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17
Some firstorder algorithms for total variation based image restoration
, 2009
"... This paper deals with firstorder numerical schemes for image restoration. These schemes rely on a dualitybased algorithm proposed in 1979 by Bermùdez and Moreno. This is an old and forgotten algorithm that is revealed wider than recent schemes (such as the Chambolle projection algorithm) and able ..."
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Cited by 38 (2 self)
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This paper deals with firstorder numerical schemes for image restoration. These schemes rely on a dualitybased algorithm proposed in 1979 by Bermùdez and Moreno. This is an old and forgotten algorithm that is revealed wider than recent schemes (such as the Chambolle projection algorithm) and able to improve contemporary schemes. Total variation regularization and smoothed total variation regularization are investigated. Algorithms are presented for such regularizations in image restoration. We prove the convergence of all the proposed schemes. We illustrate our study with numerous numerical examples. We make some comparisons with a class of efficient algorithms (proved to be optimal among firstorder numerical schemes) recently introduced by Y. Nesterov.
Multiplicative noise removal using variable splitting and constrained optimization
 IEEE Transactions on Image Processing
, 2010
"... Abstract—Multiplicative noise (also known as speckle noise) models are central to the study of coherent imaging systems, such as synthetic aperture radar and sonar, and ultrasound and laser imaging. These models introduce two additional layers of difficulties with respect to the standard Gaussian ad ..."
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Cited by 21 (1 self)
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Abstract—Multiplicative noise (also known as speckle noise) models are central to the study of coherent imaging systems, such as synthetic aperture radar and sonar, and ultrasound and laser imaging. These models introduce two additional layers of difficulties with respect to the standard Gaussian additive noise scenario: 1) the noise is multiplied by (rather than added to) the original image; 2) the noise is not Gaussian, with Rayleigh and Gamma being commonly used densities. These two features of multiplicative noise models preclude the direct application of most stateoftheart algorithms, which are designed for solving unconstrained optimization problems where the objective has two terms: a quadratic data term (loglikelihood), reflecting the additive and Gaussian nature of the noise, plus a convex (possibly nonsmooth) regularizer (e.g., a total variation or waveletbased regularizer/prior). In this paper, we address these difficulties by: 1) converting the multiplicative model into an additive one by taking logarithms, as proposed by some other authors; 2) using variable splitting to obtain an equivalent constrained problem; and 3) dealing with this optimization problem using the augmented Lagrangian framework. A set of experiments shows that the proposed method, which we name MIDAL (multiplicative image denoising by augmented Lagrangian), yields stateoftheart results both in terms of speed and denoising performance. Index Terms—Augmented Lagrangian, Douglas–Rachford splitting, multiplicative noise, speckled images, synthetic aperture
Minimization and parameter estimation for seminorm regularization models with Idivergence constraints
, 2012
"... In this papers we analyze the minimization of seminorms ‖L · ‖ on R n under the constraint of a bounded Idivergence D(b,H·) for rather general linear operators H and L. The Idivergence is also known as KullbackLeibler divergence and appears in many models in imaging science, in particular when d ..."
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Cited by 13 (2 self)
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In this papers we analyze the minimization of seminorms ‖L · ‖ on R n under the constraint of a bounded Idivergence D(b,H·) for rather general linear operators H and L. The Idivergence is also known as KullbackLeibler divergence and appears in many models in imaging science, in particular when dealing with Poisson data. Often H represents, e.g., a linear blur operator and L is some discrete derivative or frame analysis operator. We prove relations between the the parameters of Idivergence constrained and penalized problems without assuming the uniqueness of their minimizers. To solve the Idivergence constrained problem we apply firstorder primaldual algorithms which reduce the problem to the solution of certain proximal minimization problems in each iteration step. One of these proximation problems is an Idivergence constrained least squares problem which can be solved based on Morosov’s discrepancy principle by a Newton method. Interestingly, the algorithm produces not only a sequence of vectors which converges to a minimizer of the constrained problem but also a sequence of parameters which convergences to a regularization parameter so that the corresponding penalized problem has the same solution as our constrained one. We demonstrate the performance of various algorithms for different image restoration tasks both for images corrupted by Poisson noise and multiplicative Gamma noise. 1
Mathematical modeling of textures: Application to color image decomposition with a projected gradient algorithm
 JMIV
"... 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 contai ..."
