Abstract
Let I :Ω→ℜ be a given bounded image function, where Ω is an open and bounded domain which belongs to ℜn. Let us consider n=2 for the purpose of illustration. Also, let S={xi}i∈Ω be a finite set of given points. We would like to find a contour Γ⊂Ω, such that Γ is an object boundary interpolating the points from S. We combine the ideas of the geodesic active contour (cf. Caselles et al. [7,8]) and of interpolation of points (cf. Zhao et al. [40]) in a level set approach developed by Osher and Sethian [33]. We present modelling of the proposed method, both theoretical results (viscosity solution) and numerical results are given.
Similar content being viewed by others
References
D. Adalsteinsson and J.A. Sethian, A fast level set method for propagating interfaces, J. Comput. Phys. 118(2) (1995) 269–277.
L. Alvarez, P.-L. Lions and J.-M. Morel, Image selective smoothing and edge detection by nonlinear diffusion, SIAM J. Numer. Anal. 29(3) (1992) 845–866.
D. Apprato, J.B. Betbeder, C. Gout and A. Vieira-Testé, Segmentation method under geometric constraints after pre-processing, in: Curves and Surfaces, Vol. IV, eds. A. Cohen, C. Rabut and L.L. Schumaker (Vanderbilt Univ. Press, Nashville, TN, 2000) pp. 9–18.
G. Barles, Nonlinear Neumann Boundary Conditions for Quasilinear Degenerate Elliptic Equations and Applications, First version.
G. Barles, Solutions de Viscosité des Équations de Hamilton–Jacobi (Springer, Berlin, 1994).
V. Caselles, F. Catté, C. Coll and F. Dibos, A geometric model for active contours in image processing, Numer. Math. 66 (1993) 1–31.
V. Caselles, R. Kimmel and G. Sapiro, Geodesic active contours, in: Proc. of the Fifth Internat. Conf. on Computer Vision (20–23 June 1995) pp. 694–699.
V. Caselles, R. Kimmel and G. Sapiro, Geodesic active contours, Internat. J. Comput. Vision 22(1) (1997) 61–87.
T.F. Chan, B.Y. Sandberg and L.A. Vese, Active contours without edges for vector-valued images, J. Visual Communication Image Representation 11(2) (2000) 130.
T.F. Chan, J. Shen and L. Vese, Variational PDE models in image processing, Notices Amer. Math. Soc. 50(1) (2003) 14–26.
T.F. Chan and L.A. Vese, An efficient variational multiphase motion for the Mumford–Shah segmentation mode, in: IEEE Asilomar Conf. on Signals Systems and Computers, Vol. 1 (2000) pp. 490–494.
T. Chan and L. Vese, Active contours without edges, IEEE Trans. Image Process. 10(2) (2001) 266–277.
I. Cohen and L.D. Cohen, Deformable models for 3D medical images using finite element and balloons, in: Proc. of IEEE Conf. on Computer Vision and Pattern Recognition, Champaign (IEEE, Los Alamitos, 1992) pp. 592–598.
I. Cohen, L.D. Cohen and N. Ayache, Using deformable surfaces to segment 3D images and infer differential structures, Computer Vision 56(2) (1992) 242–263.
L.D. Cohen, On active contours models and balloons, Computer Vision 53(2) (1991) 211–218.
L.D. Cohen, E. Bardinet and N. Ayache, Surface reconstruction using active contour models, INRIA, Rapport de Recherche (1992).
M.G. Crandall, H. Ishii and P.-L. Lions, User’s guide to viscosity solutions of second order partial differential equations, Bull. Amer. Math. Soc. 27(1) (1992) 1–69.
B.A. Dubrovin, A.T. Fomenko and S.P. Novikov, Modern geometry methods and applications, Part I, in: The Geometry of Surfaces, Transformation Groups, and Fields (Springer, Berlin, 1992).
A. Friedman, Partial Differential Equations of Parabolic Type (Prentice-Hall, Englewood Cliffs, NJ, 1964).
Y. Giga and M.-H. Sato, Generalized interface evolution with boundary condition, Proc. Japan Acad. Ser. A 67 (1991) 263–266.
