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Image segmentation, level set, regularised Riemannian mean curvature flow equation, finite volume method, approximation of the nonlinear smoothing term
Purpose of the paper is to study nonlinear smoothing term initiated in [3], [4], [6] and [7] for problems of image segmentation and missing boundaries completion. The generalization of approach presented in [1] is proposed and applied in the field of image segmentation. So called regularised Riemannian mean curvature flow equation is studied 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 [2], is used. We mention two different approaches for the approximation of the nonlinear smoothing term in the equation and known theoretical results for both of them. We provide the numerical tests for both schemes. It the last section we discuss obtained results and propose possibilities for the future research.
[1] Eymard, R., Handlovičová, A., Mikula, K.: Study of a finite volume scheme for regularised mean curvature flow level set equation. IMA J. on Numerical Analysis, Vol. 31, 813-846, 2011. DOI 10.1093/imanum/drq025 | MR 2832781
[2] Osher, S., A., J. Sethian: Fronts propagating with curvature-dependent speed: Algorithms basedon Hamilton-Jacobi formulations. J. Comput. Phys., 79(1):12-49, 1988. MR 0965860
[3] Mikula, K., Sarti, A., Sgallarri, A.: Co-volume method for Riemannian mean curvature flow in subjective surfaces multiscale segmentation. Computing and Visualization in Science, Vol. 9, No. 1, 23-31, 2006. DOI 10.1007/s00791-006-0014-0 | MR 2214835
[4] Mikula, K., Sarti, A., Sgallari, F.: Co-volume level set method in subjective surface based medicalimage segmentation. in: Handbook of Medical Image Analysis: Segmentation and Registration Models (J.Suri et al., Eds.), Springer, New York, 583-626, 2005.
[5] Handlovičová, A., Tibenský, M.: Convergence of the numerical scheme for regularised Riemannian mean curvature flow equation. submitted to Tatra Mountains Mathematical Publications, 2017. MR 3939443
[6] Mikula, K., Ramarosy, N.: Semi-implicit finite volume scheme for solving nonlinear diffusion equations in image processing. Numerische Mathematik 89, No. 3, 561-590, 2001. DOI 10.1007/PL00005479 | MR 1864431
[7] Tibenský, M.: Využitie metód založených na level set rovnici v spracovaní obrazu. Faculty of Mathematics, Physics and Informatics, Comenius University, 2016.
[8] Droniou, J., Nataraj, N.: Improved $L^2$ estimate for gradient schemes, and super-convergence of the TPFA finite volume scheme. IMA Journal of Numerical Analysis 2017, 2016. MR 3829161
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