame theory, reproducing kernel Hilbert spaces, approximation theory, amalgam spaces, and sampling. Inspired by applications taken from communication, astronomy, and medicine, the following aspects will be emphasized: (a) The sampling problem is well defined within the setting of shiftinvariant spaces. (b) The general theory works in arbitrary dimension and for a broad class of generators. (c) The reconstruction of a function from any sufficiently dense nonuniform sampling set is obtained by efficient iterative algorithms. These algorithms converge geometrically and are robust in the presence of noise. (d) To model the natural decay conditions of real signals and images, the sampling theory is developed in weighted Lpspaces.
Systematic design of unitary spacetime constellations
 IEEE TRANS. INFORM. THEORY
, 2000
"... We propose a systematic method for creating constellations of unitary space–time signals for multipleantenna communication links. Unitary space–time signals, which are orthonormal in time across the antennas, have been shown to be welltailored to a Rayleigh fading channel where neither the transm ..."
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Cited by 196 (10 self)
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We propose a systematic method for creating constellations of unitary space–time signals for multipleantenna communication links. Unitary space–time signals, which are orthonormal in time across the antennas, have been shown to be welltailored to a Rayleigh fading channel where neither the transmitter nor the receiver knows the fading coefficients. The signals can achieve low probability of error by exploiting multipleantenna diversity. Because the fading coefficients are not known, the criterion for creating and evaluating the constellation is nonstandard and differs markedly from the familiar maximumEuclideandistance norm. Our construction begins with the first signal in the constellation—an oblong complexvalued matrix whose columns are orthonormal—and systematically produces the remaining signals by successively rotating this signal in a highdimensional complex space. This construction easily produces large constellations of highdimensional signals. We demonstrate its efficacy through examples involving one, two, and three transmitter antennas.
Framelets: MRABased Constructions of Wavelet Frames
, 2001
"... We discuss wavelet frames constructed via multiresolution analysis (MRA), with emphasis on tight wavelet frames. In particular, we establish general principles and specific algorithms for constructing framelets and tight framelets, and we show how they can be used for systematic constructions of spl ..."
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Cited by 194 (72 self)
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We discuss wavelet frames constructed via multiresolution analysis (MRA), with emphasis on tight wavelet frames. In particular, we establish general principles and specific algorithms for constructing framelets and tight framelets, and we show how they can be used for systematic constructions of spline, pseudospline tight frames and symmetric biframes with short supports and high approximation orders. Several explicit examples are discussed. The connection of these frames with multiresolution analysis guarantees the existence of fast implementation algorithms, which we discuss briefly as well.
Oversampled Filter Banks
 IEEE Trans. Signal Processing
, 1998
"... Perfect reconstruction oversampled filter banks are equivalent to a particular class of frames in ` (Z). These frames are the subject of this paper. First, necessary and sufficient conditions on a filter bank for implementing a frame or a tight frame expansion are established, as well as a neces ..."
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Cited by 127 (2 self)
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Perfect reconstruction oversampled filter banks are equivalent to a particular class of frames in ` (Z). These frames are the subject of this paper. First, necessary and sufficient conditions on a filter bank for implementing a frame or a tight frame expansion are established, as well as a necessary and sufficient condition for perfect reconstruction using FIR filters after an FIR analysis. Complete parameterizations of oversampled filter banks satisfying these conditions are given. Further, we study the condition under which the frame dual to the frame associated with an FIR filter bank is also FIR and give a parameterization of a class of filter banks satisfying this property. Then, we focus on nonsubsampled filter banks. Nonsubsampled filter banks implement transforms similar to continuoustime transforms and allow for very flexible design. We investigate relations of these filter banks to continuoustime filtering and illustrate the design flexibility by giving a procedure for designing maximally flat twochannel filter banks that yield highly regular wavelets with a given number of vanishing moments.
Frames and Stable Bases for ShiftInvariant Subspaces of . . .
, 1994
"... Let X be a countable fundamental set in a Hilbert space H, and let T be the operator T : ` 2 (X) ! H : c 7! X x2X c(x)x: Whenever T is welldefined and bounded, X is said to be a Bessel sequence. If, in addition, ran T is closed, then X is a frame. Finally, a frame whose corresponding T is inje ..."
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Cited by 127 (29 self)
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Let X be a countable fundamental set in a Hilbert space H, and let T be the operator T : ` 2 (X) ! H : c 7! X x2X c(x)x: Whenever T is welldefined and bounded, X is said to be a Bessel sequence. If, in addition, ran T is closed, then X is a frame. Finally, a frame whose corresponding T is injective is a stable basis (also known as a Riesz basis). This paper considers the above three properties for subspaces H of L 2 (IR d ), and for sets X of the form X = fOE(\Delta \Gamma ff) : OE 2 \Phi; ff 2 ZZ d g; with \Phi either a singleton, a finite set, or, more generally, a countable set. The analysis is performed on the Fourier domain, where the two operators TT and T T are decomposed into a collection of simpler "fiber" operators. The main theme of the entire analysis is the characterization of each of the above three properties in terms of the analogous property of these simpler operators. AMS (MOS) Subject Classifications: 42C15 Key Words: Riesz bases, stable bases, shif...
Images as embedding maps and minimal surfaces: Movies, color, texture, and volumetric medical images
 INT. J. COMPUT. VIS
, 2000
"... We extend the geometric framework introduced in Sochen et al. (IEEE Trans. on Image Processing, 7(3):310–318, 1998) for image enhancement. We analyze and propose enhancement techniques that selectively smooth images while preserving either the multichannel edges or the orientationdependent textu ..."
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Cited by 110 (23 self)
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We extend the geometric framework introduced in Sochen et al. (IEEE Trans. on Image Processing, 7(3):310–318, 1998) for image enhancement. We analyze and propose enhancement techniques that selectively smooth images while preserving either the multichannel edges or the orientationdependent texture features in them. Images are treated as manifolds in a featurespace. This geometrical interpretation lead to a general way for grey level, color, movies, volumetric medical data, and colortexture image enhancement. We first review our framework in which the Polyakov action from highenergy physics is used to develop a minimization procedure through a geometric flow for images. Here we show that the geometric flow, based on manifold volume minimization, yields a novel enhancement procedure for color images. We apply the geometric framework and the general Beltrami flow to featurepreserving denoising of images in various spaces. Next, we introduce a new method for color and texture enhancement. Motivated by Gabor’s geometric image sharpening method (Gabor, Laboratory Investigation, 14(6):801–807, 1965), we present a geometric sharpening procedure for color images with texture. It is based on inverse diffusion across the multichannel edge, and diffusion along the edge.