C. Gout, C. Le Guyader and L. Vese, Image segmentation under interpolation conditions, Preprint, CAM-IPAM, University of California at Los Angeles (2003) 44 pages.
C. Gout and S. Vieira-Testé, An algorithm for contrast enhancement and segmentation of images, in: IEEE Internat. Conf. on Image Processing, Vol. 2, Vancouver, Canada (2001) pp. 716–719.
H. Ishii and M.-H. Sato, Nonlinear oblique derivative problems for singular degenerate parabolic equations on a general domain, Preprint, Waseda (2001).
M. Kass, A. Witkin and D. Terzopoulos, Snakes: Active contour models, Internat. J. Comput. Vision 1(4) (1987) 133–144.
S. Kichenassamy, A. Kumar, P. Olver, A. Tannenbaum and A. Yezzi, Gradient flows and geometric active contour models, in: Proc. of the Fifth Internat. Conf. on Computer Vision (20–23 June 1995) pp. 810–815.
S. Kichenassamy, A. Kumar, P. Olver, A. Tannenbaum and A. Yezzi, Conformal curvature flows: From phase transitions to active vision, Archive Rational Mech. Anal. 134(3) (1996) 275–301.
O.A. Ladyzhenskaya, V.A. Solonnikov and N.N. Ural’tseva, Linear and Quasilinear Equations of Parabolic Type (Amer. Math. Soc., Providence, RI, 1968).
C. Le Guyader, Imagerie mathématique: Théory et applications, Thèse de doctorat, INSA Rouen (2004) en cours de rédaction.
C. Le Guyader, D. Apprato and C. Gout, Using a level set approach for image segmentation under interpolation conditions, Numer. Algorithms (2003) submitted.
R. Malladi and J.A. Sethian, Image processing via level curvature flow, Proc. National Acad. Sci. 92(15) (1995) 7046–7050.
R. Malladi, J.A. Sethian and B.C. Vermuri, A fast level set based algorithm for topology independent shape modeling and recovery, in: Proc. of the 3rd ECCV, Stockholm, Sweden (1994) pp. 3–13.
S. Osher and R. Fedkiw, Level Set Methods and Dynamic Implicit Surfaces (Springer, Berlin, 2003).
S. Osher and J.A. Sethian, Fronts propagation with curvature dependent speed: Algorithms based on Hamilton–Jacobi formulations, J. Comput. Phys. 79 (1988) 12–49.
J.A. Sethian, Level Set Methods and Fast Marching Methods: Evolving Interfaces in Computational Geometry, Fluid Mechanics, Computer Vision and Material Science (Cambridge Univ. Press, London, 1999).
J.A. Sethian, Evolution, implementation and application of level set and fast marching methods for advancing fronts, J. Comput. Phys. 169(2) (2001) 503–555.
J.A. Sethian, A review of recent numerical algorithms for surface s moving with curvature dependent flows, J. Differential Geometry 31 (1989) 131–161.
L. Vese, A method to convexify functions via curve evolution, Commun. Partial Differential Equations 24(9/10) (1999) 1573.
L.A. Vese, Study in the BV space of a denoising–deblurring variational problem, Appl. Math. Optim. 44(2) (2001) 131–162.
L.A. Vese and T.F. Chan, A multiphase level set framework for image segmentation using the Mumford and Shah model, Internat. J. Computer Vision 50(3) (2002) 271–293.
H.-K. Zhao, S. Osher, B. Merriman and M. Kang, Implicit and nonparametric shape reconstruction from unorganized data using a variational level set method, Computer Vision Image Understanding 80(3) (2000) 295–314.
H.-K. Zhao, T. Chan, B. Merriman and S. Osher, A variational level set approach to multiphase motion, J. Comput. Phys. 127 (1996) 179–195.
Author information
Authors and Affiliations
Corresponding author
Additional information
AMS subject classification
49L25, 74G65, 68U10
Rights and permissions
About this article
Cite this article
Gout, C., Le Guyader, C. & Vese, L. Segmentation under geometrical conditions using geodesic active contours and interpolation using level set methods. Numer Algor 39, 155–173 (2005). https://doi.org/10.1007/s11075-004-3627-8
Received:
Accepted:
Issue Date:
DOI: https://doi.org/10.1007/s11075-004-3627-8