Abstract. Motion of curves governed by geometric evolution laws, such as mean curvature flow and surface diffusion, is the basis for many algorithms in image processing. If the images to be processed are defined on nonplanar surfaces, the geometric evolution laws have to be restricted to the surface and turn into geodesic evolution laws. In this paper we describe efficient algorithms for geodesic mean curvature flow and geodesic surface diffusion within a level-set approach. Thereby we compare approaches with an explicit representation of the surface by a triangulated surface mesh and an implicit surface representation as the zero-level surface of a level-set function. As an application we present the numerical treatment of the classical model of Rudin, Osher, and Fatemi to denoise images on surfaces.Key words. image denoising, PDEs on surfaces, higher order equations, geodesic evolution laws, level-set method, finite elements, adaptivity
AMS subject classifications. 35K55, 53C44, 53D25DOI. 10.1137/0706996401. Introduction. Planar motion of curves is the basis for many image processing algorithms. Examples include algorithms for image denoising, image restoration, and image decomposition; see, e.g., [23,29] and [35,20,30,6] and the references therein for a review. The underlying geometric evolution laws in many of these models are typically of second order and are versions of mean curvature flow. Various numerical approaches have been developed for these equations and today are widely used in image processing for planar images. If images defined on nonplanar surfaces have to be processed, the developed evolution laws for planar images can easily be modified by replacing the operators by their geometric counterparts, e.g., the Laplacian by the surface Laplacian. Instead of planar motion of curves we now have to deal with the evolution of curves which are restricted to surfaces. The problem of mean curvature flow, for example, becomes a geodesic mean curvature flow problem. Analytical results and computational algorithms for the equations on surfaces require more care, which is due to the additional nonlinearity of the surface operators.Different approaches have been developed to deal with the motion of curves in image processing on nonplanar surfaces. One requires the representation of the surface by a surface mesh. For example, [10,13,27,15] use this approach to solve (an)isotropic geometric diffusion