Local and Nonlocal Discrete Regularization on Weighted Graphs for Image and Mesh Processing

Abstract

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.

Generalised Nonlocal Image Smoothing

“…Our discrete variational framework includes as special cases patch-based generalisations of M-smoothers and bilateral filtering, as well as the NL-means filter of Buades et al [15]. Other related approaches due to Kervrann et al [45], Bougleux et al [10] and Gilboa et al [36,37] could also be derived from our energy model by employing a different similarity measure and/or by redefining the spatial weight functions that we use as search windows. In this work we have mainly exploited the use of the weighted 2 norm to compute patch distances.…”

“…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]:…”

Morphological PDEs on Graphs for Image Processing on Surfaces and Point Clouds

“…This constitutes the basis of the framework of PdEs [35] that enables transposing PDEs on graphs. All of these definitions are borrowed from [6,9,36].…”

Efficient Algorithms for Image and High Dimensional Data Processing Using Eikonal Equation on Graphs

“…These methods have been applied for image and data processing [12]. With these approaches, the authors in [13] have proposed an adaptation of (3) on weighted graphs of arbitrary topology.…”