Abstract
In this paper we propose a modification in the usual numerical method for computing the solutions of the curvature equation in the plane . This modification takes place near the singularities of the image. We propose to use zero as the vertical speed at a saddle point and, at an extremum, the geometric mean of the eigenvalues of the Hessian matrix. This modification is theoretically justified and the preliminary experimental results show that it makes the algorithm more reliable.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
L. Alvarez, F. Guichard, P.L. Lions, and J.M. Morel, “Axioms and fundamental equations of image processing,” Archive for Rational Mechanics and Analysis, Vol. 16, No. 9, pp. 200–257, 1993.
Yun-Gang Chen, Yoshikazu Giga, and Shun’ichi Goto, “Uniqueness and existence of viscosity solutions of generalized mean curvature flow equations,” J. Diff. Geometry, Vol. 33, pp. 749–786, 1991.
M. Craizer, “Evolution of critical points in curvature and affine morphological scale spaces,” in Proc. Scale Space Methods in Computer Vision, 2003, LNCS2695, pp. 523–536.
M. Craizer and R. Teixeira, “Evolution of an extremum by curvature motion,” J. Math. Anal. Appl., Vol. 293, pp. 721–737, 2004.
L.C. Evans and J. Spruck, “Motion of level sets by mean curvature I,” J. Diff. Geometry, Vol. 33, pp. 635–681, 1991.
L.C. Evans and J. Spruck, “Motion of level sets by mean curvature II,” Trans. American Math. Soc., Vol. 330, No. 1, 1992.
L.C. Evans and J. Spruck, “Motion of level sets by mean curvature III,” J. GeometricAnalysis, Vol. 2, No. 2, 1992.
M.E. Gage, “Curve shortening makes convex curves circular,” Inv. Mathematicae, Vol. 76, pp. 357–364, 1984.
M.E. Gage and R.S. Hamilton, “The heat equation shrinking convex plane curves,” J. Diff. Geometry, Vol. 23, pp. 69–96, 1986.
M.A. Grayson, “The heat equation shrinks embedded plane curves to round points,” J. Diff. Geometry, Vol. 26, pp. 285–314, 1987.
L.D. Griffin, “Mean, median and mode filtering of images,” Proc. Royal Soc. Lond. A, Vol. 456, pp. 2995–3004, 2000.
F. Guichard and J.M. Morel, Image Iterative Smoothing and P.D.E.’s, pre-print.
T. Lindeberg, Scale-Space Theory in Computer Vision, Kluwer Academic Publishers, 1994.
A. Mackworth and F. Mokhtarian, “A theory of multiscale, curvature-based shape representation for planar curves,” IEEE-PAMI, Vol. 14, pp. 789–805, 1992.
P. Perona and J. Malik, “A scale space and edge detection using anisotropic diffusion,” in Proc. IEEE Comp. Soc. Workshop Computer Vision, 1987.
J.A. Sethian, “Numerical algorithms for propagating interfaces: Hamilton-Jacobi equations and conservation laws,” J. Differential Geometry, Vol. 31, pp. 131–161, 1990.
J.A. Sethian, Level Set Methods and Fast Marching Methods, Cambridge University Press, 1999.
Software Megawave, http://www.cmla.ens-cachan.fr/Cmla/Megawave/presentation.html
R.C. Teixeira, “Introduçao aos espaços de escala,” IMPA, 2001, in portuguese.
Author information
Authors and Affiliations
Corresponding author
Additional information
Marcos Craizer has a degree in mathematics from UFRJ (Rio de Janeiro), a M.Sc. from IMPA (Rio de Janeiro) and received his Ph.D. in mathematics also from IMPA, in 1989. His research interests in image processing includes image representation, curve evolution and PDE applications. Since 1988, he has been working at the math department of PUC-Rio, Brazil.
Sinésio Pesco is an Assistant Professor of the Department of Mathematics at Pontifical Catholic University of Rio de Janeiro (PUCRio). He received his Ph.D. and MS degree in Applied Mathematics at PUC-Rio and a B.S. degree in mathematics from State University of Maringa Brasil. He has visiting positions at Lawrence Livermore National Laboratory, CSE/OGI School of Science and Engineering (Oregon Health & Science University) and Scientific Computation and Imaging Insititute (University of UTAH). His main research interests are in Computational Topology, Image Processing and Scientific Visualization. Since 1991, he has been working in the development of a CAD system for petroleum reservoir modeling.
Ralph Teixeira has a degree in Computer Engineering from IME (in Rio), a M.Sc. from IMPA (also in Rio) and received his Ph.D. in Mathematics from Harvard University in 1998. His research interests in Computer Vision include shape representations by skeletons (medial axis and similar objects), curve evolutions and PDE applications. Since 2001, he has been working at Fundação Getulio Vargas in Rio de Janeiro, Brazil.
Rights and permissions
About this article
Cite this article
Craizer, M., Pesco, S. & Teixeira, R. A Numerical Scheme for the Curvature Equation Near the Singularities. J Math Imaging Vis 22, 89–95 (2005). https://doi.org/10.1007/s10851-005-4784-7
Issue Date:
DOI: https://doi.org/10.1007/s10851-005-4784-7