Oracle Inequalities and Minimax Rates for Nonlocal Means and Related Adaptive Kernel-Based Methods

This paper describes a novel theoretical characterization of the performance of nonlocal means (NLM) for noise removal. NLM has proved 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 its parameters 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, 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 two-dimensional images corrupted by additive white Gaussian noise.