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580
A fast iterative shrinkagethresholding algorithm with application to . . .
, 2009
"... We consider the class of Iterative ShrinkageThresholding Algorithms (ISTA) for solving linear inverse problems arising in signal/image processing. This class of methods is attractive due to its simplicity, however, they are also known to converge quite slowly. In this paper we present a Fast Iterat ..."
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Cited by 365 (4 self)
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We consider the class of Iterative ShrinkageThresholding Algorithms (ISTA) for solving linear inverse problems arising in signal/image processing. This class of methods is attractive due to its simplicity, however, they are also known to converge quite slowly. In this paper we present a Fast Iterative ShrinkageThresholding Algorithm (FISTA) which preserves the computational simplicity of ISTA, but with a global rate of convergence which is proven to be significantly better, both theoretically and practically. Initial promising numerical results for waveletbased image deblurring demonstrate the capabilities of FISTA.
Waveletbased statistical signal processing using hidden Markov models
 IEEE Transactions on Signal Processing
, 1998
"... Abstract — Waveletbased statistical signal processing techniques such as denoising and detection typically model the wavelet coefficients as independent or jointly Gaussian. These models are unrealistic for many realworld signals. In this paper, we develop a new framework for statistical signal pr ..."
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Cited by 325 (52 self)
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Abstract — Waveletbased statistical signal processing techniques such as denoising and detection typically model the wavelet coefficients as independent or jointly Gaussian. These models are unrealistic for many realworld signals. In this paper, we develop a new framework for statistical signal processing based on waveletdomain hidden Markov models (HMM’s) that concisely models the statistical dependencies and nonGaussian statistics encountered in realworld signals. Waveletdomain HMM’s are designed with the intrinsic properties of the wavelet transform in mind and provide powerful, yet tractable, probabilistic signal models. Efficient expectation maximization algorithms are developed for fitting the HMM’s to observational signal data. The new framework is suitable for a wide range of applications, including signal estimation, detection, classification, prediction, and even synthesis. To demonstrate the utility of waveletdomain HMM’s, we develop novel algorithms for signal denoising, classification, and detection. Index Terms — Hidden Markov model, probabilistic graph, wavelets.
Minimax Estimation via Wavelet Shrinkage
, 1992
"... We attempt to recover an unknown function from noisy, sampled data. Using orthonormal bases of compactly supported wavelets we develop a nonlinear method which works in the wavelet domain by simple nonlinear shrinkage of the empirical wavelet coe cients. The shrinkage can be tuned to be nearly minim ..."
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Cited by 246 (32 self)
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We attempt to recover an unknown function from noisy, sampled data. Using orthonormal bases of compactly supported wavelets we develop a nonlinear method which works in the wavelet domain by simple nonlinear shrinkage of the empirical wavelet coe cients. The shrinkage can be tuned to be nearly minimax over any member of a wide range of Triebel and Besovtype smoothness constraints, and asymptotically minimax over Besov bodies with p q. Linear estimates cannot achieve even the minimax rates over Triebel and Besov classes with p <2, so our method can signi cantly outperform every linear method (kernel, smoothing spline, sieve,:::) in a minimax sense. Variants of our method based on simple threshold nonlinearities are nearly minimax. Our method possesses the interpretation of spatial adaptivity: it reconstructs using a kernel which mayvary in shape and bandwidth from point to point, depending on the data. Least favorable distributions for certain of the Triebel and Besov scales generate objects with sparse wavelet transforms. Many real objects have similarly sparse transforms, which suggests that these minimax results are relevant for practical problems. Sequels to this paper discuss practical implementation, spatial adaptation properties and applications to inverse problems.
Wavelet shrinkage: asymptopia
 Journal of the Royal Statistical Society, Ser. B
, 1995
"... Considerable e ort has been directed recently to develop asymptotically minimax methods in problems of recovering in nitedimensional objects (curves, densities, spectral densities, images) from noisy data. A rich and complex body of work has evolved, with nearly or exactly minimax estimators bein ..."
