scholarly journals An obstacle problem arising in large exponent limit of power mean curvature flow equation

2019 ◽  
Vol 372 (3) ◽  
pp. 2103-2141 ◽  
Author(s):  
Qing Liu ◽  
Naoki Yamada
Keyword(s):  
Mean Curvature ◽  
Obstacle Problem ◽  
Mean Curvature Flow ◽  
Curvature Flow ◽  
Flow Equation ◽  
Power Mean ◽  
Large Exponent

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

2018 ◽  
Vol 72 (1) ◽  
pp. 123-140
Author(s):  
Matúš Tibenský ◽  
Angela Handlovičová
Keyword(s):  
Image Segmentation ◽  
Finite Volume ◽  
Level Set ◽  
Mean Curvature ◽  
Numerical Scheme ◽  
Mean Curvature Flow ◽  
Curvature Flow ◽  
Flow Equation ◽  
Finite Volume Scheme ◽  
Volume Scheme

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.

scholarly journals Viscosity solutions for the crystalline mean curvature flow with a nonuniform driving force term

2020 ◽  
Vol 1 (6) ◽  
Author(s):  
Yoshikazu Giga ◽  
Norbert Požár
Keyword(s):  
Level Set ◽  
Mean Curvature ◽  
Driving Force ◽  
Mean Curvature Flow ◽  
Approximate Solutions ◽  
Curvature Flow ◽  
Flow Equation ◽  
Force Term ◽  
Existence Of A Solution ◽  
Unique Existence

Abstract A general purely crystalline mean curvature flow equation with a nonuniform driving force term is considered. The unique existence of a level set flow is established when the driving force term is continuous and spatially Lipschitz uniformly in time. By introducing a suitable notion of a solution a comparison principle of continuous solutions is established for equations including the level set equations. An existence of a solution is obtained by stability and approximation by smoother problems. A necessary equi-continuity of approximate solutions is established. It should be noted that the value of crystalline curvature may depend not only on the geometry of evolving surfaces but also on the driving force if it is spatially inhomogeneous.

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