Viscosity Solutions of a Level-Set Method for AnisotropicGeometric Diffusion in Image Processing

摘要

We discuss the existence of viscosity solutions fora class of anisotropic level-set methods which can be seen asan extension of the mean-curvature motion with a nonlinearanisotropic diffusion tensor. In an earlier work (Mikula etal. in Comput. Vis. Sci. 6(4):197-209, 2004; Preusser andRumpf in SIAM J. Appl. Math. 62(5):1772-1793, 2002) wehave applied such methods for the denoising and enhance-ment of static images and image sequences. The models arecharacterized by the fact that—unlike the mean-curvaturemotion—they are capable of retaining important geometricstructures like edges and corners of the level-sets. The ar-ticle reviews the definition of the model and discusses itsgeometric behavior. The proof of the existence of viscositysolutions for these models is based on a fixed point argu-ment which utilizes a compactness property of the diffusiontensor. For the application to image processing suitable reg-ularizations of the diffusion tensor are presented for whichthe compactness assumptions of the existence proof hold.Finally, we consider the half relaxed limits of the solutionsof auxiliary problems to show the compactness of the so-lution operator and thus the existence of a solution to theoriginal problem.