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20
Nested iterative algorithms for convex constrained image recovery problems
 IEEE Journal of Selected Topics in Signal Processing
, 2007
"... The objective of this paper is to develop methods for solving image recovery problems subject to constraints on the solution. More precisely, we will be interested in problems which can be formulated as the minimization over a closed convex constraint set of the sum of two convex functions f and g, ..."
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Cited by 21 (6 self)
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The objective of this paper is to develop methods for solving image recovery problems subject to constraints on the solution. More precisely, we will be interested in problems which can be formulated as the minimization over a closed convex constraint set of the sum of two convex functions f and g, where f may be nonsmooth and g is differentiable with a Lipschitzcontinuous gradient. To reach this goal, we derive two types of algorithms that combine forwardbackward and DouglasRachford iterations. The weak convergence of the proposed algorithms is proved. In the case when the Lipschitzcontinuity property of the gradient of g is not satisfied, we also show that, under some assumptions, it remains possible to apply these methods to the considered optimization problem by making use of a quadratic extension technique. The effectiveness of the algorithms is demonstrated for two waveletbased image restoration problems involving a signaldependent Gaussian noise and a Poisson noise, respectively. 1
A Fast Multilevel Algorithm for WaveletRegularized Image Restoration
 IEEE Trans. Image Processing
"... Abstract—We present a multilevel extension of the popular “thresholded Landweber ” algorithm for waveletregularized image restoration that yields an order of magnitude speed improvement over the standard fixedscale implementation. The method is generic and targeted towards largescale linear inver ..."
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Cited by 13 (5 self)
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Abstract—We present a multilevel extension of the popular “thresholded Landweber ” algorithm for waveletregularized image restoration that yields an order of magnitude speed improvement over the standard fixedscale implementation. The method is generic and targeted towards largescale linear inverse problems, such as 3D deconvolution microscopy. The algorithm is derived within the framework of bound optimization. The key idea is to successively update the coefficients in the various wavelet channels using fixed, subbandadapted iteration parameters (step sizes and threshold levels). The optimization problem is solved efficiently via a proper chaining of basic iteration modules. The higher level description of the algorithm is similar to that of a multigrid solver for PDEs, but there is one fundamental difference: the latter iterates though a sequence of multiresolution versions of the original problem, while, in our case, we cycle through the wavelet subspaces corresponding to the difference between successive approximations. This strategy is motivated by the special structure of the problem and the preconditioning properties of the wavelet representation. We establish that the solution of the restoration problem corresponds to a fixed point of our multilevel optimizer. We also provide experimental evidence that the improvement in convergence rate is essentially determined by the (unconstrained) linear part of the algorithm, irrespective of the type of wavelet. Finally, we illustrate the technique with some image deconvolution examples, including some real 3D fluorescence microscopy data. Index Terms—Bound optimization, confocal, convergence acceleration, deconvolution, fast, fluorescence, inverse problems,regularization, majorizeminimize, microscopy, multigrid, multilevel, multiresolution, multiscale, nonlinear, optimization transfer, preconditioning, reconstruction, restoration, sparsity,
A SURE Approach for Digital Signal/Image Deconvolution Problems
, 2009
"... In this paper, we are interested in the classical problem of restoring data degraded by a convolution and the addition of a white Gaussian noise. The originality of the proposed approach is twofold. Firstly, we formulate the restoration problem as a nonlinear estimation problem leading to the mini ..."
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Cited by 7 (2 self)
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In this paper, we are interested in the classical problem of restoring data degraded by a convolution and the addition of a white Gaussian noise. The originality of the proposed approach is twofold. Firstly, we formulate the restoration problem as a nonlinear estimation problem leading to the minimization of a criterion derived from Stein’s unbiased quadratic risk estimate. Secondly, the deconvolution procedure is performed using any analysis and synthesis frames that can be overcomplete or not. New theoretical results concerning the calculation of the variance of the Stein’s risk estimate are also provided in this work. Simulations carried out on natural images show the good performance of our method w.r.t. conventional waveletbased restoration methods.
A fast waveletbased reconstruction method for magnetic resonance imaging
 IEEE Trans. Med. Imag
, 2011
"... Abstract—In this work, we exploit the fact that wavelets can represent magnetic resonance images well, with relatively few coefficients. We use this property to improve magnetic resonance imaging (MRI) reconstructions from undersampled data with arbitrary kspace trajectories. Reconstruction is pose ..."
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Cited by 6 (2 self)
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Abstract—In this work, we exploit the fact that wavelets can represent magnetic resonance images well, with relatively few coefficients. We use this property to improve magnetic resonance imaging (MRI) reconstructions from undersampled data with arbitrary kspace trajectories. Reconstruction is posed as an optimization problem that could be solved with the iterative shrinkage/thresholding algorithm (ISTA) which, unfortunately, converges slowly. To make the approach more practical, we propose a variant that combines recent improvements in convex optimization and that can be tuned to a given specific kspace trajectory. We present a mathematical analysis that explains the performance of the algorithms. Using simulated and in vivo data, we show that our nonlinear method is fast, as it accelerates ISTA by almost two orders of magnitude. We also show that it remains competitive with TV regularization in terms of image quality. Index Terms—Compressed sensing, fast iterative shrinkage/ thresholding algorithm (FISTA), fast weighted iterative shrinkage/ thresholding algorithm (FWISTA), iterative shrinkage/thresholding algorithm (ISTA), magnetic resonance imaging (MRI), nonCartesian, nonlinear reconstruction, sparsity, thresholded Landweber, total variation, undersampled spiral, wavelets. I.
