Nonlocal Discrete Regularization on Weighted Graphs: A Framework for Image and Manifold Processing

Abstract

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.

PCA using graph total variation

“…Finally, our model can be solved efficiently, in time linear in the number of data samples, with a forward-backward based primal dual algorithm. We point out here that the G-TV framework has been used in previous works, such as [9], [10]. However, to the best of our knowledge it has never been used in the context of PCA.…”

Generalised Nonlocal Image Smoothing

“…The discretisations involve pixel differences that are weighted by a patch-based similarity between pixels as in [15]. Bougleux et al [9,33,10] designed a discrete graph regularisation framework that can be seen as a digital extension of the continuous framework [38] employing a -Dirichlet regulariser. The same discrete framework has been applied in image segmentation tasks [73].…”

“…Other definitions of the weighted gradient norm are possible using alternative weighted difference operators (see [41] and references therein). This regulariser has been used in [90,9,33,10] for regularisation on arbitrary graphs. In particular, the following energy functionals have been proposed in [10]:…”

Graph-based tools for microscopic cellular image segmentation

“…The reader can note that this isotropic regularization corresponds to the weighted discrete transcription of the regularization functional in the continuous case. The interest reader can refer to [13,21] for more details on the formulation and the connections with other formalisms. Moreover, in [22], we have extended this discrete isotropic regularization to a discrete anisotropic regularization framework for image and data processing.…”

“…In this work, we use our recently proposed discrete regularization framework based on weighted graph [13] to address the image segmentation problem. This framework is inspired by continuous regularization and data dependent function analysis methods.…”