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From Sparse Solutions of Systems of Equations to Sparse Modeling of Signals and Images
, 2007
"... A fullrank matrix A ∈ IR n×m with n < m generates an underdetermined system of linear equations Ax = b having infinitely many solutions. Suppose we seek the sparsest solution, i.e., the one with the fewest nonzero entries: can it ever be unique? If so, when? As optimization of sparsity is combinato ..."
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Cited by 202 (31 self)
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A fullrank matrix A ∈ IR n×m with n < m generates an underdetermined system of linear equations Ax = b having infinitely many solutions. Suppose we seek the sparsest solution, i.e., the one with the fewest nonzero entries: can it ever be unique? If so, when? As optimization of sparsity is combinatorial in nature, are there efficient methods for finding the sparsest solution? These questions have been answered positively and constructively in recent years, exposing a wide variety of surprising phenomena; in particular, the existence of easilyverifiable conditions under which optimallysparse solutions can be found by concrete, effective computational methods. Such theoretical results inspire a bold perspective on some important practical problems in signal and image processing. Several wellknown signal and image processing problems can be cast as demanding solutions of undetermined systems of equations. Such problems have previously seemed, to many, intractable. There is considerable evidence that these problems often have sparse solutions. Hence, advances in finding sparse solutions to underdetermined systems energizes research on such signal and image processing problems – to striking effect. In this paper we review the theoretical results on sparse solutions of linear systems, empirical
Sparse representation for color image restoration
 the IEEE Trans. on Image Processing
, 2007
"... Sparse representations of signals have drawn considerable interest in recent years. The assumption that natural signals, such as images, admit a sparse decomposition over a redundant dictionary leads to efficient algorithms for handling such sources of data. In particular, the design of well adapted ..."
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Cited by 106 (27 self)
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Sparse representations of signals have drawn considerable interest in recent years. The assumption that natural signals, such as images, admit a sparse decomposition over a redundant dictionary leads to efficient algorithms for handling such sources of data. In particular, the design of well adapted dictionaries for images has been a major challenge. The KSVD has been recently proposed for this task [1], and shown to perform very well for various grayscale image processing tasks. In this paper we address the problem of learning dictionaries for color images and extend the KSVDbased grayscale image denoising algorithm that appears in [2]. This work puts forward ways for handling nonhomogeneous noise and missing information, paving the way to stateoftheart results in applications such as color image denoising, demosaicing, and inpainting, as demonstrated in this paper. EDICS Category: COLCOLR (Color processing) I.
Learning multiscale sparse representations for image and video restoration
, 2007
"... Abstract. This paper presents a framework for learning multiscale sparse representations of color images and video with overcomplete dictionaries. A singlescale KSVD algorithm was introduced in [1], formulating sparse dictionary learning for grayscale image representation as an optimization proble ..."
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Cited by 51 (17 self)
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Abstract. This paper presents a framework for learning multiscale sparse representations of color images and video with overcomplete dictionaries. A singlescale KSVD algorithm was introduced in [1], formulating sparse dictionary learning for grayscale image representation as an optimization problem, efficiently solved via Orthogonal Matching Pursuit (OMP) and Singular Value Decomposition (SVD). Following this work, we propose a multiscale learned representation, obtained by using an efficient quadtree decomposition of the learned dictionary, and overlapping image patches. The proposed framework provides an alternative to predefined dictionaries such as wavelets, and shown to lead to stateoftheart results in a number of image and video enhancement and restoration applications. This paper describes the proposed framework, and accompanies it by numerous examples demonstrating its strength. Key words. Image and video processing, sparsity, dictionary, multiscale representation, denoising, inpainting, interpolation, learning. AMS subject classifications. 49M27, 62H35
Orthogonal bandlet bases for geometric images approximation
 Com. Pure and Applied Mathematics
, 2006
"... This paper introduces orthogonal bandlet bases to approximate images having some geometrical regularity. These bandlet bases are computed by applying parameterized Alpert transform operators over an orthogonal wavelet basis. These bandletization operators depend upon a multiscale geometric flow that ..."
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Cited by 18 (8 self)
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This paper introduces orthogonal bandlet bases to approximate images having some geometrical regularity. These bandlet bases are computed by applying parameterized Alpert transform operators over an orthogonal wavelet basis. These bandletization operators depend upon a multiscale geometric flow that is adapted to the image, at each wavelet scale. This bandlet construction has a hierarchical structure over wavelet coefficients taking advantage of existing regularity among these coefficients. It is proved that C α images having singularities along C α curves are approximated in a best orthogonal bandlet basis with an optimal asymptotic error decay. Fast algorithms and compression applications are described. c ○ 2000 Wiley Periodicals, Inc. I
Locally parallel texture modeling
 SIAM Journal on Imaging Sciences
, 2011
"... Abstract. 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 weight ..."
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Cited by 3 (1 self)
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Abstract. 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.
Numerical issues when using wavelets
 In Encyclopedia of Complexity and Systems Science
, 2007
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Learning Hierarchical and Topographic Dictionaries with Structured Sparsity
"... Recent work in signal processing and statistics have focused on defining new regularization functions, which not only induce sparsity of the solution, but also take into account the structure of the problem. 1–7 We present in this paper a class of convex penalties introduced in the machine learning ..."
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Recent work in signal processing and statistics have focused on defining new regularization functions, which not only induce sparsity of the solution, but also take into account the structure of the problem. 1–7 We present in this paper a class of convex penalties introduced in the machine learning community, which take the form of a sum of ℓ2 and ℓ∞norms over groups of variables. They extend the classical groupsparsity regularization8–10 in the sense that the groups possibly overlap, allowing more flexibility in the group design. We review efficient optimization methods to deal with the corresponding inverse problems, 11–13 and their application to the problem of learning dictionaries of natural image patches: 14–18 On the one hand, dictionary learning has indeed proven effective for various signal processing tasks. 17, 19 On the other hand, structured sparsity provides a natural framework for modeling dependencies between dictionary elements. We thus consider a structured sparse regularization to learn dictionaries embedded in a particular structure, for instance a tree11 or a twodimensional grid. 20 In the latter case, the results we obtain are similar to the dictionaries produced by topographic independent component analysis. 21
MODELING OCCLUSION AND SCALING IN NATURAL IMAGES YANN GOUSSEAU ∗
"... Abstract. The dead leaves model, introduced by the Mathematical Morphology school, consists of the superposition of random closed sets (the objects), and enables to model the occlusion phenomena. When combined with specific size distributions for objects, one obtains random fields providing adequate ..."
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Abstract. The dead leaves model, introduced by the Mathematical Morphology school, consists of the superposition of random closed sets (the objects), and enables to model the occlusion phenomena. When combined with specific size distributions for objects, one obtains random fields providing adequate models for natural images. However, this framework imposes bounds on the sizes of objects. We consider the limits of these random fields when letting the cutoff sizes tend to zero and infinity. As a result we obtain a random field that contains homogeneous regions, satisfies scaling properties and is statistically relevant for modeling natural images. We then investigate the combined effect of these features on the regularity of images in the framework of Besov spaces.