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Cited by 239 (35 self)
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Considerable e ort has been directed recently to develop asymptotically minimax methods in problems of recovering in nitedimensional objects (curves, densities, spectral densities, images) from noisy data. A rich and complex body of work has evolved, with nearly or exactly minimax estimators being obtained for a variety of interesting problems. Unfortunately, the results have often not been translated into practice, for a variety of reasons { sometimes, similarity to known methods, sometimes, computational intractability, and sometimes, lack of spatial adaptivity. We discuss a method for curve estimation based on n noisy data; one translates the empirical wavelet coe cients towards the origin by an amount p p 2 log(n) = n. The method is di erent from methods in common use today, is computationally practical, and is spatially adaptive; thus it avoids a number of previous objections to minimax estimators. At the same time, the method is nearly minimax for a wide variety of loss functions { e.g. pointwise error, global error measured in L p norms, pointwise and global error in estimation of derivatives { and for a wide range of smoothness classes, including standard Holder classes, Sobolev classes, and Bounded Variation. This is amuch broader nearoptimality than anything previously proposed in the minimax literature. Finally, the theory underlying the method is interesting, as it exploits a correspondence between statistical questions and questions of optimal recovery and informationbased complexity.
An EM Algorithm for WaveletBased Image Restoration
, 2002
"... This paper introduces an expectationmaximization (EM) algorithm for image restoration (deconvolution) based on a penalized likelihood formulated in the wavelet domain. Regularization is achieved by promoting a reconstruction with lowcomplexity, expressed in terms of the wavelet coecients, taking a ..."
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Cited by 233 (21 self)
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This paper introduces an expectationmaximization (EM) algorithm for image restoration (deconvolution) based on a penalized likelihood formulated in the wavelet domain. Regularization is achieved by promoting a reconstruction with lowcomplexity, expressed in terms of the wavelet coecients, taking advantage of the well known sparsity of wavelet representations. Previous works have investigated waveletbased restoration but, except for certain special cases, the resulting criteria are solved approximately or require very demanding optimization methods. The EM algorithm herein proposed combines the efficient image representation oered by the discrete wavelet transform (DWT) with the diagonalization of the convolution operator obtained in the Fourier domain. The algorithm alternates between an Estep based on the fast Fourier transform (FFT) and a DWTbased Mstep, resulting in an ecient iterative process requiring O(N log N) operations per iteration. Thus, it is the rst image restoration algorithm that optimizes a waveletbased penalized likelihood criterion and has computational complexity comparable to that of standard wavelet denoising or frequency domain deconvolution methods. The convergence behavior of the algorithm is investigated, and it is shown that under mild conditions the algorithm converges to a globally optimal restoration. Moreover, our new approach outperforms several of the best existing methods in benchmark tests, and in some cases is also much less computationally demanding.
Translationinvariant denoising
, 1995
"... DeNoising with the traditional (orthogonal, maximallydecimated) wavelet transform sometimes exhibits visual artifacts; we attribute some of these – for example, Gibbs phenomena in the neighborhood of discontinuities – to the lack of translation invariance of the wavelet basis. One method to suppre ..."
