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Bregmanized Nonlocal Regularization for Deconvolution and Sparse Reconstruction ∗
"... We propose two algorithms based on Bregman iteration and operator splitting technique for nonlocal TV regularization problems. The convergence of the algorithms is analyzed and applications to deconvolution and sparse reconstruction are presented. 1 ..."
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Cited by 88 (9 self)
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We propose two algorithms based on Bregman iteration and operator splitting technique for nonlocal TV regularization problems. The convergence of the algorithms is analyzed and applications to deconvolution and sparse reconstruction are presented. 1
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.
Higher order learning with graphs
 In ICML ’06: Proceedings of the 23rd international conference on Machine learning
, 2006
"... Recently there has been considerable interest in learning with higher order relations (i.e., threeway or higher) in the unsupervised and semisupervised settings. Hypergraphs and tensors have been proposed as the natural way of representing these relations and their corresponding algebra as the nat ..."
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Cited by 42 (0 self)
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Recently there has been considerable interest in learning with higher order relations (i.e., threeway or higher) in the unsupervised and semisupervised settings. Hypergraphs and tensors have been proposed as the natural way of representing these relations and their corresponding algebra as the natural tools for operating on them. In this paper we argue that hypergraphs are not a natural representation for higher order relations, indeed pairwise as well as higher order relations can be handled using graphs. We show that various formulations of the semisupervised and the unsupervised learning problem on hypergraphs result in the same graph theoretic problem and can be analyzed using existing tools. 1.
Regularization on graphs with functionadapted diffusion process
, 2006
"... Harmonic analysis and diffusion on discrete data has been shown to lead to stateoftheart algorithms for machine learning tasks, especially in the context of semisupervised and transductive learning. The success of these algorithms rests on the assumption that the function(s) to be studied (learn ..."
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Cited by 38 (8 self)
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Harmonic analysis and diffusion on discrete data has been shown to lead to stateoftheart algorithms for machine learning tasks, especially in the context of semisupervised and transductive learning. The success of these algorithms rests on the assumption that the function(s) to be studied (learned, interpolated, etc.) are smooth with respect to the geometry of the data. In this paper we present a method for modifying the given geometry so the function(s) to be studied are smoother with respect to the modified geometry, and thus more amenable to treatment using harmonic analysis methods. Among the many possible applications, we consider the problems of image denoising and transductive classification. In both settings, our approach improves on standard diffusion based methods.
A VARIATIONAL FRAMEWORK FOR EXEMPLARBASED IMAGE INPAINTING
, 2010
"... 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, a known weakness of purely local methods. Local methods on the other hand have demonstrated to be very appropriate for the r ..."
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Cited by 24 (2 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, a known weakness of purely local methods. Local methods on the other hand have demonstrated to be very appropriate for the recovering of geometric structures 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 nonlocal image inpainting, from which important and representative 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.
Geometry of probability spaces
 Constr. Approx
"... Partial differential equations and the Laplacian operator on domains in Euclidean spaces have played a central role in understanding natural phenomena. However this avenue has been limited in many areas where calculus is obstructed as in singular spaces, and function spaces of functions on a space X ..."
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Cited by 21 (1 self)
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Partial differential equations and the Laplacian operator on domains in Euclidean spaces have played a central role in understanding natural phenomena. However this avenue has been limited in many areas where calculus is obstructed as in singular spaces, and function spaces of functions on a space X where X itself is a function space. Examples of the last
IMAGE PROCESSING WITH NONLOCAL SPECTRAL BASES
, 2008
"... This article studies regularization schemes that are defined using a lifting of the image pixels in a high dimensional space. For some specific classes of geometric images, this discrete set of points is sampled along a low dimensional smooth manifold. The construction of differential operators on ..."
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Cited by 16 (1 self)
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This article studies regularization schemes that are defined using a lifting of the image pixels in a high dimensional space. For some specific classes of geometric images, this discrete set of points is sampled along a low dimensional smooth manifold. The construction of differential operators on this lifted space allows one to compute PDE flows and perform variational optimizations. All these schemes lead to regularizations that exploit the manifold structure of the lifted image. Depending on the specific definition of the lifting, one recovers several wellknown semilocal and nonlocal denoising algorithms that can be interpreted as local estimators over a semilocal or a nonlocal manifold. This framework also allows one to define thresholding operators in adapted orthogonal bases. These bases are eigenvectors of the discrete Laplacian on a manifold adapted to the geometry of the image. Numerical results compare the efficiency of PDE flows, energy minimizations and thresholdings in the semilocal and nonlocal settings. The superiority of the nonlocal computations is studied through the performance of nonlinear approximation in orthogonal bases.
