A 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. With the increasing use of imaging in medicine and sciences it might be considered inevitable that researchers will try to apply such noise filtering schemes in quantitative analysis. In order to do this with confidence we need to develop an understanding of the noise removal process that goes beyond subjective appearance. The purpose of this paper is to develop and test a statistical basis of NLM, in order to try to understand the conditions required for its use. The theory is illustrated on synthetic data and real MR images of the brain.