Oracle inequalities and minimax rates for non-local means and related adaptive kernel-based methods

TL;DR

This paper provides a theoretical analysis of non-local means (NLM) for image noise removal, comparing its performance and error decay rates with classical methods, and validating findings through simulations.

Contribution

It offers the first comprehensive theoretical characterization of NLM's performance, including error rates and parameter choices, and compares it with other established denoising techniques.

Findings

NLM achieves favorable error decay rates for cartoon images.

Theoretical insights into patch size and parameter selection.

Validation of theoretical results through image simulations.

Abstract

This paper describes a novel theoretical characterization of the performance of non-local means (NLM) for noise removal. NLM has proven effective in a variety of empirical studies, but little is understood fundamentally about how it performs relative to classical methods based on wavelets or how various parameters (e.g., patch size) should be chosen. For cartoon images and images which may contain thin features and regular textures, the error decay rates of NLM are derived and compared with those of linear filtering, oracle estimators, variable-bandwidth kernel methods, Yaroslavsky's filter and wavelet thresholding estimators. The trade-off between global and local search for matching patches is examined, and the bias reduction associated with the local polynomial regression version of NLM is analyzed. The theoretical results are validated via simulations for 2D images corrupted by additive white Gaussian noise.