We propose a discrete variational approach for image smoothing consisting of nonlocal data and smoothness contraints that penalise general dissimilarity measures defined on image patches. One of such dissimilarity measures is the weighted 2 distance between patches. In such a case we derive an iterative neighbourhood filter that induces a new similarity measure in the photometric domain. It can be regarded as an extended patch similarity measure that evaluates not only the patch similarity of two chosen pixels, but also the similarity of their corresponding neighbours. This leads to a more robust smoothing process since the pixels selected for averaging are more coherent with the local image structure. The suggested approach includes two recently proposed filters as special cases: The NLmeans filter of Buades et al. and the NDS filter of Mrázek et al. In fact, the approach introduced here can be considered as a generalisation of the latter filter. We evaluate our method for the task of denoising greyscale and colour images degraded by Gaussian and impulse noise, demonstrating that it compares very well to other more sophisticated patch-based approaches.
Summary. Image simplification and smoothing is a very important basic ingredient of a lot of practical applications. In this paper we compare different numerical approaches to solve this image approximation task within a unifying variational approach presented in . For methods based on fixed point iterations we show the existence of fixed points. To speed up the convergence we also use two approaches involving Newton's method which is only applicable for convex penalisers. The running time in practice is studied with numerical examples in 1-D and 2-D.
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