  We first establish a law of large numbers and a convergence theorem in
distribution to show the rate of convergence of the non-local means filter for
removing Gaussian noise. We then introduce the notion of degree of similarity
to measure the role of similarity for the non-local means filter. Based on the
convergence theorems, we propose a patch-based weighted means filter for
removing impulse noise and its mixture with Gaussian noise by combining the
essential idea of the trilateral filter and that of the non-local means filter.
Our experiments show that our filter is competitive compared to recently
proposed methods.
