Perturbation of the Eigenvectors of the Graph Laplacian: Application to Image Denoising

TL;DR

This paper investigates how noise affects the eigenvectors of the graph Laplacian in image patches and introduces a denoising algorithm leveraging the robustness of low-index eigenvectors to improve image quality.

Contribution

It provides a new understanding of noise influence on graph Laplacian eigenvectors and proposes an algorithm for image denoising based on these insights.

Findings

The low-index eigenvectors are highly robust to noise perturbations.

Patches from smooth image regions can be reconstructed with few eigenvectors.

The proposed denoising algorithm outperforms standard methods.

Abstract

The original contributions of this paper are twofold: a new understanding of the influence of noise on the eigenvectors of the graph Laplacian of a set of image patches, and an algorithm to estimate a denoised set of patches from a noisy image. The algorithm relies on the following two observations: (1) the low-index eigenvectors of the diffusion, or graph Laplacian, operators are very robust to random perturbations of the weights and random changes in the connections of the patch-graph; and (2) patches extracted from smooth regions of the image are organized along smooth low-dimensional structures in the patch-set, and therefore can be reconstructed with few eigenvectors. Experiments demonstrate that our denoising algorithm outperforms the denoising gold-standards.