Statistical interpretation of non-local means

Noise filtering is a common step in image processing, and is particularly effective in improving the subjective quality of images. A large number of techniques have been developed, many of which concentrate on the problem of removing noise without damaging small structures such as edges. One recent approach that demonstrates empirical merit is the non-local means (NLM) algorithm. However, in order to use noise filtering algorithms in quantitative or clinical image analysis tasks an understanding of their behaviour that goes beyond subjective appearance must be developed. The purpose of this study is to investigate the statistical basis of NLM in order to attempt to understand the conditions required for its use. The theory is illustrated on synthetic data and clinical magnetic resonance images of the brain.