Results 1  10
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183
Compressed sensing
 IEEE Trans. Inform. Theory
"... Abstract—Suppose is an unknown vector in (a digital image or signal); we plan to measure general linear functionals of and then reconstruct. If is known to be compressible by transform coding with a known transform, and we reconstruct via the nonlinear procedure defined here, the number of measureme ..."
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Cited by 1730 (18 self)
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Abstract—Suppose is an unknown vector in (a digital image or signal); we plan to measure general linear functionals of and then reconstruct. If is known to be compressible by transform coding with a known transform, and we reconstruct via the nonlinear procedure defined here, the number of measurements can be dramatically smaller than the size. Thus, certain natural classes of images with pixels need only = ( 1 4 log 5 2 ()) nonadaptive nonpixel samples for faithful recovery, as opposed to the usual pixel samples. More specifically, suppose has a sparse representation in some orthonormal basis (e.g., wavelet, Fourier) or tight frame (e.g., curvelet, Gabor)—so the coefficients belong to an ball for 0 1. The most important coefficients in that expansion allow reconstruction with 2 error ( 1 2 1
From Sparse Solutions of Systems of Equations to Sparse Modeling of Signals and Images
, 2007
"... A fullrank matrix A ∈ IR n×m with n < m generates an underdetermined system of linear equations Ax = b having infinitely many solutions. Suppose we seek the sparsest solution, i.e., the one with the fewest nonzero entries: can it ever be unique? If so, when? As optimization of sparsity is combinato ..."
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Cited by 202 (31 self)
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A fullrank matrix A ∈ IR n×m with n < m generates an underdetermined system of linear equations Ax = b having infinitely many solutions. Suppose we seek the sparsest solution, i.e., the one with the fewest nonzero entries: can it ever be unique? If so, when? As optimization of sparsity is combinatorial in nature, are there efficient methods for finding the sparsest solution? These questions have been answered positively and constructively in recent years, exposing a wide variety of surprising phenomena; in particular, the existence of easilyverifiable conditions under which optimallysparse solutions can be found by concrete, effective computational methods. Such theoretical results inspire a bold perspective on some important practical problems in signal and image processing. Several wellknown signal and image processing problems can be cast as demanding solutions of undetermined systems of equations. Such problems have previously seemed, to many, intractable. There is considerable evidence that these problems often have sparse solutions. Hence, advances in finding sparse solutions to underdetermined systems energizes research on such signal and image processing problems – to striking effect. In this paper we review the theoretical results on sparse solutions of linear systems, empirical
Wavelet Threshold Estimators for Data With Correlated Noise
, 1994
"... Wavelet threshold estimators for data with stationary correlated noise are constructed by the following prescription. First, form the discrete wavelet transform of the data points. Next, apply a leveldependent soft threshold to the individual coefficients, allowing the thresholds to depend on the l ..."
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Cited by 182 (13 self)
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Wavelet threshold estimators for data with stationary correlated noise are constructed by the following prescription. First, form the discrete wavelet transform of the data points. Next, apply a leveldependent soft threshold to the individual coefficients, allowing the thresholds to depend on the level in the wavelet transform. Finally, transform back to obtain the estimate in the original domain. The threshold used at level j is s j p 2 log n, where s j is the standard deviation of the coefficients at that level, and n is the overall sample size. The minimax properties of the estimators are investigated by considering a general problem in multivariate normal decision theory, concerned with the estimation of the mean vector of a general multivariate normal distribution subject to squared error loss. An ideal risk is obtained by the use of an `oracle' that provides the optimum diagonal projection estimate. This `benchmark' risk can be considered in its own right as a measure of the s...
Modeling the Joint Statistics of Images in the Wavelet Domain
 IN PROC SPIE, 44TH ANNUAL MEETING
, 1999
"... I describe a statistical model for natural photographic images, when decomposed in a multiscale wavelet basis. In particular, I examine both the marginal and pairwise joint histograms of wavelet coefficients at adjacent spatial locations, orientations, and spatial scales. Although the histograms ar ..."
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Cited by 98 (3 self)
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I describe a statistical model for natural photographic images, when decomposed in a multiscale wavelet basis. In particular, I examine both the marginal and pairwise joint histograms of wavelet coefficients at adjacent spatial locations, orientations, and spatial scales. Although the histograms are highly nonGaussian, they are nevertheless well described using fairly simple parameterized density models.
Wavelet estimators in nonparametric regression: a comparative simulation study
 Journal of Statistical Software
, 2001
"... OATAO is an open access repository that collects the work of Toulouse researchers and makes it freely available over the web where possible. ..."
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Cited by 72 (9 self)
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OATAO is an open access repository that collects the work of Toulouse researchers and makes it freely available over the web where possible.
