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.

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.

In this paper, we study the ability of the cooperation of two-color pixel classification schemes (Bayesian and K-means classification) with color watershed. Using color pixel classification alone does not sufficiently accurately extract color regions so we suggest to use a strategy based on three steps: simplification, classification, and color watershed. Color watershed is based on a new aggregation function using local and global criteria. The strategy is performed on microscopic images. Quantitative measures are used to evaluate the resulting segmentations according to a learning set of reference images.

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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.

The generalization of mathematical morphology to multivariate vector spaces is addressed in this paper. The proposed approach is fully unsupervised and consists in learning a complete lattice from an image as a nonlinear bijective mapping, interpreted in the form of a learned rank transformation together with an ordering of vectors. This unsupervised ordering of vectors relies on three steps: dictionary learning, manifold learning and out of sample extension. In addition to providing an efficient way to construct a vectorial ordering, the proposed approach can become a supervised ordering by the integration of pairwise constraints. The performance of the approach is illustrated with color image processing examples.

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.

This paper proposes a technical review of exemplar-based inpainting approaches with a particular focus on greedy methods. Several comparative and illustrative experiments are provided to deeply explore and enlighten these methods, and to have a better understanding on the state-of-the-art improvements of these approaches. From this analysis, three improvements over Criminisi et al. algorithm are then presented and detailed: 1) a tensor-based data term for a better selection of pixel candidates to fill in; 2) a fast patch lookup strategy to ensure a better global coherence of the reconstruction; and 3) a novel fast anisotropic spatial blending algorithm that reduces typical block artifacts using tensor models. Relevant comparisons with the state-of-the-art inpainting methods are provided that exhibit the effectiveness of our contributions.