×

zbMATH — the first resource for mathematics

Geometrical image segmentation by the Allen-Cahn equation. (English) Zbl 1055.94502
Summary: We present an algorithm of pattern recovery (image segmentation) based on the solution of the Allen-Cahn equation. The approach is usually understood as a regularization of the level-set motion by mean curvature where we impose a special forcing term which lets the initial level set closely surround the pattern in question. We show convergence of the numerical scheme and demonstrate how the algorithm functions on several artificial as well as real examples.

MSC:
94A08 Image processing (compression, reconstruction, etc.) in information and communication theory
68U10 Computing methodologies for image processing
65L12 Finite difference and finite volume methods for ordinary differential equations
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Alvarez, L.; Lions, P.L.; Morel, J.M., Image selective smoothing and edge detection by nonlinear diffusion II, SIAM J. numer. anal, 29, 845-866, (1992) · Zbl 0766.65117
[2] Barles, G.; Soner, H.M.; Souganidis, P.E., Front propagation and phase field theory, SIAM J. control optim, 31, 439-469, (1993) · Zbl 0785.35049
[3] Beneš, M., Mathematical analysis of phase-field equations with numerically efficient coupling terms, Interfaces and free boundaries, 3, 201-221, (2001) · Zbl 0986.35116
[4] Beneš, M., Mathematical and computational aspects of solidification of pure substances, Acta math. univ. Comenian, 70, 1, 123-152, (2001) · Zbl 0990.80006
[5] Beneš, M., Diffuse-interface treatment of the anisotropic Mean-curvature flow, Appl. math, 48, 6, 437-453, (2003) · Zbl 1099.53044
[6] Beneš, M.; Mikula, K., Simulation of anisotropic motion by Mean curvature – comparison of phase-field and sharp-interface approaches, Acta math. univ. Comenian, 67, 1, 17-42, (1998) · Zbl 0963.80004
[7] Brenner, S.C.; Scott, L.R., The mathematical theory of finite element methods, (1994), Springer New York · Zbl 0804.65101
[8] Caginalp, G., An analysis of a phase field model of a free boundary, Arch. rational mech. anal, 92, 205-245, (1986) · Zbl 0608.35080
[9] Caselles, V.; Kimmel, R.; Sapiro, G., Geodesic active contours, Internat. J. comput. vision, 22, 61-79, (1997) · Zbl 0894.68131
[10] Catté, F.; Lions, P.L.; Morel, J.M.; Coll, T., Image selective smoothing and edge detection by nonlinear diffusion, SIAM J. numer. anal, 29, 182-193, (1992) · Zbl 0746.65091
[11] Chen, Y.-G.; Giga, Y.; Goto, S., Uniqueness and existence of viscosity solutions of generalized Mean curvature flow equations, J. differential geom, 33, 749-786, (1991) · Zbl 0696.35087
[12] Handlovičová, A.; Mikula, K.; Sgallari, F., Semi-implicit complementary volume scheme for solving level-set like equations in image processing and curve evolution, Numer. math, 93, 675-695, (2003) · Zbl 1065.65105
[13] ()
[14] ()
[15] C.M. Elliott, A.R. Gardiner. Double-obstacle phase-field computations of dendritic growth, Technical Report, CMAIA, University of Sussex, Brighton, 1996, Report No. 96/19
[16] Elliott, C.M.; Paolini, M.; Scha̋tzle, R., Interface estimates for the fully anisotropic allen – cahn equation and anisotropic Mean curvature flow, Math. models methods appl. sci, 6, 1103-1118, (1996) · Zbl 0873.35039
[17] Evans, L.C.; Soner, H.M.; Souganidis, P.E., Phase transitions and generalized motion by Mean curvature, Comm. pure appl. math, 45, 1097-1123, (1992) · Zbl 0801.35045
[18] Evans, L.C.; Spruck, J., Motion of level sets by Mean curvature I, J. differential geom, 33, 635-681, (1991) · Zbl 0726.53029
[19] Feng, X.B.; Prohl, A., Numerical analysis of the allen – cahn equation and approximation for Mean curvature flows, Numer. math, 94, 1, 33-65, (2003) · Zbl 1029.65093
[20] Fučı́k, S.; Kufner, A., Nonlinear differential equations, (1978), TKI SNTL Prague, (in Czech) · Zbl 0647.35001
[21] Kačur, J., Method of rothe in evolution equations, Teubner-texte zur Mathematik, (1985), Teubner Leipzig · Zbl 0582.65084
[22] Kačur, J.; Mikula, K., Solution of nonlinear diffusion appearing in image smoothing and edge detection, Appl. numer. math, 17, 47-59, (1995) · Zbl 0823.65089
[23] Kichenassamy, S.; Kumar, A.; Olver, P.; Tannenbaum, A.; Yezzi, A., Conformal curvature flows, Arch. rational mech. anal, 134, 275-301, (1996) · Zbl 0937.53029
[24] Kühn, T., Convergence of a fully discrete approximation for advected Mean curvature flows, IMA J. numer. anal, 18, 595-634, (1998) · Zbl 0934.76043
[25] Lions, J.L., Quelques Méthodes de Résolution des problèmes aux limites non linéaires, (1969), Dunod, Gauthiers-Villars Paris · Zbl 0189.40603
[26] Malladi, R.; Sethian, J.; Vemuri, B., Shape modeling with front propagation: a level set approach, IEEE trans. pattern anal. Mach. intel, 17, 158-174, (1995)
[27] Mikula, K.; Kačur, J., Evolution of convex plane curves describing anisotropic motions of phase interfaces, SIAM J. sci. comput, 17, 1302-1327, (1996) · Zbl 0868.35063
[28] Nochetto, R.H.; Paolini, M.; Rovida, S.; Verdi, C., Variational approximation of the geometric motion of fronts, (), 124-149 · Zbl 0814.35051
[29] Osher, S.; Fedkiw, R., Levelset methods and dynamic implicit surface, (2003), Springer Berlin
[30] Paolini, M., An efficient algorithm for computing anisotropic evolution by Mean curvature, () · Zbl 0838.73079
[31] Paolini, M.; Verdi, C., Asymptotic and numerical analyses of the Mean curvature flow with a space-dependent relaxation parameter, Asymptotic anal, 5, 553-574, (1992) · Zbl 0757.65078
[32] Perona, P.; Malik, J., Scale space and edge detection using anisotropic diffusion, IEEE trans. pattern anal. Mach. intell, 12, 629-639, (1990)
[33] Samarskii, A.A., Theory of difference schemes, (1977), Nauka Moscow · Zbl 0368.65031
[34] Samarskii, A.A.; Nikolaev, Y.S., Numerical solution of large sparse systems, (1984), Academia Prague, Czech translation
[35] Sethian, J.A., Level set methods, (1996), Cambridge University Press New York · Zbl 0859.76004
[36] Visintin, A., Models of phase transitions, (1996), Birkhäuser Boston, MA · Zbl 0882.35004
[37] Zhao, H.K.; Osher, S.; Chan, T.; Merriman, B., A variational level set approach to multiphase motion, J. comput. phys, 127, 179-195, (1996) · Zbl 0860.65050
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.