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281
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.
Texture analysis and classification with treestructured wavelet transform
 IEEE TRANS. IMAGE PROCESSING
, 1993
"... One difficulty of texture analysis in the past was the lack of adequate tools to characterize different scales of textures effectively. Recent developments in multiresolution analysis such as the Gabor and wavelet transforms help to overcome this difficulty. In this research, we propose a multiresol ..."
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Cited by 319 (1 self)
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One difficulty of texture analysis in the past was the lack of adequate tools to characterize different scales of textures effectively. Recent developments in multiresolution analysis such as the Gabor and wavelet transforms help to overcome this difficulty. In this research, we propose a multiresolution approach based on a modified wavelet transform called the treestructured wavelet transform or wavelet packets for texture analysis and classification. The development of this new transform is motivated by the observation that a large class of natural textures can be modeled as quasiperiodic signals whose dominant frequencies are located in the middle frequency channels. With the transform, we are able to zoom into any desired frequency channels for further decomposition. In contrast, the conventional pyramidstructured wavelet transform performs further decomposition only in low frequency channels. We develop a progressive texture classification algorithm which is not only computationally attractive but also has excellent performance. The performance of our new method is compared with that of several other methods using the DCT, DST, DHT, pyramidstructured wavelet transforms, Gabor filters, and Laws filters.
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 218 (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.
Basis pursuit.
 In IEEE the TwentyEighth Asilomar Conference onSignals, Systems and Computers,
, 1994
"... ..."
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 133 (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...
Fast algorithms for discrete and continuous wavelet transforms
 IEEE TRANSACTIONS ON INFORMATION THEORY
, 1992
"... Several algorithms are reviewed for computing various types of wavelet transforms: the Mallat algorithm, the “a trous” algorithm and their generalizations by Shensa. The goal is 1) to develop guidelines for implementing discrete and continuous wavelet transforms efficiently, 2) to compare the variou ..."
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Cited by 131 (1 self)
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Several algorithms are reviewed for computing various types of wavelet transforms: the Mallat algorithm, the “a trous” algorithm and their generalizations by Shensa. The goal is 1) to develop guidelines for implementing discrete and continuous wavelet transforms efficiently, 2) to compare the various algorithms obtained and give an idea of possible gains by providing operation counts. The computational structure of the algorithms rather than the mathematical relationship between transforms and algorithms, is focused upon. Most wavelet transform algorithms compute sampled coefficients of the continuous wavelet transform using the filter bank structure of the discrete wavelet transform. Although this general method is already efficient, it is shown that noticeable computational savings can be obtained by applying known fast convolution techniques (such as the FFT) in a suitable manner. The modified algorithms are termed “fast” because of their ability to reduce the computational complexity per computed coefficient from L to log L (within a small constant factor) for large filter lengths L. For short filters, we obtain smaller gains: “fast running FIR filtering” techniques allow one to achieve typically 30 % save in computations. This is still of practical interest when heavy computation of wavelet transforms is required, and the resulting algorithms remain easy to implement.
Frametheoretic analysis of oversampled filter banks
 IEEE Trans. Sign. Proc
"... Abstract—We provide a frametheoretic analysis of oversampled finite impulse response (FIR) and infinite impulse response (IIR) uniform filter banks (FB’s). Our analysis is based on a new relationship between the FB’s polyphase matrices and the frame operator corresponding to an FB. For a given over ..."
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Cited by 98 (5 self)
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Abstract—We provide a frametheoretic analysis of oversampled finite impulse response (FIR) and infinite impulse response (IIR) uniform filter banks (FB’s). Our analysis is based on a new relationship between the FB’s polyphase matrices and the frame operator corresponding to an FB. For a given oversampled analysis FB, we present a parameterization of all synthesis FB’s providing perfect reconstruction. We find necessary and sufficient conditions for an oversampled FB to provide a frame expansion. A new frametheoretic procedure for the design of paraunitary FB’s from given nonparaunitary FB’s is formulated. We show that the frame bounds of an FB can be obtained by an eigenanalysis of the polyphase matrices. The relevance of the frame bounds as a characterization of important numerical properties of an FB is assessed by means of a stochastic sensitivity analysis. We consider special cases in which the calculation of the frame bounds and synthesis filters is simplified. Finally, simulation results are presented. Index Terms — Filter banks, frames, oversampling, polyphase representation.
Quantized Overcomplete Expansions in R^N: Analysis, Synthesis, and Algorithms
 IEEE TRANSACTIONS ON INFORMATION THEORY
, 1998
"... Coefficient quantization has peculiar qualitative effects on representations of vectors in IR with respect to overcomplete sets of vectors. These effects are investigated in two settings: frame expansions (representations obtained by forming inner products with each element of the set) and matchi ..."
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Cited by 94 (15 self)
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Coefficient quantization has peculiar qualitative effects on representations of vectors in IR with respect to overcomplete sets of vectors. These effects are investigated in two settings: frame expansions (representations obtained by forming inner products with each element of the set) and matching pursuit expansions (approximations obtained by greedily forming linear combinations). In both cases, based on the concept of consistency, it is shown that traditional linear reconstruction methods are suboptimal, and better consistent reconstruction algorithms are given. The proposed consistent reconstruction algorithms were in each case implemented, and experimental results are included. For frame expansions, results are proven to bound distortion as a function of frame redundancy r and quantization step size for linear, consistent, and optimal reconstruction methods. Taken together, these suggest that optimal reconstruction methods will yield O(1=r ) meansquared error (MSE), and that consistency is sufficient to insure this asymptotic behavior. A result on the asymptotic tightness of random frames is also proven. Applicability of quantized matching pursuit to lossy vector compression is explored. Experiments demonstrate the likelihood that a linear reconstruction is inconsistent, the MSE reduction obtained with a nonlinear (consistent) reconstruction algorithm, and generally competitive performance at low bit rates.
Wavelets on graphs via spectral graph theory
, 2009
"... We propose a novel method for constructing wavelet transforms of functions defined on the vertices of an arbitrary finite weighted graph. Our approach is based on defining scaling using the the graph analogue of the Fourier domain, namely the spectral decomposition of the discrete graph Laplacian L. ..."
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Cited by 90 (8 self)
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We propose a novel method for constructing wavelet transforms of functions defined on the vertices of an arbitrary finite weighted graph. Our approach is based on defining scaling using the the graph analogue of the Fourier domain, namely the spectral decomposition of the discrete graph Laplacian L. Given a wavelet generating kernel g and a scale parameter t, we define the scaled wavelet operator T t g = g(tL). The spectral graph wavelets are then formed by localizing this operator by applying it to an indicator function. Subject to an admissibility condition on g, this procedure defines an invertible transform. We explore the localization properties of the wavelets in the limit of fine scales. Additionally, we present a fast Chebyshev polynomial approximation algorithm for computing the transform that avoids the need for diagonalizing L. We highlight potential applications of the transform through examples of wavelets on graphs corresponding to a variety of different problem domains.