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Cited by 228 (8 self)
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DeNoising with the traditional (orthogonal, maximallydecimated) wavelet transform sometimes exhibits visual artifacts; we attribute some of these – for example, Gibbs phenomena in the neighborhood of discontinuities – to the lack of translation invariance of the wavelet basis. One method to suppress such artifacts, termed “cycle spinning ” by Coifman, is to “average out ” the translation dependence. For a range of shifts, one shifts the data (right or left as the case may be), DeNoises the shifted data, and then unshifts the denoised data. Doing this for each of a range of shifts, and averaging the several results so obtained, produces a reconstruction subject to far weaker Gibbs phenomena than thresholding based DeNoising using the traditional orthogonal wavelet transform. CycleSpinning over the range of all circulant shifts can be accomplished in order nlog 2(n) time; it is equivalent to denoising using the undecimated or stationary wavelet transform. Cyclespinning exhibits benefits outside of wavelet denoising, for example in cosine packet denoising, where it helps suppress ‘clicks’. It also has a counterpart in frequency domain denoising, where the goal of translationinvariance is replaced by modulation invariance, and the central shiftDeNoiseunshift operation is replaced by modulateDeNoisedemodulate. We illustrate these concepts with extensive computational examples; all figures presented here are reproducible using the WaveLab software package. 1
Wavelet Thresholding via a Bayesian Approach
 J. R. STATIST. SOC. B
, 1996
"... We discuss a Bayesian formalism which gives rise to a type of wavelet threshold estimation in nonparametric regression. A prior distribution is imposed on the wavelet coefficients of the unknown response function, designed to capture the sparseness of wavelet expansion common to most applications. ..."
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Cited by 204 (27 self)
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We discuss a Bayesian formalism which gives rise to a type of wavelet threshold estimation in nonparametric regression. A prior distribution is imposed on the wavelet coefficients of the unknown response function, designed to capture the sparseness of wavelet expansion common to most applications. For the prior specified, the posterior median yields a thresholding procedure. Our prior model for the underlying function can be adjusted to give functions falling in any specific Besov space. We establish a relation between the hyperparameters of the prior model and the parameters of those Besov spaces within which realizations from the prior will fall. Such a relation gives insight into the meaning of the Besov space parameters. Moreover, the established relation makes it possible in principle to incorporate prior knowledge about the function's regularity properties into the prior model for its wavelet coefficients. However, prior knowledge about a function's regularity properties might b...
Nonlinear Wavelet Image Processing: Variational Problems, Compression, and Noise Removal through Wavelet Shrinkage
 IEEE Trans. Image Processing
, 1996
"... This paper examines the relationship between waveletbased image processing algorithms and variational problems. Algorithms are derived as exact or approximate minimizers of variational problems; in particular, we show that wavelet shrinkage can be considered the exact minimizer of the following pro ..."
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Cited by 193 (11 self)
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This paper examines the relationship between waveletbased image processing algorithms and variational problems. Algorithms are derived as exact or approximate minimizers of variational problems; in particular, we show that wavelet shrinkage can be considered the exact minimizer of the following problem: given an image F defined on a square I, minimize over all g in the Besov space B 1 1 (L1 (I)) the functional #F  g# 2 L 2 (I) + ##g# B 1 1 (L 1 (I)) .Weusethetheoryof nonlinear wavelet image compression in L2 (I) to derive accurate error bounds for noise removal through wavelet shrinkage applied to images corrupted with i.i.d., mean zero, Gaussian noise. A new signaltonoise ratio, which we claim more accurately reflects the visual perception of noise in images, arises in this derivation. We present extensive computations that support the hypothesis that nearoptimal shrinkage parameters can be derived if one knows (or can estimate) only two parameters about an image F:thelarge...
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...
Analysis Of Multiresolution Image Denoising Schemes Using GeneralizedGaussian Priors
 IEEE TRANS. INFO. THEORY
, 1998
"... In this paper, we investigate various connections between wavelet shrinkage methods in image processing and Bayesian estimation using Generalized Gaussian priors. We present fundamental properties of the shrinkage rules implied by Generalized Gaussian and other heavytailed priors. This allows us to ..."
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Cited by 177 (9 self)
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In this paper, we investigate various connections between wavelet shrinkage methods in image processing and Bayesian estimation using Generalized Gaussian priors. We present fundamental properties of the shrinkage rules implied by Generalized Gaussian and other heavytailed priors. This allows us to show a simple relationship between differentiability of the logprior at zero and the sparsity of the estimates, as well as an equivalence between universal thresholding schemes and Bayesian estimation using a certain Generalized Gaussian prior.