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A multiscale method in surface processing is presented which carries over image processing methodology based on nonlinear diffusion equations to the fairing of noisy, textured, parametric surfaces. The aim is to smooth noisy, triangulated surfaces and accompanying noisy textures-as they are delivered by new scanning technology-while enhancing geometric and texture features. For an initial textured surface a fairing method is described which simultaneously processes the texture and the surface. Considering an appropriate coupling of the two smoothing processes one can take advantage of the frequently present strong correlation between edge features in the texture and on the surface edges. The method is based on an anisotropic curvature evolution of the surface itself and an anisotropic diffusion on the processed surface applied to the texture. Here, the involved diffusion tensors depends on a regularized shape operator of the evolving surface and on regularized texture gradients. A spatial finite element discretization on arbitrary unstructured triangular grids and a semi-implicit finite difference discretization in time are the building blocks of the corresponding numerical algorithm. A normal projection is applied to the discrete propagation velocity to avoid tangential drifting in the surface evolution. Different applications underline the efficiency and flexibility of the presented surface processing tool.