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152
Inpainting and zooming using sparse representations
 The Computer Journal
"... Representing the image to be inpainted in an appropriate sparse representation dictionary, and combining elements from Bayesian statistics and modern harmonic analysis, we introduce an expectation maximization (EM) algorithm for image inpainting and interpolation. From a statistical point of view, t ..."
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Cited by 57 (9 self)
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Representing the image to be inpainted in an appropriate sparse representation dictionary, and combining elements from Bayesian statistics and modern harmonic analysis, we introduce an expectation maximization (EM) algorithm for image inpainting and interpolation. From a statistical point of view, the inpainting/interpolation can be viewed as an estimation problem with missing data. Toward this goal, we propose the idea of using the EM mechanism in a Bayesian framework, where a sparsity promoting prior penalty is imposed on the reconstructed coefficients. The EM framework gives a principled way to establish formally the idea that missing samples can be recovered/ interpolated based on sparse representations. We first introduce an easy and efficient sparserepresentationbased iterative algorithm for image inpainting. Additionally, we derive its theoretical convergence properties. Compared to its competitors, this algorithm allows a high degree of flexibility to recover different structural components in the image (piecewise smooth, curvilinear, texture, etc.). We also suggest some guidelines to automatically tune the regularization parameter.
Nonlocal Regularization of Inverse Problems
, 2008
"... This article proposes a new framework to regularize linear inverse problems using the total variation on nonlocal graphs. This nonlocal graph allows to adapt the penalization to the geometry of the underlying function to recover. A fast algorithm computes iteratively both the solution of the regul ..."
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Cited by 56 (3 self)
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This article proposes a new framework to regularize linear inverse problems using the total variation on nonlocal graphs. This nonlocal graph allows to adapt the penalization to the geometry of the underlying function to recover. A fast algorithm computes iteratively both the solution of the regularization process and the nonlocal graph adapted to this solution. We show numerical applications of this method to the resolution of image processing inverse problems such as inpainting, superresolution and compressive sampling.
Inpainting Surface Holes
 In Int. Conference on Image Processing
, 2003
"... An algorithm for fillingin surface holes is introduced in this paper. The basic idea is to represent the surface of interest in implicit form, and fillin the holes with a system of geometric partial differential equations derived from image inpainting algorithms. The framework and examples with sy ..."
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Cited by 51 (2 self)
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An algorithm for fillingin surface holes is introduced in this paper. The basic idea is to represent the surface of interest in implicit form, and fillin the holes with a system of geometric partial differential equations derived from image inpainting algorithms. The framework and examples with synthetic and real data are presented.
Total Variation Wavelet Inpainting
 J. Math. Imaging Vision
, 2006
"... We consider the problem of filling in missing or damaged wavelet coe#cients due to lossy image transmission or communication. The task is closely related to classical inpainting problems, but also remarkably di#ers in that the inpainting regions are in the wavelet domain. New challenges include that ..."
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Cited by 49 (4 self)
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We consider the problem of filling in missing or damaged wavelet coe#cients due to lossy image transmission or communication. The task is closely related to classical inpainting problems, but also remarkably di#ers in that the inpainting regions are in the wavelet domain. New challenges include that the resulting inpainting regions in the pixel domain are usually not well defined, as well as that degradation is often spatially inhomogeneous. Two novel variational models are proposed to meet such challenges, which combine the total variation (TV) minimization technique with wavelet representations. The associated EulerLagrange equations lead to nonlinear partial di#erential equations (PDE's) in the wavelet domain, and proper numerical algorithms and schemes are designed to handle their computation. The proposed models can have e#ective and automatic control over geometric features of the inpainted images, including the sharpness and curvature information of edges.
Noise removal using smoothed normals and surface fitting
 IEEE T. Image Process
"... Abstract—In this work, we use partial differential equation techniques to remove noise from digital images. The removal is done in two steps. We first use a totalvariation filter to smooth the normal vectors of the level curves of a noise image. After this, we try to find a surface to fit the smoot ..."
