Abstract-We introduce a nonlocal discrete regularization framework on weighted graphs of the arbitrary topologies for image and manifold processing. The approach considers the problem as a variational one, which consists in minimizing a weighted sum of two energy terms: a regularization one that uses a discrete weighted p-Dirichlet energy, and an approximation one. This is the discrete analogue of recent continuous Euclidean nonlocal regularization functionals. The proposed formulation leads to a family of simple and fast nonlinear processing methods based on the weighted p-Laplace operator, parameterized by the degree p of regularity, the graph structure and the graph weight function. These discrete processing methods provide a graph-based version of recently proposed semi-local or nonlocal processing methods used in image and mesh processing, such as the bilateral filter, the TV digital filter or the nonlocal means filter. It works with equal ease on regular 2D-3D images, manifolds or any data. We illustrate the abilities of the approach by applying it to various types of images, meshes, manifolds and data represented as graphs.
International audienceIn this paper we introduce a new family of partial difference operators on graphs and study equations involving these operators. This family covers local variational $p$-Laplacian, $\infty$-Laplacian, nonlocal $p$-Laplacian and $\infty$-Laplacian, $p$-Laplacian with gradient terms, and gradient operators used in morphology based on the partial differential equation. We analyze a corresponding parabolic equation involving these operators which enables us to interpolate adaptively between $p$-Laplacian diffusion-based filtering and morphological filtering, i.e., erosion and dilation. Then, we consider the elliptic partial difference equation with its corresponding Dirichlet problem and we prove the existence and uniqueness of respective solutions. For $p=\infty$, we investigate the connection with Tug-of-War games. Finally, we demonstrate the adaptability of this new formulation for different tasks in image and point cloud processing, such as filtering, segmentation, clustering, and inpainting
In this paper we propose an adaptation of the Eikonal equation on weighted graphs, using the framework of Partial difference Equations, and with the motivation of extending this equation's applications to any discrete data that can be represented by graphs. This adaptation leads to a finite difference equation defined on weighted graphs and a new efficient algorithm for multiple labels simultaneous propagation on graphs, based on such equation. We will show that such approach enables the resolution of many applications in image and high dimensional data processing using a unique framework. Keywords Eikonal equation • weighted graph • non-local image processing • active contour • PdE • fast marching • high dimensional data This work was supported under a doctoral grant of the Conseil Régional de Basse-Normandie and the Coeur et Cancer association.
We propose a discrete regularization framework on weighted graphs of arbitrary topology, which unifies local and nonlocal processing of images, meshes, and more generally discrete data. The approach considers the problem as a variational one, which consists in minimizing a weighted sum of two energy terms: a regularization one that uses the discrete p-Dirichlet form, and an approximation one. The proposed model is parametrized by the degree p of regularity, by the graph structure and by the weight function. The minimization solution leads to a family of simple linear and nonlinear processing methods. In particular, this family includes the exact expression or the discrete version of several neighborhood filters, such as the bilateral and the nonlocal means filter. In the context of images, local and nonlocal regularizations, based on the total variation models, are the continuous analog of the proposed model. Indirectly and naturally, it provides a discrete extension of these regularization methods for any discrete data or functions.
Abstract. We propose a discrete regularization framework on weighted graphs of arbitrary topology, which unifies image and mesh filtering. The approach considers the problem as a variational one, which consists in minimizing a weighted sum of two energy terms: a regularization one that uses the discrete p-Laplace operator, and an approximation one. This formulation leads to a family of simple nonlinear filters, parameterized by the degree p of smoothness and by the graph weight function. Some of these filters provide a graph-based version of well-known filters used in image and mesh processing, such as the bilateral filter, the TV digital filter or the nonlocal mean filter.
Partial difference equations (PDEs) and variational methods for image processing on Euclidean domains spaces are very well established because they permit to solve a large range of real computer vision problems. With the recent advent of many 3D sensors, there is a growing interest in transposing and solving PDEs on surfaces and point clouds. In this paper, we propose a simple method to solve such PDEs using the framework of PDEs on graphs. This latter approach enables us to transcribe, for surfaces and point clouds, many models and algorithms designed for image processing. To illustrate our proposal, three problems are considered: (1) p -Laplacian restoration and inpainting; (2) PDEs mathematical morphology; and (3) active contours segmentation.
In this paper, we introduce a new class of nonlocal p-Laplacian operators that interpolate between non-local Laplacian and infinity Laplacian. These operators are discrete analogous of the game p-laplacian operators on Euclidean spaces, and involve discrete morphological gradient on graphs. We study the Dirichlet problem associated with the new p-Laplacian equation and prove existence and uniqueness of it's solution. We also consider non-local diffusion on graphs involving these operators. Finally, we propose to use these operators as a unified framework for solution of many inverse problems in image processing and machine learning. Index Terms-p-Laplacian, PDEs-based morphology on graphs, image processing, machine learning, Tug-of-war games.
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