Convergence of the Numerical Scheme for Regularised Riemannian Mean Curvature Flow Equation

Abstract

The aim of the paper is to study problem of image segmentation and missing boundaries completion introduced in [Mikula, K.—Sarti, A.––Sgallarri, A.: Co-volume method for Riemannian mean curvature flow in subjective surfaces multiscale segmentation, Comput. Vis. Sci. 9 (2006), 23–31], [Mikula, K.—Sarti, A.—Sgallari, F.: Co-volume level set method in subjective surface based medical image segmentation, in: Handbook of Medical Image Analysis: Segmentation and Registration Models (J. Suri et al., eds.), Springer, New York, 583–626, 2005], [Mikula, K.—Ramarosy, N.: Semi-implicit finite volume scheme for solving nonlinear diffusion equations in image processing, Numer. Math. 89 (2001), 561–590] and [Tibenský, M.: VyužitieMetód Založených na Level Set Rovnici v Spracovaní Obrazu, Faculty of mathematics, physics and informatics, Comenius University, Bratislava, 2016]. We generalize approach presented in [Eymard, R.—Handlovičová, A.—Mikula, K.: Study of a finite volume scheme for regularised mean curvature flow level set equation, IMA J. Numer. Anal. 31 (2011), 813–846] and apply it in the field of image segmentation. The so called regularised Riemannian mean curvature flow equation is presented and the construction of the numerical scheme based on the finite volume method approach is explained. The principle of the level set, for the first time given in [Osher, S.—Sethian, J. A.: Fronts propagating with curvature-dependent speed: Algorithms based on Hamilton-Jacobi formulations, J. Comput. Phys. 79 (1988), 12–49] is used. Based on the ideas from [Eymard, R.—Handlovičová, A.– –Mikula, K.: Study of a finite volume scheme for regularised mean curvature flow level set equation, IMA J. Numer. Anal. 31 (2011), 813–846] we prove the stability estimates on the numerical solution and the uniqueness of the numerical solution. In the last section, there is a proof of the convergence of the numerical scheme to the weak solution of the regularised Riemannian mean curvature flow equation and the proof of the convergence of the approximation of the numerical gradient is mentioned as well.