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Cited by 39 (11 self)
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Abstract—In this work, we use partial differential equation techniques to remove noise from digital images. The removal is done in two steps. We first use a totalvariation filter to smooth the normal vectors of the level curves of a noise image. After this, we try to find a surface to fit the smoothed normal vectors. For each of these two stages, the problem is reduced to a nonlinear partial differential equation. Finite difference schemes are used to solve these equations. A broad range of numerical examples are given in the paper. Index Terms—Anisotropic diffusion, image denoising, nonlinear partial differential equations (PDEs), normal processing. I.
Manifold models for signals and images
 COMPUTER VISION AND IMAGE UNDERSTANDING
, 2009
"... This article proposes a new class of models for natural signals and images. The set of patches extracted from the data to analyze is constrained to be close to a low dimensional manifold. This manifold structure is detailed for various ensembles suitable for natural signals, images and textures mode ..."
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Cited by 29 (3 self)
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This article proposes a new class of models for natural signals and images. The set of patches extracted from the data to analyze is constrained to be close to a low dimensional manifold. This manifold structure is detailed for various ensembles suitable for natural signals, images and textures modeling. These manifolds provide a lowdimensional parameterization of the local geometry of these datasets. These manifold models can be used to regularize inverse problems in signal and image processing. The restored signal is represented as a smooth curve or surface traced on the manifold that matches the forward measurements. A manifold pursuit algorithm computes iteratively a solution of the manifold regularization problem. Numerical simulations on inpainting and compressive sensing inversion show that manifolds models bring an improvement for the recovery of data with geometrical features. Key words: signal processing, image modeling, texture, manifold. PACS: code, code Capturing the complex geometry of signals and images is at the core of recent advances in sound and natural image processing. Edges and texture patterns create complex nonlocal interactions. This paper studies these geometries for several sounds, images and textures models. The set of local patches in the dataset is modeled using smooth manifolds. These local features trace a continuous curve (resp. surface) on the manifold, which is a prior that can be used to solve inverse problems.
A variational framework for nonlocal image inpainting
 PROC. OF EMMCVPR
, 2009
"... Nonlocal methods for image denoising and inpainting have gained considerable attention in recent years. This is in part due to their superior performance in textured images and regions, a known weakness of purely local methods. Local methods on the other hand have demonstrated to be very appropriat ..."
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Cited by 28 (3 self)
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Nonlocal methods for image denoising and inpainting have gained considerable attention in recent years. This is in part due to their superior performance in textured images and regions, a known weakness of purely local methods. Local methods on the other hand have demonstrated to be very appropriate for the recovering of geometric structure such as image edges. The synthesis of both types of methods is a trend in current research. Variational analysis in particular is an appropriate tool for a unified treatment of local and nonlocal methods. In this work we propose a general variational framework for the problem of nonlocal image inpainting, from which several previous inpainting schemes can be derived, in addition to leading to novel ones. We explicitly study some of these, relating them to previous work and showing results on synthetic and real images.
A Unifying Framework for Image Inpainting
, 2009
"... Inpainting is the art of modifying an image in a form that is not detectable by an ordinary observer. There are numerous and very different approaches to tackle the inpainting problem, but we point out that the most successful inpainting algorithms are based on one or two of the following three basi ..."
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Cited by 26 (0 self)
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Inpainting is the art of modifying an image in a form that is not detectable by an ordinary observer. There are numerous and very different approaches to tackle the inpainting problem, but we point out that the most successful inpainting algorithms are based on one or two of the following three basic techniques: copyandpaste texture synthesis, geometric PDE’s, and coherence among neighboring pixels. We combine these three building blocks in a unifying variational model, and provide a working algorithm for image inpainting trying to approximate the minimum of the proposed energy functional. Our experiments show that the combination of all three terms of the proposed energy works better than taking each term separately, and the results obtained are stateoftheart. Index Terms Image inpainting, variational models, texture synthesis, PDE’s.