Shearletbased deconvolution
 IEEE Trans. Image Process
, 2009
"... Abstract—In this paper, a new type of deconvolution algorithm is proposed that is based on estimating the image from a shearlet decomposition. Shearlets provide a multidirectional and multiscale decomposition that has been mathematically shown to represent distributed discontinuities such as edges b ..."
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Cited by 4 (4 self)
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Abstract—In this paper, a new type of deconvolution algorithm is proposed that is based on estimating the image from a shearlet decomposition. Shearlets provide a multidirectional and multiscale decomposition that has been mathematically shown to represent distributed discontinuities such as edges better than traditional wavelets. Constructions such as curvelets and contourlets share similar properties, yet their implementations are significantly different from that of shearlets. Taking advantage of unique properties of a new Mchannel implementation of the shearlet transform, we develop an algorithm that allows for the approximation inversion operator to be controlled on a multiscale and multidirectional basis. A key improvement over closely related approaches such as ForWaRD is the automatic determination of the threshold values for the noise shrinkage for each scale and direction without explicit knowledge of the noise variance using a generalized cross validation (GCV). Various tests show that this method can perform significantly better than many competitive deconvolution algorithms. Index Terms—Deconvolution, generalized cross validation, shearlets, wavelets. I.
RESTORATION OF IMAGES AND 3D DATA TO HIGHER RESOLUTION BY DECONVOLUTION WITH SPARSITY REGULARIZATION
"... Image convolution is conventionally approximated by the LTI discrete model. It is well recognized that the higher the sampling rate, the better is the approximation. However sometimes images or 3D data are only available at a lower sampling rate due to physical constraints of the imaging system. In ..."
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Cited by 2 (0 self)
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Image convolution is conventionally approximated by the LTI discrete model. It is well recognized that the higher the sampling rate, the better is the approximation. However sometimes images or 3D data are only available at a lower sampling rate due to physical constraints of the imaging system. In this paper, we model the undersampled observation as the result of combining convolution and subsampling. Because the wavelet coefficients of piecewise smooth images tend to be sparse and well modelled by treelike structures, we propose the L0 reweightedL2 minimization (L0RL2) algorithm to solve this problem. This promotes sparsity by minimizing the reweighted L2 norm, which approximates the L0 norm, and by enforcing a tree model through bivariate shrinkage. We test the algorithm on 3 examples: a simple ring, the cameraman image and a 3D microscope dataset; and show that good results can be obtained.
Hessian SchattenNorm Regularization for Linear Inverse Problems
"... Abstract — We introduce a novel family of invariant, convex, and nonquadratic functionals that we employ to derive regularized solutions of illposed linear inverse imaging problems. The proposed regularizers involve the Schatten norms of the Hessian matrix, which are computed at every pixel of the ..."
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Cited by 2 (2 self)
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Abstract — We introduce a novel family of invariant, convex, and nonquadratic functionals that we employ to derive regularized solutions of illposed linear inverse imaging problems. The proposed regularizers involve the Schatten norms of the Hessian matrix, which are computed at every pixel of the image. They can be viewed as secondorder extensions of the popular totalvariation (TV) seminorm since they satisfy the same invariance properties. Meanwhile, by taking advantage of secondorder derivatives, they avoid the staircase effect, a common artifact of TVbased reconstructions, and perform well for a wide range of applications. To solve the corresponding optimization problems, we propose an algorithm that is based on a primaldual formulation. A fundamental ingredient of this algorithm is the projection of matrices onto Schatten norm balls of arbitrary radius. This operation is performed efficiently based on a direct link we provide between vector projections onto ℓq norm balls and matrix projections onto Schatten norm balls. Finally, we demonstrate the effectiveness of the proposed methods through experimental results on several inverse imaging problems with real and simulated data. Index Terms — Eigenvalue optimization, Hessian operator, image reconstruction, matrix projections, Schatten norms.
Improved Iterative Curvelet Thresholding for Compressed Sensing
"... A new theory named compressed sensing for simultaneous sampling and compression of signals has been becoming popular in the communities of signal processing, imaging and applied mathematics. In this paper, we present improved/accelerated iterative curvelet thresholding methods for compressed sensing ..."
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Cited by 1 (0 self)
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A new theory named compressed sensing for simultaneous sampling and compression of signals has been becoming popular in the communities of signal processing, imaging and applied mathematics. In this paper, we present improved/accelerated iterative curvelet thresholding methods for compressed sensing reconstruction in the fields of remote sensing. Some recent strategies including BioucasDias and Figueiredo’s twostep iteration, Beck and Teboulle’s fast method, and Osher et al’s linearized Bregman iteration are applied to iterative curvelet thresholding in order to accelerate convergence. Advantages and disadvantages of the proposed methods are studied using the socalled pseudoPareto curve in the numerical experiments on singlepixel remote sensing and Fourierdomain random imaging.
FASTL0BASEDSPARSESIGNALRECOVERY
"... This paper develops an algorithm on finding sparse signals fromlimitedobservationsofalinearsystem. Weassumean adaptive Gaussian model for sparse signals. This model resultsinaleastsquareproblemwithaniterativelyreweighted L2 penalty that approximates the L0norm. We propose a fast algorithm to solve ..."
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This paper develops an algorithm on finding sparse signals fromlimitedobservationsofalinearsystem. Weassumean adaptive Gaussian model for sparse signals. This model resultsinaleastsquareproblemwithaniterativelyreweighted L2 penalty that approximates the L0norm. We propose a fast algorithm to solve the problem within a continuation framework. Inourexamples,weshowthatthecorrectsparsity map and sparsity level are gradually learnt during the iterationsevenwhenthenumberofobservationsisreduced, or when observation noise is present. In addition, with the helpofsophisticatedinterscalesignalmodels,thealgorithm is able to recover signal to a better accuracy with reduced number of observations. 1.