Viscosity solutions of a level-set method for anisotropic geometric diffusion in image processing

We discuss the existence of viscosity solutions for a class of anisotropic level-set methods which can be seen as an extension of the mean-curvature motion with a nonlinear anisotropic diffusion tensor. In an earlier work (Mikula et al. in Comput. Vis. Sci. 6(4):197-209, [2004]; Preusser and Rumpf in SIAM J. Appl. Math. 62(5):1772-1793, [2002]) we have applied such methods for the denoising and enhancement of static images and image sequences. The models are characterized by the fact that-unlike the mean-curvature motion-they are capable of retaining important geometric structures like edges and corners of the level-sets. The article reviews the definition of the model and discusses its geometric behavior. The proof of the existence of viscosity solutions for these models is based on a fixed point argument which utilizes a compactness property of the diffusion tensor. For the application to image processing suitable regularizations of the diffusion tensor are presented for which the compactness assumptions of the existence proof hold. Finally, we consider the half relaxed limits of the solutions of auxiliary problems to show the compactness of the solution operator and thus the existence of a solution to the original problem.