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Cited by 10 (1 self)
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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.
1 NLSAR: a unified NonLocal framework for resolutionpreserving (Pol)(In)SAR denoising
"... performed to mitigate these fluctuations in homogeneous regions. Furthermore, the computation of the interferometric and polarimetric signatures of a radar scene requires estimating local covariance matrices from several pixels. Prior to their analysis, SAR images then often undergo processing steps ..."
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Cited by 5 (1 self)
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performed to mitigate these fluctuations in homogeneous regions. Furthermore, the computation of the interferometric and polarimetric signatures of a radar scene requires estimating local covariance matrices from several pixels. Prior to their analysis, SAR images then often undergo processing steps that degrade their resolution. Though a speckle reduction step and covariance estimation are unavoidable in many applications, special care must be taken to limit blurring of significant structures in SAR images. The simplest approach to speckle reduction and covariance estimation, spatial multilooking, computes a simple moving average with a (typically rectangular) window. Sufficient smoothing of homogeneous regions comes at the cost of a strong resolution loss. Several improvements to multilooking have been proposed
A Dictionary Learning Approach for Poisson Image Deblurring
, 2013
"... The restoration of images corrupted by blur and Poisson noise is a key issue in medical and biological image processing. While most existing methods are based on variational models, generally derived from a Maximum A Posteriori (MAP) formulation, recently sparse representations of images have shown ..."
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Cited by 5 (0 self)
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The restoration of images corrupted by blur and Poisson noise is a key issue in medical and biological image processing. While most existing methods are based on variational models, generally derived from a Maximum A Posteriori (MAP) formulation, recently sparse representations of images have shown to be efficient approaches for image recovery. Following this idea, we propose in this paper a model containing three terms: a patchbased sparse representation prior over a learned dictionary, the pixelbased total variation regularization term and a datafidelity term capturing the statistics of Poisson noise. The resulting optimization problem can be solved by an alternating minimization technique combined with variable splitting. Extensive experimental results suggest that in terms of visual quality, PSNR value and the method noise, the proposed algorithm outperforms stateoftheart methods.
Multiplicative noise cleaning via a variational method involving curvelet
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A convex variational model for restoring blurred images with multiplicative noise,” UCLA Camreport
, 2012
"... Abstract. In this paper, a new variational model for restoring blurred images with multiplicative noise is proposed. Based on the statistical property of the noise, a quadratic penalty function technique is utilized in order to obtain a strictly convex model under a mild condition, which guarantees ..."
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Abstract. In this paper, a new variational model for restoring blurred images with multiplicative noise is proposed. Based on the statistical property of the noise, a quadratic penalty function technique is utilized in order to obtain a strictly convex model under a mild condition, which guarantees the uniqueness of the solution and the stabilization of the algorithm. For solving the new convex variational model, a primaldual algorithm is proposed and its convergence is studied. The paper ends with a report on numerical tests for the simultaneous deblurring and denoising of images subject to multiplicative noise. A comparison with other methods is provided as well. Key words. Convexity, deblurring, multiplicative noise, primaldual algorithm, total variation regularization, variational model. AMS subject classifications. 52A41, 65K10, 65K15, 90C30, 90C47
The convex relaxation method on deconvolution model with multiplicative noise
 Commun. Comput. Phys
, 2013
"... Abstract. In this paper, we consider variational approaches to handle the multiplicative noise removal and deblurring problem. Based on rather reasonable physical blurringnoisy assumptions, we derive a new variational model for this issue. After the study of the basic properties, we propose to appr ..."
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Abstract. In this paper, we consider variational approaches to handle the multiplicative noise removal and deblurring problem. Based on rather reasonable physical blurringnoisy assumptions, we derive a new variational model for this issue. After the study of the basic properties, we propose to approximate it by a convex relaxation model which is a balance between the previous nonconvex model and a convex model. The relaxed model is solved by an alternating minimization approach. Numerical examples are presented to illustrate the effectiveness and efficiency of the proposed method.