An MBO scheme on graphs for segmentation and image processing., UCLA
, 2012
"... Abstract. In this paper we present a computationally efficient algorithm utilizing a fully or semi nonlocal graph Laplacian for solving a wide range of learning problems in data clustering and image processing. Combining ideas from L1 compressive sensing, image processing and graph methods, the diff ..."
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Cited by 10 (5 self)
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Abstract. In this paper we present a computationally efficient algorithm utilizing a fully or semi nonlocal graph Laplacian for solving a wide range of learning problems in data clustering and image processing. Combining ideas from L1 compressive sensing, image processing and graph methods, the diffuse interface model based on the GinzburgLandau functional was recently introduced to the graph community for solving problems in data classification. Here, we propose an adaptation of the classic numerical MerrimanBenceOsher (MBO) scheme for graphbased methods and also make use of fast numerical solvers for finding eigenvalues and eigenvectors of the graph Laplacian. We present various computational examples to demonstrate the performance of our model, which is successful on images with texture and repetitive structure due to its nonlocal nature. Key words. Image processing, Nyström extension, GinzburgLandau functional, MBO scheme 1. Introduction. This
UNIFYING LOCAL AND NONLOCAL PROCESSING WITH PARTIAL DIFFERENCE OPERATORS ON WEIGHTED GRAPHS
 INTERNATIONAL WORKSHOP ON LOCAL AND NONLOCAL APPROXIMATION IN IMAGE PROCESSING, SUISSE
, 2008
"... In this paper, local and nonlocal image processing are unified, within the same framework, by defining discrete derivatives on weighted graphs. These discrete derivatives allow to transcribe continuous partial differential equations and energy functionals to partial difference equations and discrete ..."
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Cited by 8 (4 self)
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In this paper, local and nonlocal image processing are unified, within the same framework, by defining discrete derivatives on weighted graphs. These discrete derivatives allow to transcribe continuous partial differential equations and energy functionals to partial difference equations and discrete functionals over weighted graphs. With this methodology, we consider two gradientbased problems: regularization and mathematical morphology. The gradientbased regularization framework allows to connect isotropic and anisotropic pLaplacians diffusions, as well as neighborhood filtering. Within the same discrete framework, we present morphological operations that allow to recover and to extend wellknown PDEsbased and algebraic operations to nonlocal configurations. Finally, experimental results show the ability and the flexibility of the proposed methodology in the context of image and unorganized data set processing.
Phase transition in the family of presistances
"... We study the family of presistances on graphs for p 1. This family generalizes the standard resistance distance. We prove that for any fixed graph, for p =1 the presistance coincides with the shortest path distance, for p =2it coincides with the standard resistance distance, and for p!1it converge ..."
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Cited by 7 (0 self)
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We study the family of presistances on graphs for p 1. This family generalizes the standard resistance distance. We prove that for any fixed graph, for p =1 the presistance coincides with the shortest path distance, for p =2it coincides with the standard resistance distance, and for p!1it converges to the inverse of the minimal stcut in the graph. Secondly, we consider the special case of random geometric graphs (such as knearest neighbor graphs) when the number n of vertices in the graph tends to infinity. We prove that an interesting phase transition takes place. There exist two critical thresholds p ⇤ and p ⇤ ⇤ such that if p<p ⇤ , then the presistance depends on meaningful global properties of the graph, whereas if p>p ⇤ ⇤ , it only depends on trivial local quantities and does not convey any useful information. We can explicitly compute the critical values: p ⇤ = 1 + 1/(d 1) and p ⇤ ⇤ = 1 + 1/(d 2) where d is the dimension of the underlying space (we believe that the fact that there is a small gap between p ⇤ and p ⇤ ⇤ is an artifact of our proofs). We also relate our findings to Laplacian regularization and suggest to use qLaplacians as regularizers, where q satisfies 1/p ⇤ +1/q =1. 1