A convolution-thresholding approximation of generalized curvature flows

SIAM Journal on Numerical Analysis

Volumn 42, Issue 6, 2005, Pages 2652-2670

  Grzhibovskis, Richards ^a   Heintz, Alexei ^a  

^a CHALMERS UNIVERSITY OF TECHNOLOGY   (Sweden)

Author keywords

Convolution thresholding scheme; Generalized curvature flow; Level set equation; Viscosity solution

Indexed keywords

CONVERGENCE OF NUMERICAL METHODS; CONVOLUTION; FUNCTIONS; VISCOSITY;

CONVOLUTION-THRESHOLD SCHEME; GENERALIZED CURVATURE FLOW; LEVEL-SET EQUATION; VISCOSITY SOLUTION;

APPROXIMATION THEORY;

EID: 29144495939     PISSN: 00361429     EISSN: None     Source Type: Journal    
DOI: 10.1137/S0036142903431316     Document Type: Article

References (31)

1. L. ALVAREZ, F. GUICHARD, P.-L. LIONS, AND J.-M. MOREL, Axioms and fundamental equations of image processing, Arch. Rational Mech. Anal., 123 (1993), pp. 199-257.
2. 3, Comm. Partial Differential Equations, 20 (1995), pp. 1937-1958.
3. G. BARLES AND C. GEORGELIN, A simple proof of convergence for an approximation scheme for computing motions by mean curvature, SIAM J. Numer. Anal., 32 (1995), pp. 484-500.
4. G. BARLES, H. M. SONER, AND P. E. SOUGANIDIS, Front propagation and phase field theory, SIAM J. Control Optim., 31 (1993), pp. 439-469.
5. G. BARLES AND P. E. SOUGANIDIS, Convergence of approximation schemes for fully nonlinear second order equations, Asymptotic Anal., 4 (1991), pp. 271-283.
6. G. BEYLKIN, On the fast Fourier transform of functions with singularities, Appl. Comput. Harmon. Anal., 2 (1995), pp. 363-381.
7. K. A. BRAKKE, The Motion of a Surface by Its Mean Curvature, Princeton University Press, Princeton, NJ, 1978.
8. L. BRONSARD AND R. V. KOHN, Motion by mean curvature as the singular limit of Ginzburg-Landau dynamics, J. Differential Equations, 90 (1991), pp. 211-237.
9. F. CAO, Partial differential equations and mathematical morphology, J. Math. Pures Appl. (9), 77 (1998), pp. 909-941.
10. F. CATTÉ, F. DIBOS, AND G. KOEPFLER, A morphological scheme for mean curvature motion and applications to anisotropic diffusion and motion of level sets, SIAM J. Numer. Anal., 32 (1995), pp. 1895-1909.
11. Y. G. CHEN, Y. GIGA, AND S. GOTO, Uniqueness and existence of viscosity solutions of generalized mean curvature flow equations, J. Differential Geom., 33 (1991), pp. 749-786.
12. M. G. CRANDALL, H. ISHII, AND P.-L. LIONS, User's guide to viscosity solutions of second order partial differential equations, Bull. AMS (N.S.), 27 (1992), pp. 1-67.
13. M. G. CRANDALL AND P.-L. LIONS, Convergent difference schemes for nonlinear parabolic equations and mean curvature motion, Numer. Math., 75 (1996), pp. 17-41.
14. E. DE GIORGI, Conjectures on limits of some quasilinear parabolic equations and flow by mean curvature, in Partial Differential Equations and Related Subjects (Trento, 1990), Longman Scientific and Technical, Harlow, 1992, pp. 85-95.
15. L. C. EVANS, Convergence of an algorithm for mean curvature motion, Indiana Univ. Math. J., 42 (1993), pp. 533-557.
16. L. C. EVANS, H. M. SONER, AND P. E. SOUGANIDIS, Phase transitions and generalized motion by mean curvature, Comm. Pure Appl. Math., 45 (1992), pp. 1097-1123.
17. L. C. EVANS AND J. SPRUCK, Motion of level sets by mean curvature. I, J. Differential Geom., 33 (1991), pp. 635-681.
18. F. GUICHARD AND J.-M. MOREL, Partial differential equations and image iterative filtering, in The State of the Art in Numerical Analysis (York, 1996), Oxford University Press, New York, 1997, pp. 525-562.
19. T. ILMANEN, Convergence of the Allen-Cahn equation to Brakke's motion by mean curvature, J. Differential Geom., 38 (1993), pp. 417-461.
20. H. ISHII, A generalization of the Bence, Merriman and Osher algorithm for motion by mean curvature, in Curvature Flows and Related Topics (Levico, 1994), Gakkötosho, Tokyo, 1995, pp. 111-127.
21. H. ISHII, G. E. PIRES, AND P. E. SOUGANIDIS, Threshold dynamics type approximation schemes for propagating fronts, J. Math. Soc. Japan, 51 (1999), pp. 267-308.
22. H. ISHII AND P. SOUGANIDIS, Generalized motion of noncompact hypersurfaces with velocity having arbitrary growth on the curvature tensor, Tohoku Math. J. (2), 47 (1995), pp. 227-250.
23. F. LEONI, Convergence of an approximation scheme for curvature-dependent motions of sets, SIAM J. Numer. Anal., 39 (2001), pp. 1115-1131.
24. W. LORENSEN AND H. CLINE, Marching cubes: A high resolution 3d surface construction algorithm, ACM Comput. Graphics, 21 (1987), pp. 163-170.
25. B. MERRIMAN, J. K. BENCE, AND S. J. OSHER, Motion of multiple junctions: A level set approach, J. Comput. Phys., 112 (1994), pp. 334-363.
26. T. OHTA, D. JASNOW, AND K. KAWASAKI, Universal scaling in the motion of random interfaces, Phys. Rev. Lett., 47 (1982), pp. 1223-1226.
27. S. OSHER AND J. A. SETHIAN, Fronts propagating with curvature-dependent speed: Algorithms based on Hamilton-Jacobi formulations, J. Comput. Phys., 79 (1988), pp. 12-49.
28. D. PASQUIGNON, Approximation of viscosity solution by morphological filters, ESAIM Control Optim. Calc. Var., 4 (1999), pp. 335-359 (electronic).
29. J. RUBINSTEIN, P. STERNBERG, AND J. B. KELLER, Fast reaction, slow diffusion, and curve shortening, SIAM J. Appl. Math., 49 (1989), pp. 116-133.
30. S. J. RUUTH, Efficient algorithms for diffusion-generated motion by mean curvature, J. Comput. Phys., 144 (1998), pp. 603-625.
31. J. A. SETHIAN, Level Set Methods and Fast Marching Methods, 2nd ed., Cambridge University Press, Cambridge, UK, 1999.