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50
A douglasRachford splitting approach to nonsmooth convex variational signal recovery
 IEEE Journal of Selected Topics in Signal Processing
, 2007
"... Abstract — Under consideration is the large body of signal recovery problems that can be formulated as the problem of minimizing the sum of two (not necessarily smooth) lower semicontinuous convex functions in a real Hilbert space. This generic problem is analyzed and a decomposition method is propo ..."
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Cited by 87 (23 self)
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Abstract — Under consideration is the large body of signal recovery problems that can be formulated as the problem of minimizing the sum of two (not necessarily smooth) lower semicontinuous convex functions in a real Hilbert space. This generic problem is analyzed and a decomposition method is proposed to solve it. The convergence of the method, which is based on the DouglasRachford algorithm for monotone operatorsplitting, is obtained under general conditions. Applications to nonGaussian image denoising in a tight frame are also demonstrated. Index Terms — Convex optimization, denoising, DouglasRachford, frame, nondifferentiable optimization, Poisson noise,
HessianBased Norm Regularization for Image Restoration With Biomedical Applications
"... methods that can be effectively used for image restoration problems in a variational framework. Motivated by the great success of the totalvariation (TV) functional, we extend it to also include secondorder differential operators. Specifically, we derive secondorder regularizers that involve matr ..."
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Cited by 20 (5 self)
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methods that can be effectively used for image restoration problems in a variational framework. Motivated by the great success of the totalvariation (TV) functional, we extend it to also include secondorder differential operators. Specifically, we derive secondorder regularizers that involve matrix norms of the Hessian operator. The definition of these functionals is based on an alternative interpretation of TV that relies on mixed norms of directional derivatives. We show that the resulting regularizers retain some of the most favorable properties of TV, i.e., convexity, homogeneity, rotation, and translation invariance, while dealing effectively with the staircase effect. We further develop an efficient minimization scheme for the corresponding objective functions. The proposed algorithm is of the iteratively reweighted leastsquare type and results from a majorization–minimization approach. It relies on a problemspecific preconditioned conjugate gradient method, which makes the overall minimization scheme very attractive since it can be applied effectively to large images in a reasonable computational time. We validate the overall proposed regularization framework through deblurring experiments under additive Gaussian noise on standard and biomedical images. Index Terms—Biomedical imaging, Frobenius norm, Hessian matrix, image deblurring, linear inverse problems, majorization–minimization (MM) algorithms, spectral norm. I.
Perturbationresilient blockiterative projection methods with application to image reconstruction from projections
, 2009
"... A blockiterative projection algorithm for solving the consistent convex feasibility problem in a finitedimensional Euclidean space that is resilient to bounded and summable perturbations (in the sense that convergence to a feasible point is retained even if such perturbations are introduced in eac ..."
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Cited by 18 (8 self)
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A blockiterative projection algorithm for solving the consistent convex feasibility problem in a finitedimensional Euclidean space that is resilient to bounded and summable perturbations (in the sense that convergence to a feasible point is retained even if such perturbations are introduced in each iterative step of the algorithm) is proposed. This resilience can be used to steer the iterative process towards a feasible point that is superior in the sense of some functional on the points in the Euclidean space having a small value. The potential usefulness of this is illustrated in image reconstruction from projections, using both total variation and negative entropy as the functional. 1
Constrained and SNRbased solutions for TVHilbert space image denoising
"... We examine the general regularization model which is based on totalvariation for the structural part and a Hilbertspace norm for the oscillatory part. This framework generalizes the RudinOsherFatemi and the OsherSoleVese models and opens way for new denoising or decomposition methods with tuna ..."
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Cited by 17 (3 self)
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We examine the general regularization model which is based on totalvariation for the structural part and a Hilbertspace norm for the oscillatory part. This framework generalizes the RudinOsherFatemi and the OsherSoleVese models and opens way for new denoising or decomposition methods with tunable norms, which are adapted to the nature of the noise or textures of the image. We give sufficient conditions and prove the convergence of an iterative numerical implementation, following Chambolle’s projection algorithm. In this paper we focus on the denoising problem. In order to provide an automatic solution, a systematic method for choosing the weight between the energies is imperative. The classical method for selecting the weight parameter according to the noise variance is reformulated in a Hilbert space sense. Moreover, we generalize a recent study of GilboaSochenZeevi where the weight parameter is selected such that the denoised result is close to optimal, in the SNR sense. A broader definition of SNR, which is frequency weighted, is formulated in the context of inner products. A necessary condition for maximal SNR is provided. Lower and upper bounds on the SNR performance of the classical and optimal strategies are established, under quite general assumptions.
Vese, Mathematical modeling of textures: Application to color image decomposition with a projected gradient algorithm
 JMIV
"... Abstract: In this paper, we are interested in texture modeling with functional analysis spaces. We focus on the case of color image processing, and in particular color image decomposition. The problem of image decomposition consists in splitting an original image f into two components u and v. u sho ..."
