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16
Uncertainty principles and ideal atomic decomposition
 IEEE Transactions on Information Theory
, 2001
"... Suppose a discretetime signal S(t), 0 t
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Cited by 361 (19 self)
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Suppose a discretetime signal S(t), 0 t<N, is a superposition of atoms taken from a combined time/frequency dictionary made of spike sequences 1ft = g and sinusoids expf2 iwt=N) = p N. Can one recover, from knowledge of S alone, the precise collection of atoms going to make up S? Because every discretetime signal can be represented as a superposition of spikes alone, or as a superposition of sinusoids alone, there is no unique way of writing S as a sum of spikes and sinusoids in general. We prove that if S is representable as a highly sparse superposition of atoms from this time/frequency dictionary, then there is only one such highly sparse representation of S, and it can be obtained by solving the convex optimization problem of minimizing the `1 norm of the coe cients among all decompositions. Here \highly sparse " means that Nt + Nw < p N=2 where Nt is the number of time atoms, Nw is the number of frequency atoms, and N is the length of the discretetime signal.
Wave atoms and sparsity of oscillatory patterns
 Appl. Comput. Harmon. Anal
, 2006
"... We introduce “wave atoms ” as a variant of 2D wavelet packets obeying the parabolic scaling wavelength ∼ (diameter) 2. We prove that warped oscillatory functions, a toy model for texture, have a significantly sparser expansion in wave atoms than in other fixed standard representations like wavelets, ..."
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Cited by 45 (5 self)
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We introduce “wave atoms ” as a variant of 2D wavelet packets obeying the parabolic scaling wavelength ∼ (diameter) 2. We prove that warped oscillatory functions, a toy model for texture, have a significantly sparser expansion in wave atoms than in other fixed standard representations like wavelets, Gabor atoms, or curvelets. We propose a novel algorithm for a tight frame of wave atoms with redundancy two, directly in the frequency plane, by the “wrapping ” technique. We also propose variants of the basic transform for applications in image processing, including an orthonormal basis, and a shiftinvariant tight frame with redundancy four. Sparsity and denoising experiments on both seismic and fingerprint images demonstrate the potential of the tool introduced.
A Wavelet Shrinkage Approach to Tomographic Image Reconstruction
 J. Amer. Statist. Assoc
, 1996
"... A method is proposed for reconstructing images from tomographic data with respect to a twodimensional wavelet basis. The WaveletVaguelette Decomposition is used as a framework within which expressions for the necessary wavelet coefficients may be derived. These coefficients are calculated using a ..."
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Cited by 24 (1 self)
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A method is proposed for reconstructing images from tomographic data with respect to a twodimensional wavelet basis. The WaveletVaguelette Decomposition is used as a framework within which expressions for the necessary wavelet coefficients may be derived. These coefficients are calculated using a version of the filtered backprojection algorithm, as a computational tool, in a multiresolution fashion. The necessary filters are defined in terms of the underlying wavelets. Denoising is accomplished through an adaptation of the Wavelet Shrinkage approach of Donoho et al., and amounts to a form of regularization. Combining the above two steps yields the proposed WVD/WS reconstruction algorithm, which is compared to the traditional filtered backprojection method in a small study. Key Words: Backprojection, WaveletVaguelette Decomposition, Tomography. 1 INTRODUCTION 1.1 The Tomography Problem Tomographic image reconstruction refers to a broad class of problems in which the goal is to rec...
Estimating The Square Root Of A Density Via Compactly Supported Wavelets
, 1997
"... This paper addresses the problem of univariate density estimation in a novel way. Our approach falls in the class of so called projection estimators, introduced by ..."
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Cited by 19 (6 self)
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This paper addresses the problem of univariate density estimation in a novel way. Our approach falls in the class of so called projection estimators, introduced by
Wavelet and Fourier Methods for Solving the Sideways Heat Equation
, 1997
"... We consider an inverse heat conduction problem, the Sideways Heat Equation, which is a model of a problem, where one wants to determine the temperature on both sides of a thick wall, but where one side is inaccessible to measurements. Mathematically it is formulated as a Cauchy problem for the heat ..."
