A direct method for solving an anisotropic mean curvature flow of plane curves with an external force

Mathematical Methods in the Applied Sciences

Volumn 27, Issue 13, 2004, Pages 1545-1565

  Mikula, Karol ^a   Ševčovič, Daniel ^b  

^a SLOVAK UNIVERSITY OF TECHNOLOGY   (Slovakia)

^b Physics and Informatics   (Slovakia)

Author keywords

Anisotropy; External force; Image segmentation; Interface; Lagrangian approach; Mean curvature flow; Semi implicit scheme

Indexed keywords

ANISOTROPY; COMPUTATIONAL COMPLEXITY; DIFFERENTIAL EQUATIONS; IMAGE SEGMENTATION; PARAMETER ESTIMATION; VECTORS; VISUALIZATION;

EXTERNAL FORCE; LAGRANGIAN APPROACH; MEAN CURVATURE FLOW; SEMI-IMPLICIT APPROACH;

PROBLEM SOLVING;

EID: 4043136634     PISSN: 01704214     EISSN: None     Source Type: Journal    
DOI: 10.1002/mma.514     Document Type: Article

References (32)

1. Alvarez L, Guichard F, Lions PL, Morel JM. Axioms and fundamental equations of image processing. Archives for Rational Mechanics and Analysis 1993; 123:200-257.
2. Sapiro G, Tannenbaum A. On affine plane curve evolution. Journal of Functional Analysis 1994; 119:79-120.
3. Caselles V, Kimmel R, Sapiro G. Geodesic active contours. International Journal of Computer Vision 1997; 22:61-79.
4. Kichenassamy S, Kumar A, Olver P, Tannenbaum A, Yezzi A, Conformal curvature flows: from phase transitions to active vision. Archives for Rational Mechanics and Analysis 1996; 134:275-301.
5. Sethian JA. Level Set Methods and Fast Marching Methods: Evolving Interfaces in Computational Geometry, Fluid Mechanics, Computer Vision, and Material Science. Cambridge University Press: New York, 1999.
6. Sapiro G. Geometric Partial Differential Equations and Image Analysis. Cambridge University Press: Cambridge, 2001.
7. Osher S, Fedkiw R. Level Set Methods and Dynamic Implicit Surfaces. Springer: Berlin, 2003.
8. Osher S, Sethian J. Fronts propagating with curvature dependent speed: algorithm based on Hamilton-Jacobi formulation, Journal of Computational Physics 1988; 79:12-49.
9. Nochetto R, Paolini M, Verdi C. Sharp error analysis for curvature dependent evolving fronts. Mathematical Models and Methods in Applied Sciences 1993; 3:711-723.
10. Beneš M. Mathematical and computational aspects of solidification of pure crystallic materials. Acta Mathematica Universitatis Comenianae 2001; 70:123-151.
11. Gage M, Hamilton RS. The heat equation shrinking convex plane curves. Journal of Differential Geometry 1986; 23:69-96.
12. Grayson M. The heat equation shrinks embedded plane curves to round points. Journal of Differential Geometry 1987; 26:285-314.
13. Angenent SB, Gurtin ME. Multiphase thermomechanics with an interfacial structure 2. Evolution of an isothermal interface. Archives for Rational Mechanics and Analysis 1989; 108:323-391.
14. Dziuk G. Convergence of a semi discrete scheme for the curve shortening flow. Mathematical Models and Methods in Applied Sciences 1994; 4:589-606.
15. Dziuk G. Discrete anisotropic curve shortening flow. SIAM Journal on Numerical Analysis 1999; 36: 1808-1830.
16. Hou TY, Lowengrub J, Shelley M. Removing the stiffness from interfacial flows and surface tension. Journal of Computational Physics 1994; 114:312-338.
17. Kimura M. Numerical analysis for moving boundary problems using the boundary tracking method. Japan Journal of Industrial and Applied Mathematics 1997; 14:373-398.
18. Deckelnick K. Weak solutions of the curve shortening flow. Calculus of Variations and Partial Differential Equations 1997; 5:489-510.
19. Hou TY, Klapper I, Si H. Removing the stiffness of curvature in computing 3-d filaments. Journal of Computational Physics 1998; 143:628-664.
20. Mikula K, Šsevčovič D. Solution of nonlinearly curvature driven evolution of plane curves. Applied Numerical Mathematics 1999; 31:191-207.
21. Mikula K, Ševčovič D. Evolution of plane curves driven by a nonlinear function of curvature and anisotropy, SIAM Journal on Applied Mathematics 2001; 61:1473-1501.
22. Mikula K, Kačur J. Evolution of convex plane curves describing anisotropic motions of phase interfaces. SIAM Journal on Scientific Computing 1996; 17:1302-1327.
23. Mikula K. Solution of nonlinear curvature driven evolution of plane convex curves. Applied Numerical Mathematics 1997; 21:1-14.
24. Schmidt A. Computation of three dimensional dendrites with finite elements. Journal of Computational Physics 1996; 125:293-312.
25. Perona P, Malik J. Scale space and edge detection using anisotropic diffusion. Proceedings of IEEE Computer Society Workshop on Computer Vision, 1987.
26. Catté V, Lions PL, Morel JM, Coll T. Image selective smoothing and edge detection by nonlinear diffusion. SIAM Journal on Numerical Analysis 1992; 29:182-193.
27. Kačur J, Mikula K. Solution of nonlinear diffusion appearing in image smoothing and edge detection. Applied Numerical Mathematics 1995; 17:47-59.
28. Angenent SB. Parabolic equations for curves on surfaces I: curves with p-integrable curvature. Annals of Mathematics 1990; 132:451-483.
29. Angenent SB. Parabolic equations for curves on surfaces II: intersections, blow-up and generalized solutions. Annals of Mathematics 1991; 133:171-215.
30. Angenent SB. Nonlinear analytic semiflows. Proceedings of Royal Society of Edinburgh, Section A 1990; 115:91-107.
31. Caselles V, Catté F, Coll T, Dibos F. A geometric model for active contours in image processing. Numerische Matematik 1993; 66:1-31.
32. Malladi R, Sethian J, Vemuri B. Shape modeling with front propagation: a level set approach. IEEE Transactions on Pattern Analysis and Machine Intelligence 1995; 17:158-174.