Graph Laplacian Regularization for Image Denoising: Analysis in the Continuous Domain

TL;DR

This paper analyzes graph Laplacian regularization for image denoising by connecting discrete graph models to continuous Riemannian manifolds, deriving an optimal metric, and interpreting it as anisotropic diffusion, leading to a competitive denoising algorithm.

Contribution

It provides a theoretical analysis linking graph Laplacian regularization to continuous domains, derives an optimal metric for denoising, and interprets the regularizer as anisotropic diffusion, with practical algorithm development.

Findings

The graph Laplacian converges to a continuous functional in the limit.

An optimal metric space for denoising is derived based on patch self-similarity.

The proposed method outperforms state-of-the-art algorithms on piecewise smooth images.

Abstract

Inverse imaging problems are inherently under-determined, and hence it is important to employ appropriate image priors for regularization. One recent popular prior---the graph Laplacian regularizer---assumes that the target pixel patch is smooth with respect to an appropriately chosen graph. However, the mechanisms and implications of imposing the graph Laplacian regularizer on the original inverse problem are not well understood. To address this problem, in this paper we interpret neighborhood graphs of pixel patches as discrete counterparts of Riemannian manifolds and perform analysis in the continuous domain, providing insights into several fundamental aspects of graph Laplacian regularization for image denoising. Specifically, we first show the convergence of the graph Laplacian regularizer to a continuous-domain functional, integrating a norm measured in a locally adaptive metric space. Focusing on image denoising, we derive an optimal metric space assuming non-local self-similarity of pixel patches, leading to an optimal graph Laplacian regularizer for denoising in the discrete domain. We then interpret graph Laplacian regularization as an anisotropic diffusion scheme to explain its behavior during iterations, e.g., its tendency to promote piecewise smooth signals under certain settings. To verify our analysis, an iterative image denoising algorithm is developed. Experimental results show that our algorithm performs competitively with state-of-the-art denoising methods such as BM3D for natural images, and outperforms them significantly for piecewise smooth images.