Wavelet Shrinkage Denoising Using the NonNegative Garrote
, 1997
"... In this paper, we combine Donoho and Johnstone's Wavelet Shrinkage denoising technique (known as WaveShrink) with Breiman's nonnegative garrote. We show that the nonnegative garrote shrinkage estimate enjoys the same asymptotic convergence rate as the hard and the soft shrinkage estimates. Simulat ..."
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Cited by 52 (1 self)
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In this paper, we combine Donoho and Johnstone's Wavelet Shrinkage denoising technique (known as WaveShrink) with Breiman's nonnegative garrote. We show that the nonnegative garrote shrinkage estimate enjoys the same asymptotic convergence rate as the hard and the soft shrinkage estimates. Simulations are used to demonstrate that garrote shrinkage offers advantages over both hard shrinkage (generally smaller meansquare error and less sensitivity to small perturbations in the data) and soft shrinkage (generally smaller bias and overall meansquareerror). The minimax thresholds for the nonnegative garrote are derived and the threshold selection procedure based on Stein's Unbiased Risk Estimate (SURE) is studied. We also propose a threshold selection procedure based on combining Coifman and Donoho's cyclespinning and SURE. The procedure is called SPINSURE. We use examples to show that SPINSURE is more stable than SURE: smaller standard deviation and smaller range. Key Words and Phra...
Novel Bayesian Multiscale Method for Speckle Removal in Medical Ultrasound Images
 IEEE TRANS. MED. IMAG
, 2001
"... A novel speckle suppression method for medical ultrasound images is presented. First, the logarithmic transform of the original image is analyzed into the multiscale wavelet domain. We show that the subband decompositions of ultrasound images have significantly nonGaussian statistics that are best ..."
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Cited by 51 (9 self)
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A novel speckle suppression method for medical ultrasound images is presented. First, the logarithmic transform of the original image is analyzed into the multiscale wavelet domain. We show that the subband decompositions of ultrasound images have significantly nonGaussian statistics that are best described by families of heavytailed distributions such as the alphastable. Then, we design a Bayesian estimator that exploits these statistics. We use the alphastable model to develop a blind noiseremoval processor that performs a nonlinear operation on the data. Finally, we compare our technique with current stateoftheart soft and hard thresholding methods applied on actual ultrasound medical images and we quantify the achieved performance improvement.
Wavelet Processes and Adaptive Estimation of the Evolutionary Wavelet Spectrum
, 1998
"... This article defines and studies a new class of nonstationary random processes constructed from discrete nondecimated wavelets which generalizes the Cramer (Fourier) representation of stationary time series. We define an evolutionary wavelet spectrum (EWS) which quantifies how process power va ..."
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Cited by 48 (28 self)
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This article defines and studies a new class of nonstationary random processes constructed from discrete nondecimated wavelets which generalizes the Cramer (Fourier) representation of stationary time series. We define an evolutionary wavelet spectrum (EWS) which quantifies how process power varies locally over time and scale. We show how the EWS may be rigorously estimated by a smoothed wavelet periodogram and how both these quantities may be inverted to provide an estimable timelocalized autocovariance. We illustrate our theory with a pedagogical example based on discrete nondecimated Haar wavelets and also a real medical time series example.
A Joint Inter and Intrascale Statistical Model for Bayesian Wavelet Based Image Denoising
 IEEE Trans. Image Proc
, 2002
"... This paper presents a new waveletbased image denoising method, which extends a recently emerged "geometrical" Bayesian framework. The new method combines these criteria for distinguishing supposedly useful coefficient from noise coefficient magnit:q54 tgni evolut47 across scales and spatA5 clust:q5 ..."
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Cited by 47 (5 self)
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This paper presents a new waveletbased image denoising method, which extends a recently emerged "geometrical" Bayesian framework. The new method combines these criteria for distinguishing supposedly useful coefficient from noise coefficient magnit:q54 tgni evolut47 across scales and spatA5 clust:q5A of large coefficients near image edges. These three crit546 are combined in a Bayesian framework. The spatD5 clust:q5] propert:5 are expressed in a prior model. Thest6[]A:q5D propertAA concerning coefficient magnit[:q andt:55 evolut4[ across scales are expressed in a joint condit:q]6 model. The three main noveltAA with respect to relat[ approaches are:(1)t he int760C7:q]0056: of wavelet coefficient are st0057:q]005 charact:q]55C and different local crit44C for dist]6:q]55C5 useful coefficient from noise are evaluat]6 (2) a joint condit:q]7 model is introduced, and (3) a novel anisot:q]7 Markov Random Field prior model is proposed. The results demonstrate an improved denoising performance over related earlier techniques.