A new similarity measure for nonlocal filtering in the presence of multiplicative noise

A new similarity measure and nonlocal filters for images corrupted by multiplicative noise are presented. The considered filters are generalizations of the nonlocal means filter of Buades et al., which is known to be well suited for removing additive Gaussian noise. To adapt this filter to different noise models, the involved patch comparison has first of all to be performed by a suitable noise dependent similarity measure. For this purpose, a recently proposed probabilistic measure for general noise models by Deledalle et al. is studied. This measure is analyzed in the context of conditional density functions and its properties are examined for data corrupted by additive and multiplicative noise. Since it turns out to have unfavorable properties for multiplicative noise, a new similarity measure is deduced consisting of a probability density function specially chosen for this type of noise. The properties of this new measure are studied theoretically as well as by numerical experiments. To finally obtain nonlocal filters, a weighted maximum likelihood estimation framework is applied, which also incorporates the noise statistics. Moreover, the weights occurring in these filters are defined using the new similarity measure and different adaptations are proposed to further improve the results. Finally, restoration results for images corrupted by multiplicative Gamma and Rayleigh noise are presented to demonstrate the very good performance of these nonlocal filters.