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155
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 ..."
Abstract

Cited by 264 (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.
Basis Pursuit
, 1994
"... The TimeFrequency and TimeScale communities have recently developed an enormous number of overcomplete signal dictionaries  wavelets, wavelet packets, cosine packets, wilson bases, chirplets, warped bases, and hyperbolic cross bases being a few examples. Basis Pursuit is a technique for decompos ..."
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Cited by 119 (15 self)
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The TimeFrequency and TimeScale communities have recently developed an enormous number of overcomplete signal dictionaries  wavelets, wavelet packets, cosine packets, wilson bases, chirplets, warped bases, and hyperbolic cross bases being a few examples. Basis Pursuit is a technique for decomposing a signal into an "optimal" superposition of dictionary elements. The optimization criterion is the l 1 norm of coefficients. The method has several advantages over Matching Pursuit and Best Ortho Basis, including superresolution and stability. 1 Introduction Over the last five years or so, there has been an explosion of awareness of alternatives to traditional signal representations. Instead of just representing objects as superpositions of sinusoids (the traditional Fourier representation) we now have available alternate dictionaries  signal representation schemes  of which the Wavelets dictionary is only the most wellknown. Wavelet dictionaries, Gabor dictionaries, Multiscale...
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 75 (22 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...
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 75 (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.
Filter Bank Frame Expansions with Erasures
, 2002
"... We study frames for robust transmission over the Internet. In our previous work, we used quantized finitedimensional frames to achieve resilience to packet losses; here, we allow the input to be a sequence in ` 2 (Z) and focus on a filterbank implementation of the system. We present results in par ..."
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Cited by 50 (4 self)
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We study frames for robust transmission over the Internet. In our previous work, we used quantized finitedimensional frames to achieve resilience to packet losses; here, we allow the input to be a sequence in ` 2 (Z) and focus on a filterbank implementation of the system. We present results in parallel, R N or C N versus ` 2 (Z), and show that uniform tight frames, as well as newly introduced strongly uniform tight frames, provide the best performance.
Unitary Equivalence: A New Twist On Signal Processing
, 1995
"... Unitary similarity transformations furnish a powerful vehicle for generating infinite generic classes of signal analysis and processing tools based on concepts different from time, frequency, and scale. Implementation of these new tools involves simply preprocessing the signal by a unitary transfo ..."
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Cited by 48 (15 self)
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Unitary similarity transformations furnish a powerful vehicle for generating infinite generic classes of signal analysis and processing tools based on concepts different from time, frequency, and scale. Implementation of these new tools involves simply preprocessing the signal by a unitary transformation, performing standard processing techniques on the transformed signal, and then (in some cases) transforming the resulting output. The resulting unitarily equivalent systems focus on the critical signal characteristics in large classes of signals and, hence, prove useful for representing and processing signals that are not well matched by current techniques. As specific examples of this procedure, we generalize linear timeinvariant systems, orthonormal basis and frame decompositions, and joint timefrequency and timescale distributions, illustrating the utility of the unitary equivalence concept for uniting seemingly disparate approaches proposed in the literature. This work...
Density, overcompleteness, and localization of frames
 I. THEORY, J. FOURIER ANAL. APPL
, 2005
"... This work presents a quantitative framework for describing the overcompleteness of a large class of frames. It introduces notions of localization and approximation between two frames F = {fi}i∈I and E = {ej}j∈G (G a discrete abelian group), relating the decay of the expansion of the elements of F in ..."
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Cited by 45 (18 self)
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This work presents a quantitative framework for describing the overcompleteness of a large class of frames. It introduces notions of localization and approximation between two frames F = {fi}i∈I and E = {ej}j∈G (G a discrete abelian group), relating the decay of the expansion of the elements of F in terms of the elements of E viaamapa: I → G. A fundamental set of equalities are shown between three seemingly unrelated quantities: the relative measure of F, the relative measure of E—both of which are determined by certain averages of inner products of frame elements with their corresponding dual frame elements—and the density of the set a(I) inG. Fundamental new results are obtained on the excess and overcompleteness of frames, on the relationship between frame bounds and density, and on the structure of the dual frame of a localized frame. These abstract results yield an array of new implications for irregular Gabor frames. Various Nyquist density results for Gabor frames are recovered as special cases, but in the process both their meaning and implications are clarified. New results are obtained on the excess and overcompleteness of Gabor frames, on the relationship between frame bounds and density, and on the structure of the dual frame of an irregular Gabor frame. More generally, these results apply both to Gabor frames and to systems of Gabor molecules, whose elements share only a common envelope of concentration in the timefrequency plane.
Optimal tight frames and quantum measurement
 IEEE Trans. Inform. Theory
, 2002
"... Tight frames and rankone quantum measurements are shown to be intimately related. In fact, the family of normalized tight frames for the space in which a quantum mechanical system lies is precisely the family of rankone generalized quantum measurements (POVMs) on that space. Using this relationshi ..."
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Cited by 41 (10 self)
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Tight frames and rankone quantum measurements are shown to be intimately related. In fact, the family of normalized tight frames for the space in which a quantum mechanical system lies is precisely the family of rankone generalized quantum measurements (POVMs) on that space. Using this relationship, frametheoretical analogues of various quantummechanical concepts and results are developed. The analogue of a leastsquares quantum measurement is a tight frame that is closest in a leastsquares sense to a given set of vectors. The leastsquares tight frame is found for both the case in which the scaling of the frame is specified (constrained leastsquares frame (CLSF)) and the case in which the scaling is free (unconstrained leastsquares frame (ULSF)). The wellknown canonical frame is shown to be proportional to the ULSF and to coincide with the CLSF with a certain scaling. Finally, the canonical frame vectors corresponding to a geometrically uniform vector set are shown to be geometrically uniform and to have the same symmetries as the original vector set.
Density of Gabor Frames
, 1999
"... this paper is somewhat dierent, in that we are concerned with the connection between density properties of and frame properties of S(g; ), and the analogous problem for systems T (g; ) of pure translates. For the case of Gabor systems, there is a rich literature on this subject, especially when is ..."
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Cited by 39 (14 self)
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this paper is somewhat dierent, in that we are concerned with the connection between density properties of and frame properties of S(g; ), and the analogous problem for systems T (g; ) of pure translates. For the case of Gabor systems, there is a rich literature on this subject, especially when is the rectangular lattice = aZ