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A geometric model for active contours in image processing. (English) Zbl 0804.68159
Summary: We propose a new model for active contours based on a geometric partial differential equation. Our model is intrinsic, stable (satisfies the maximum principle) and permits a rigorous mathematical analysis. It enables us to extract smooth shapes (we cannot retrieve angles) and it can be adapted to find several contours simultaneously. Moreover, as a consequence of the stability, we can design robust algorithms which can be engineered with no parameters in applications. Numerical experiments are presented.

MSC:
68U10Image processing (computing aspects)
65D18Computer graphics, image analysis, and computational geometry
References:
[1]Alvarez, L., Lions, P.L., Morel, J.M. (1991): Image selective smoothing and edge detection by nonlinear diffusion (II). Cahier du CEREMADE no 9046, Univ. Paris IX-Dauphine, Paris
[2]Amini, A.A., Tehrani, S., Weymouth, T.E. (1988): Using dynamic programming for minimizing the energy of active contours in the presence of hard constraints. Proc. Second ICCV. 95-99
[3]Ayache, N., Boissonat, J.D., Brunet, E., Cohen, L., Chièze, J.P., Geiger, B., Monga, O., Rocchisani, J.M., Sander, P. (1989): Building highly structured volume representations in 3d medical images. Computer Aided Radiology. Berlin
[4]Barles, G. (1985): Remarks on a flame propagation model. Rapport INRIA,464, 1-38
[5]Berger, M.O. (1990): Snake growing. O. Faugeras, ed., Computer Vision-ECCV90. Lect. Notes Comput. Sci.427, 570-572 · doi:10.1007/BFb0014909
[6]Berger, M.O., Mohr, R. (1990): Towards Autonomy in Active Contour Models. Proc. 10th Int. Conf. Patt. Recogn. Atlantic City, NY, vol1, 847-851
[7]Blake, A., Zisserman, A. (1987): Visual Reconstruction. MIT Press, Cambridge, MA
[8]Chen, Y.-G., Giga, Y., Goto, S. (1989): Uniqueness and Existence of Viscosity Solutions of Generalized Mean Curvature Flow Equations. Preprint Series in Math. Ser. 57. July, Hokkaido University, Sapporo, Japan
[9]Cinquin, P. (1986): Un modèle pour la représentation d’images médicales 3d: Proceedings Euromédicine. (Sauramps Médical)86, 57-61
[10]Cinquin, P. (1987): Application des Fonctions Spline au Traitement d’Images Numériques. Université Joseph Fourier, Grenoble
[11]Cinquin, P., Goret, C., Marque, I., Lavallee, S. (1987): Morphoscopie et modélisation continue d’images 3d. Conférence AFCET IA & Reconnaissance des Formes. AFCET pp. 907-922, Paris
[12]Cohen, L.D. (1991): On active Contour Models and Balloons. CVGIP: Image Understanding53, 211-218 · Zbl 0774.68111 · doi:10.1016/1049-9660(91)90028-N
[13]Cohen, L.D., Cohen, I. (1990): A finite element method applied to new active contour models and 3D reconstruction from cross sections. Proc. Third ICCV, 587-591
[14]Cohen, L.D. (1989): On active Contour Models. Technical Report 1035, INRIA, Rocquencourt, Le Chesnay, France
[15]Crandall, M.G., Lions, P.L. (1983): Viscosity Solutions of Hamilton-Jacobi equations. Trans. Amer. Math. Soc.277, 1-42 · doi:10.1090/S0002-9947-1983-0690039-8
[16]Crandall, M.G., Evans, L.C., Lions, P.L. (1984): Some properties of Viscosity Solutions of Hamilton-Jacobi equations. Trans. Amer. Math. Soc.282, 487-502 · doi:10.1090/S0002-9947-1984-0732102-X
