Universität Des Saarlandes Fachrichtung 6.1 – Mathematik Generalised Nonlocal Image Smoothing Generalised Nonlocal Image Smoothing

We propose a discrete variational approach for image smoothing consisting of nonlocal data and smoothness constraints that penalise general dissimilarity measures defined on image patches. One of such dissimilarity measures is the weighted L2 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. By slightly modifying the way the similarities are computed we obtain two related filters: The NL-means filter of Buades et al. (SIAM Multiscale Model. Simul. 4(2):490–530, 2005b) and the NDS filter of Mrázek et al. (Geometric Properties for Incomplete Data, Computational Imaging and Vision, vol. 31, pp. 335–352, Springer, Dordrecht, 2006). In fact, the proposed approach can be considered as a generalisation of the latter filter to the space of patches. We also provide novel insights into relations of the NDS filter with diffusion/regularisation methods as well as with some recently proposed graph regularisation techniques. We evaluate our method for the task of denoising greyscale and colour images degraded with Gaussian and salt-and-pepper noise, demonstrating that it compares very well to other more sophisticated approaches.