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.


68U10 Computing methodologies for image processing
65D18 Numerical aspects of computer graphics, image analysis, and computational geometry
Full Text: DOI EuDML


[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
[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 · Zbl 0735.35082
[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
[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 · Zbl 0599.35024
[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 · Zbl 0543.35011
[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 · Zbl 0755.35015
[18] Evans, L.C., Spruck, J. (1991): Motion of level sets by mean curvature I. J. Diff. Geometry,33, 635-681 · Zbl 0726.53029
[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
[21] Gage, M. (1984): Cuve shortening makes convex curves circular. Invent. Math.76, 357-364 · Zbl 0542.53004
[22] Gage, M., Hamilton, R.S. (1986): The heat equation shrinking convex plane curves. J. Diff. Geom.23, 69-96 · Zbl 0621.53001
[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 · Zbl 0836.35009
[24] Grayson, M.A. (1987): The heat equation shrinks embedded plane curves to round points. J. Diff. Geom.26, 285-314 · Zbl 0667.53001
[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 · Zbl 0646.68105
[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 · Zbl 0497.35001
[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
[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
[34] Poggio, T., Torre, V., Koch, C. (1985): Computational vision and regularization theory. Nature,317 (6035), 314-319
[35] Terzopoulos, D. (1986): Regularization of inverse visual problems involving discontinuities. IEEE Trans. Pattern Anal. Mach. Intell.8: 413-424
[36] Terzopoulos, D. (1988): The computation of visible surface representations. IEEE Trans. Pattern Anal. Mach. Intell.10(4), 417-438 · Zbl 0669.65009
[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
[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)
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.