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Cited by 15 (8 self)
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We consider an inverse heat conduction problem, the Sideways Heat Equation, which is a model of a problem, where one wants to determine the temperature on both sides of a thick wall, but where one side is inaccessible to measurements. Mathematically it is formulated as a Cauchy problem for the heat equation in a quarter plane, with data given along the line x = 1, where the solution is wanted for 0 x ! 1. The problem is illposed, in the sense that the solution (if it exists) does not depend continuously on the data. We consider stabilizations based on replacing the time derivative in the heat equation by waveletbased approximations or a Fourierbased approximation. The resulting problem is an initial value problem for an ordinary differential equation, which can be solved by standard numerical methods, e.g. a RungeKutta method. We discuss the numerical implementation of Fourier and wavelet methods for solving the sideways heat equation. Theory predicts that the Fourier method...
Directional Dyadic Wavelet Transforms: Design and Algorithms
 IEEE TRANS. IMAGE PROCESSING
, 2002
"... We propose a simple and efficient technique for designing translation invariant dyadic wavelet transforms (DWTs) in two dimensions. Our technique relies on an extension of the work of DuvalDestin et al. [1], [2] where dyadic decompositions are constructed starting from the continuous wavelet transf ..."
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Cited by 8 (0 self)
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We propose a simple and efficient technique for designing translation invariant dyadic wavelet transforms (DWTs) in two dimensions. Our technique relies on an extension of the work of DuvalDestin et al. [1], [2] where dyadic decompositions are constructed starting from the continuous wavelet transfrom. The main advantage of this framework is that it allows for a lot of freedom in designing twodimensional (2D) dyadic wavelets. We use this property to construct directional wavelets, whose orientation filtering capabilities are very important in image processing. We address the efficient implementation of these decompositions by constructing approximate QMFs through an L²optimization. We also propose and study an efficient implementation in the Fourier domain for dealing with large filters.
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 5 (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.
15 Years of Reproducible Research in Computational Harmonic Analysis
, 2008
"... Scientific Computation is emerging as absolutely central to the scientific method. Unfortunately, it is errorprone and currently immature: traditional scientific publication is incapable of finding and rooting out errors in scientific computation; this must be recognized as a crisis. Reproducible c ..."
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Cited by 3 (0 self)
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Scientific Computation is emerging as absolutely central to the scientific method. Unfortunately, it is errorprone and currently immature: traditional scientific publication is incapable of finding and rooting out errors in scientific computation; this must be recognized as a crisis. Reproducible computational research, in which the full computational environment that produces a result is published along with the article, is an important recent development, and a necessary response to this crisis. We have been practicing reproducible computational research for 15 years and integrated it with our scientific research, and with doctoral and postdoctoral education. In this article, we review our approach, how the approach has spread over time, and how science funding agencies could help spread the idea more rapidly. 1
On convergence rates equivalency and sampling strategies in in functional deconvolution models,” The Annals of Statistics
, 2010
"... Using the asymptotical minimax framework, we examine convergence rates equivalency between a continuous functional deconvolution model and its reallife discrete counterpart over a wide range of Besov balls and for the L 2risk. For this purpose, all possible models are divided into three groups. Fo ..."
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Cited by 2 (0 self)
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Using the asymptotical minimax framework, we examine convergence rates equivalency between a continuous functional deconvolution model and its reallife discrete counterpart over a wide range of Besov balls and for the L 2risk. For this purpose, all possible models are divided into three groups. For the models in the first group, which we call uniform, the convergence rates in the discrete and the continuous models coincide no matter what the sampling scheme is chosen, and hence the replacement of the discrete model by its continuous counterpart is legitimate. For the models in the second group, to which we refer as regular, one can point out the best sampling strategy in the discrete model, but not every sampling scheme leads to the same convergence rates; there are at least two sampling schemes which deliver different convergence rates in the discrete model (i.e., at least one of the discrete models leads to convergence rates that are different from the convergence rates in the continuous model). The third group consists of models for which, in general, it is impossible to devise the best sampling strategy; we call these