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13
ATOMIC DECOMPOSITION BY BASIS PURSUIT
, 1995
"... The TimeFrequency and TimeScale 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 d ..."
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Cited by 2741 (61 self)
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The TimeFrequency and TimeScale 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 l 1 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 illposed problems, in abstract harmonic analysis, total variation denoising, and multiscale edge denoising. Basis Pursuit in highly overcomplete dictionaries leads to largescale 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 interiorpoint methods. We obtain reasonable success with a primaldual logarithmic barrier method and conjugategradient solver.
Improved linear discrimination using timefrequency dictionaries
, 1995
"... We consider linear discriminant analysis in the setting where the objects (signals/images) have many dimensions (samples/pixels) and there are relatively few training samples. We discuss ways that time frequency dictionaries can be used to adaptively select a small set of derived features which lead ..."
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Cited by 32 (0 self)
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We consider linear discriminant analysis in the setting where the objects (signals/images) have many dimensions (samples/pixels) and there are relatively few training samples. We discuss ways that time frequency dictionaries can be used to adaptively select a small set of derived features which lead to improved misclassi cation rates.
Fast Solution Methods in Electromagnetics
, 1997
"... Various methods for efficiently solving electromagnetic problems are presented. Electromagnetic scattering problems can be roughly classified into surface and volume problems, while fast methods are either differential or integral equation based. The resultant systems of linear equations are either ..."
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Various methods for efficiently solving electromagnetic problems are presented. Electromagnetic scattering problems can be roughly classified into surface and volume problems, while fast methods are either differential or integral equation based. The resultant systems of linear equations are either solved directly or iteratively. A review of various differential equation solvers, their complexities, and memory requirements is given. The issues of grid dispersion and hybridization with integral equation solvers are discussed. Several fast integral equation solvers for surface and volume scatterers are presented. These solvers have reduced computational complexities and memory requirements. 1. Introduction Computational electromagnetics is a fascinating discipline that has drawn the attention of mathematicians, engineers, physicists, and computer scientists alike. It is a discipline that creates a symbiotic marriage between mathematics, physics, computer science, and various applicatio...
Lv volume quantification via spatiotemporal analysis of realtime 3d echocardiography
 IEEE trans. on Medical Imaging
"... Abstract—This paper presents a method of fourdimensional (4D) (3D + Time) space–frequency analysis for directional denoising and enhancement of realtime threedimensional (RT3D) ultrasound and quantitative measures in diagnostic cardiac ultrasound. Expansion of echocardiographic volumes is perfo ..."
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Cited by 31 (11 self)
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Abstract—This paper presents a method of fourdimensional (4D) (3D + Time) space–frequency analysis for directional denoising and enhancement of realtime threedimensional (RT3D) ultrasound and quantitative measures in diagnostic cardiac ultrasound. Expansion of echocardiographic volumes is performed with complex exponential waveletlike basis functions called brushlets. These functions offer good localization in time and frequency and decompose a signal into distinct patterns of oriented harmonics, which are invariant to intensity and contrast range. Deformablemodel segmentation is carried out on denoised data after thresholding of transform coefficients. This process attenuates speckle noise while preserving cardiac structure location. The superiority of 4D over 3D analysis for decorrelating additive white noise and multiplicative speckle noise on a 4D phantom volume expanding in time is demonstrated. Quantitative validation, computed for contours and volumes, is performed on in vitro balloon phantoms. Clinical applications of this spaciotemporal analysis tool are reported for six patient cases providing measures of left ventricular volumes and ejection fraction. Index Terms—Echocardiography, LV volume, spaciotemporal analysis, speckle denoising. I.
Estimating Covariances of Locally Stationary Processes: Rates of Convergence of Best Basis Methods
, 1998
"... Mallat, Papanicolaou and Zhang [MPZ98] recently proposed a method for approximating the covariance of a locally stationary process by a covariance which is diagonal in a specially constructed CoifmanMeyer basis of cosine packets. In this paper we extend this approach to estimating the covariance ..."
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Cited by 28 (10 self)
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Mallat, Papanicolaou and Zhang [MPZ98] recently proposed a method for approximating the covariance of a locally stationary process by a covariance which is diagonal in a specially constructed CoifmanMeyer basis of cosine packets. In this paper we extend this approach to estimating the covariance from sampled data. Our method combines both wavelet shrinkage and cosinepacket bestbasis selection in a simple and natural way. The resulting algorithm is fast and automatic. The method has an interpretation as a nonlinear, adaptive form of anisotropic timefrequency smoothing. We introduce a new class of locally stationary processes which exhibits a form of inhomogeneous nonstationarity; our processes have covariances which typically change little from row to row, but might occasionally change abruptly. We study performance in an asymptotic setting involving triangular arrays of processes which are becoming increasingly stationary, and are able to prove rates of convergence results for our...