[17]Crandall, M.G., Ishii, I., Lions, P.L. (1991): User’s guide to Viscosity Solutions of Second Order Partial Differential Equations. Cahier du CEREMADE no 9039. Univ. Paris IX-Dauphine, Paris
[18]Evans, L.C., Spruck, J. (1991): Motion of level sets by mean curvature I. J. Diff. Geometry,33, 635-681
[19]Friedman, A. (1982): Variational Principles and Free Boundary Problems. Wiley, New York
[20]Gage, M. (1983): An isoperimetric inequality with applications to curve shortening. Duke Math. J.50, 1225-1229 · Zbl 0534.52008 · doi:10.1215/S0012-7094-83-05052-4
[21]Gage, M. (1984): Cuve shortening makes convex curves circular. Invent. Math.76, 357-364 · Zbl 0542.53004 · doi:10.1007/BF01388602
[22]Gage, M., Hamilton, R.S. (1986): The heat equation shrinking convex plane curves. J. Diff. Geom.23, 69-96
[23]Giga, Y., Goto, S., Ishii, I., Sato, M.-H. (1990): Comparison Principle and Convexity Preserving Properties of Singular Degenerate Parabolic Equations on Unbounded Domains. Preprint Hokkaido University, 1-32, Sapporo, Japan
[24]Grayson, M.A. (1987): The heat equation shrinks embedded plane curves to round points. J. Diff. Geom.26, 285-314
[25]Hirsch, M. (1976): Differential Topology. Springer, Berlin Heidelberg New York
[26]Kass, M., Witkin, A., Terzopoulos, D. (1988): Snakes: active contour models. Int. J. Comput. Vision.1, 321-331 · doi:10.1007/BF00133570
[27]Kass, M., Witkin, A., Terzopoulos, D. (1987): Snakes: active contour models. Proc. First ICCV, 259-267
[28]Ladyzhenskaja, O.A., Solonnikov, V.A., Ural’tseva, N.N. (1968): Linear and Quasilinear Equations of Parabolic Type. American Mathematical Society, Providence, R.I.
[29]Leroy, B. (1991): Etude de quelques propriétés des modèles de contours actifs (?snakes?). Rapport de stage de D.E.A. Univ. Paris-IX Dauphine, Septembre
[30]Lions, P.L. (1982). Generalized Solutions of Hamilton-Jacobi Equations. Research Notes in Mathematics69, Pitman, Boston
[31]Marr, D. (1982): Vision. Freeman, San Francisco
[32]Marr, D., Hildreth, E. (1980): A theory of edge detection. Proc. R. Soc. Lond. B207, 187-217 · doi:10.1098/rspb.1980.0020
[33]Osher, S., Sethian, J.A. (1988): Fronts propagating with curvature dependent speed: algorithms based on Hamilton-Jacobi formulations. Comput. J. Physics.79, 12-49 · Zbl 0659.65132 · doi:10.1016/0021-9991(88)90002-2
[34]Poggio, T., Torre, V., Koch, C. (1985): Computational vision and regularization theory. Nature,317 (6035), 314-319 · doi:10.1038/317314a0
[35]Terzopoulos, D. (1986): Regularization of inverse visual problems involving discontinuities. IEEE Trans. Pattern Anal. Mach. Intell.8: 413-424 · doi:10.1109/TPAMI.1986.4767807
[36]Terzopoulos, D. (1988): The computation of visible surface representations. IEEE Trans. Pattern Anal. Mach. Intell.10(4), 417-438 · doi:10.1109/34.3908
[37]Terzopoulos, D., Witkin, A., Kass, M. (1987): Symmetry seeking models for 3d object reconstruction. Proc. First ICCV, 269-276
[38]Terzopoulos, D., Witkin, A., Kass, M. (1988): Constraints on deformable models: recovering 3d shape and nonrigid motion. Artif. Intell.36, 91-123 · Zbl 0646.68105 · doi:10.1016/0004-3702(88)90080-X
[39]Zucker, S., David, C., Dobbins, A., Iverson, L. (1988): The Organization of Curve Detection: Coarse Tangent Fields and Fine Spline Coverings. In Second International Conference on Computer Vision. pp. 568-577, Tampa Florida (USA)