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Cited by 9 (1 self)
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Abstract: In this paper, we are interested in texture modeling with functional analysis spaces. We focus on the case of color image processing, and in particular color image decomposition. The problem of image decomposition consists in splitting an original image f into two components u and v. u should contain the geometric information of the original image, while v should be made of the oscillating patterns of f, such as textures. We propose here a scheme based on a projected gradient algorithm to compute the solution of various decomposition models for color images or vectorvalued images. We provide a direct convergence proof of the scheme, and we give some analysis on color texture modeling. Keywords: Texture modeling, color image decomposition, projected gradient algorithm, color texture modeling. 1
Rank related properties for basis pursuit and total variation regularization
, 2006
"... This paper focuses on optimization problems containing an l 1 kind of regularity criterion and a smooth data fidelity term. A general theorem is applied in this context; it gives an estimate of the distribution law of the “rank ” of the solution to optimization problems, when the initial datum follo ..."
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Cited by 8 (6 self)
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This paper focuses on optimization problems containing an l 1 kind of regularity criterion and a smooth data fidelity term. A general theorem is applied in this context; it gives an estimate of the distribution law of the “rank ” of the solution to optimization problems, when the initial datum follows a uniform (in a convex compact set) distribution law. It says that, asymptotically, solutions with a large rank are more and more likely. The main goal of this paper is to understand the meaning of this notion of rank for some energies which are commonly used in image processing. We study in detail the energy whose level sets are defined as the convex hull of a finite subset of R N (c.f. Basis Pursuit) and the total variation. For these energies, the notion of rank relates respectively to sparse representation and staircasing. In all cases but the 2D total variation, we are able to adapt the general theorem mentioned above to the energies under consideration. Key words: basis pursuit, sparse representation, total variation, regularization, polytopes
Dual constrained TVbased regularization on graphs
 SIAM J. on Imaging Sciences
, 2013
"... Abstract. Algorithms based on Total Variation (TV) minimization are prevalent in image processing. They play a key role in a variety of applications such as image denoising, compressive sensing and inverse problems in general. In this work, we extend the TV dual framework that includes Chambolle’s a ..."
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Cited by 8 (5 self)
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Abstract. Algorithms based on Total Variation (TV) minimization are prevalent in image processing. They play a key role in a variety of applications such as image denoising, compressive sensing and inverse problems in general. In this work, we extend the TV dual framework that includes Chambolle’s and GilboaOsher’s projection algorithms for TV minimization. We use a flexible graph data representation that allows us to generalize the constraint on the projection variable. We show how this new formulation of the TV problem may be solved by means of fast parallel proximal algorithms. On denoising and deblurring examples, the proposed approach is shown not only to perform better than recent TVbased approaches, but also to perform well on arbitrary graphs instead of regular grids. The proposed method consequently applies to a variety of other inverse problems including image fusion and mesh filtering.
Approximation of maximal cheeger sets by projection
, 2008
"... This article deals with the numerical computation of the Cheeger constant and the approximation of the maximal Cheeger set of a given subset of Rd. This problem is motivated by landslide modelling as well as by the continuous maximal flow problem. Using the fact that the maximal Cheeger set can be a ..."
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Cited by 7 (1 self)
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This article deals with the numerical computation of the Cheeger constant and the approximation of the maximal Cheeger set of a given subset of Rd. This problem is motivated by landslide modelling as well as by the continuous maximal flow problem. Using the fact that the maximal Cheeger set can be approximated by solving a rather simple projection problem, we propose a numerical strategy to compute maximal Cheeger sets and Cheeger constants.
Locally parallel texture modeling
 SIAM JOURNAL ON IMAGING SCIENCES
, 2011
"... This article presents a new adaptive framework for locally parallel texture modeling. Oscillating patterns are modeled with functionals that constrain the local Fourier decomposition of the texture. We first introduce a convex texture functional which is a weighted Hilbert norm. The weights on the ..."
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Cited by 6 (1 self)
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This article presents a new adaptive framework for locally parallel texture modeling. Oscillating patterns are modeled with functionals that constrain the local Fourier decomposition of the texture. We first introduce a convex texture functional which is a weighted Hilbert norm. The weights on the local Fourier atoms are optimized to match the local orientation and frequency of the texture. This adaptive convex model is used to solve image processing inverse problems, such as image decomposition and inpainting. The local orientation and frequency of the texture component are adaptively estimated during the minimization process. To improve inpainting performances over large missing regions, we introduce a nonconvex generalization of our texture model. This new model constrains the amplitude of the texture and allows one to impose an arbitrary oscillation profile. This nonconvex model bridges the gap between regularization methods for image restoration and patchbased synthesis approaches that are successful in texture synthesis. Numerical results show that our method improves state of the art algorithms for locally parallel textures.
A Convex Programming Algorithm for Noisy Discrete Tomography
"... Summary. A convex programming approach to discrete tomographic image reconstruction in noisy environments is proposed. Conventional constraints are mixed with noisebased constraints on the sinogram and a binaritypromoting total variation constraint. The noisebased constraints are modeled as confi ..."
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Cited by 5 (0 self)
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Summary. A convex programming approach to discrete tomographic image reconstruction in noisy environments is proposed. Conventional constraints are mixed with noisebased constraints on the sinogram and a binaritypromoting total variation constraint. The noisebased constraints are modeled as confidence regions that are constructed under a Poisson noise assumption. A convex objective is then minimized over the resulting feasibility set via a parallel blockiterative method. Applications to binary tomographic reconstruction are demonstrated. 1