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21
Restoration of Poissonian images using alternating direction optimization
 IEEE Trans. Image Process
, 2010
"... Abstract—Much research has been devoted to the problem of restoring Poissonian images, namely for medical and astronomical applications. However, the restoration of these images using stateoftheart regularizers (such as those based upon multiscale representations or total variation) is still an a ..."
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Abstract—Much research has been devoted to the problem of restoring Poissonian images, namely for medical and astronomical applications. However, the restoration of these images using stateoftheart regularizers (such as those based upon multiscale representations or total variation) is still an active research area, since the associated optimization problems are quite challenging. In this paper, we propose an approach to deconvolving Poissonian images, which is based upon an alternating direction optimization method. The standard regularization [or maximum a posteriori (MAP)] restoration criterion, which combines the Poisson loglikelihood with a (nonsmooth) convex regularizer (logprior), leads to hard optimization problems: the loglikelihood is nonquadratic and nonseparable, the regularizer is nonsmooth, and there is a nonnegativity constraint. Using standard convex analysis tools, we present sufficient conditions for existence and uniqueness of solutions of these optimization problems, for several types of regularizers: totalvariation, framebased analysis, and framebased synthesis. We attack these problems with an instance of the alternating direction method of multipliers (ADMM), which belongs to the family of augmented Lagrangian algorithms. We study sufficient conditions for convergence and show that these are satisfied, either under totalvariation or framebased (analysis and synthesis) regularization. The resulting algorithms are shown to outperform alternative stateoftheart methods, both in terms of speed and restoration accuracy. Index Terms—Alternating direction methods, augmented Lagrangian, convex optimization, image deconvolution, image restoration, Poisson images. I.
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
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
COMPRESSIVE SAMPLING WITH UNKNOWN BLURRING FUNCTION: APPLICATION TO PASSIVE MILLIMETERWAVE IMAGING
"... We propose a novel blind image deconvolution (BID) regularization framework for compressive passive millimeterwave (PMMW) imaging systems. The proposed framework is based on the variablesplitting optimization technique, which allows us to utilize existing compressive sensing reconstruction algorith ..."
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Cited by 3 (3 self)
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We propose a novel blind image deconvolution (BID) regularization framework for compressive passive millimeterwave (PMMW) imaging systems. The proposed framework is based on the variablesplitting optimization technique, which allows us to utilize existing compressive sensing reconstruction algorithms in compressive BID problems. In addition, a nonconvex lp quasinorm with 0 < p < 1 is employed as a regularization term for the image, while a simultaneous autoregressive (SAR) regularization term is utilized for the blur. Furthermore, the proposed framework is very general and it can be easily adapted to other stateoftheart BID approaches that utilize different image/blur regularization terms. Experimental results, obtained with simulations using a synthetic image and real PMMW images, show the advantage of the proposed approach compared to existing ones. Index Terms — Variablesplitting, inverse methods, compressive sensing, blind image deconvolution.
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
Gradient Histogram Estimation and Preservation for Texture Enhanced Image Denoising
, 2013
"... Natural image statistics plays an important role in image denoising, and various natural image priors, including gradient based, sparse representation based and nonlocal selfsimilarity based ones, have been widely studied and exploited for noise removal. In spite of the great success of many denois ..."
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Cited by 2 (0 self)
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Natural image statistics plays an important role in image denoising, and various natural image priors, including gradient based, sparse representation based and nonlocal selfsimilarity based ones, have been widely studied and exploited for noise removal. In spite of the great success of many denoising algorithms, they tend to smooth the fine scale image textures when removing noise, degrading the image visual quality. To address this problem, in this paper we propose a texture enhanced image denoising method by enforcing the gradient histogram of the denoised image to be close to a reference gradient histogram of the original image. Given the reference gradient histogram, a novel gradient histogram preservation (GHP) algorithm is developed to enhance the texture structures while removing noise. Two regionbased variants of GHP are proposed for the denoising of images consisting of regions with different textures. An algorithm is also developed to effectively estimate the reference gradient histogram from the noisy observation of the unknown image. Our experimental results demonstrate that the proposed GHP algorithm can well preserve the texture appearance in the denoised images, making them look more natural.
To cite this version:
, 2014
"... HAL is a multidisciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers. L’archive ouverte p ..."
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HAL is a multidisciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers. L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et a ̀ la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés. 1NLSAR: a unified NonLocal framework for resolutionpreserving (Pol)(In)SAR denoising CharlesAlban Deledalle, Loı̈c Denis, Florence Tupin, Andreas Reigber, and Marc Jäger
Gradient algorithms for . . .
, 2011
"... Partial overview of some techniques from computational optimization of possible relevance to sparse reconstruction. Illustrating the effectiveness of ℓ1 for sparsity. Other regularizers for other structures. ProxLinear methods Implementation for different regularizers ..."
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Partial overview of some techniques from computational optimization of possible relevance to sparse reconstruction. Illustrating the effectiveness of ℓ1 for sparsity. Other regularizers for other structures. ProxLinear methods Implementation for different regularizers
COMPONENTBASED RESTORATION OF SPECKLED IMAGES
"... Many coherent imaging modalities are often characterized by a multiplicative noise, known as speckle which often makes the interpretation of data difficult. In this paper, we present a speckle reduction algorithm based on separating the structure and texture components of SAR images. An iterative al ..."
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Many coherent imaging modalities are often characterized by a multiplicative noise, known as speckle which often makes the interpretation of data difficult. In this paper, we present a speckle reduction algorithm based on separating the structure and texture components of SAR images. An iterative algorithm based on surrogate functionals is presented that solves the component optimization formulation. Experiments indicate this proposed method performs favorably compared to stateoftheart despeckling methods. Index Terms — Synthetic Aperture Radar, speckle, multiplicative noise, image restoration.
Accelerated FirstOrder Stochastic Gradient Augmented Lagrangian
, 2011
"... Many applications need structured, approximate solutions of optimization formulations, rather than exact solutions. More Useful, More Credible Structured solutions are easier to understand. They correspond better to prior knowledge about the solution. They may be easier to use and actuate. Extract j ..."
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Many applications need structured, approximate solutions of optimization formulations, rather than exact solutions. More Useful, More Credible Structured solutions are easier to understand. They correspond better to prior knowledge about the solution. They may be easier to use and actuate. Extract just the essential meaning from the data set, not the less important effects. Less Data Needed Structured solution lies in lowerdimensional spaces ⇒ need to gather / sample less data to capture it. Choose good structure instead of “overfitting ” to a particular sample. The structural requirements have deep implications for how we formulate and solve these problems. Stephen Wright (UWMadison) Sparse Optimization SIAMOPT, May 2011 3 / 44ℓ1 and Sparsity A common type of desired structure is sparsity: We would like the approx solution x ∈ R n to have few nonzero components. A sparse formulation of “minx f (x) ” could be Find an approximate minimizer ¯x ∈ R n of f such that ‖x‖0 ≤ k, where ‖x‖0 denotes cardinality: the number of nonzeros in x. Too Hard! Use of ‖x‖1 has long been known to promote sparsity in x. Also, Can solve without discrete variables; It maintains convexity. Stephen Wright (UWMadison) Sparse Optimization SIAMOPT, May 2011 4 / 44Regularized Formulations with ℓ1