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

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