Fast search for best representations in multitree dictionaries
 In Wavelet Applications in Signal and Image Processing VIII, Proc. SPIE 4119, 2000. [7] S.G. Mallat. A Wavelet Tour of Signal Processing, Second Edition
, 2006
"... Abstract—We address the best basis problem—or, more generally, the best representation problem: Given a signal, a dictionary of representations, and an additive cost function, the aim is to select the representation from the dictionary which minimizes the cost for the given signal. We develop a new ..."
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Cited by 13 (4 self)
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Abstract—We address the best basis problem—or, more generally, the best representation problem: Given a signal, a dictionary of representations, and an additive cost function, the aim is to select the representation from the dictionary which minimizes the cost for the given signal. We develop a new framework of multitree dictionaries, which includes some previously proposed dictionaries as special cases. We show how to efficiently find the best representation in a multitree dictionary using a recursive treepruning algorithm. We illustrate our framework through several examples, including a novel block image coder, which significantly outperforms both the standard JPEG and quadtreebased methods and is comparable to embedded coders such as JPEG2000 and SPIHT. Index Terms—Best basis, grammar, image compression, JPEG. I.
N.: Best basis search in lapped dictionaries
 IEEE Trans. Signal Process
, 2006
"... Abstract—This paper proposes, analyzes, and illustrates several best basis search algorithms for dictionaries consisting of lapped orthogonal bases. It improves upon the best local cosine basis selection based on a dyadic tree [10], [11] by considering larger dictionaries of bases. It is shown that ..."
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Cited by 6 (2 self)
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Abstract—This paper proposes, analyzes, and illustrates several best basis search algorithms for dictionaries consisting of lapped orthogonal bases. It improves upon the best local cosine basis selection based on a dyadic tree [10], [11] by considering larger dictionaries of bases. It is shown that this can result in sparser representations and approximate shift invariance. An algorithm that is strictly shift invariant is also provided. The experiments in this paper suggest that the new dictionaries can be advantageous for timefrequency analysis, compression, and noise removal. Accelerated versions of the basic algorithm are provided that explore various tradeoffs between computational efficiency and adaptability. It is shown that the proposed algorithms are in fact applicable to any finite dictionary comprised of lapped orthogonal bases. One such novel dictionary is proposed that constructs the best local cosine representation in the frequency domain, and it is shown that the new dictionary is better suited for representing certain types of signals. Index Terms—Best basis, lapped transforms, timefrequency analysis. I.
Arbitrary tilings of the timefrequency plane using local bases
 IEEE Transactions on Signal Processing
, 1999
"... Abstract—We show how to design filters given a prescribed tiling of the time–frequency plane. Moreover, we impose on these filters the structure of local orthogonal bases. These bases were recently constructed as a generalization of the cosinemodulated filter banks in discrete time and local trigon ..."
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Cited by 3 (0 self)
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Abstract—We show how to design filters given a prescribed tiling of the time–frequency plane. Moreover, we impose on these filters the structure of local orthogonal bases. These bases were recently constructed as a generalization of the cosinemodulated filter banks in discrete time and local trigonometric bases in continuous time. They have been found to be of considerable practical importance due to their simplicity (all filters are obtained from a single prototype) and low computational complexity. We show examples of design, in particular, that of a criticalband system for use in audio coding. I.
Sparse decompositions for ventricular and atrial activity separation
 M.S. thesis, Signal Processing Institute, Ecole Polytechnique Fédérale de
, 2005
"... En primer lugar quiero dar las gracias, de una forma especial, a mis padres, Juan José y Nati, por sus consejos y su apoyo incondicional. Gracias por todo lo que me habeis dado. A mi hermano Ignacio, quiero agradecerle su dedicación siempre que lo he necesitado. Siempre has sido y serás mi referenci ..."
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En primer lugar quiero dar las gracias, de una forma especial, a mis padres, Juan José y Nati, por sus consejos y su apoyo incondicional. Gracias por todo lo que me habeis dado. A mi hermano Ignacio, quiero agradecerle su dedicación siempre que lo he necesitado. Siempre has sido y serás mi referencia. También quiero dar las gracias a Cristina por haber estado a mi lado en todo momento. Of course, I would like to thank Dr. Òscar Divorra Escoda, Lorenzo Granai and Mathieu Lemay for guiding me during these months. I have learnt a lot with you. Thank you very much! I am also grateful to Prof. Pierre Vandergheynst and Dr. JeanMarc Vesin for giving me the possibility to do my master’s thesis at EPFL. Thanks also to all my colleagues and friends in LTS for sharing with me these unforgettable months: Bruno, David Bayona, David Marimón, Mireia, Roger, Thomas,... Many thanks to my roommates: this time with you! Álex, Catalina, Daniel, Eric and Martin. I will never forget To all of you, thanks a lot! i Contents List of Figures vii