Abstract
It is well known that a conveniently rescaled iterated convo- lution of a linear positive kernel converges to a Gaussian. Therefore, all iterative linear smoothing methods of a signal or an image boils down to the application to the signal of the Heat Equation. In this survey, we explain how a similar analysis can be performed for image iterative smoothing by contrast invariant monotone operators. In particular, we prove that all iterated affine and contrast invariant monotone opera- tors are equivalent to the unique affine invariant curvature motion. We also prove that under very broad conditions, weighted median filters are equivalent to the Mean Curvature Motion Equation.
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References
L. Alvarez, F. Guichard, P.L. Lions, and J.M. Morel. Axiomatisation et nouveaux opérateurs de la morphologie mathématique. Compte rendu de l’Académie des Sciences de Paris, 1(315):265–268, 1992.
L. Alvarez, F. Guichard, P.L. Lions, and J.M. Morel. Axioms and fundamental equations of image processing. Arch. for Rat. Mech., 123(3):199–257, 1993.
G. Barles and C. Georgelin. A simple proof of convergence for an approximation scheme for computing motion by mean curvature. SIAM J.Numer.Anal., 32(2):484–500, 1995.
G. Barles and P.M. Souganidis. Convergence of approximation schemes for fully nonlinear second order equations. Asymptotic Analysis, 4:271–283, 1991.
J. Bence, B. Merriman, and S. Osher. Diffusion motion generated by mean curvature. CAM Report 92-18. Dept of Mathematics. University of California Los Angeles, April 1992.
F. Cao. Partial differential equations and mathematical morphology. Journal de Mathématiques Pures et Appliquées, 77(9):909–941, 1998.
F. Catté. Convergence of iterated affine and morphological filters by nonlinear semi-groups theory. Proceedings of ICAOS-96 INRIA-CEREMADE, June 1996.
F. Catté, F. Dibos, and G. Koepfler. A morphological scheme for mean curvature motion and application to anisotropic diffusion and motion of level sets. SIAM Jour. of Numer. Anal., 32(6):1895–1909, Dec 1995.
Y.G. Chen, Y. Giga, and S. Goto. Uniqueness and existence of viscosity solutions of generalized mean curvature flow equations. J. Diff. Geometry, 33:749–786, 1991.
M.G. Crandall and P.L. Lions. Convergent difference schemes for nonlinear parabolic equations and mean curvature motion. Numerische Mathematik, 75:17–41, 1996.
L.C. Evans. Convergence of an algorithm for mean curvature motion. Indiana Univ. Math. Journal, 42:533–557, 1993.
L.C. Evans and J. Spruck. Motion of level sets by mean curvature. I. J. of Differential Geometry, 33:635–681, 1991.
F. Guichard. Axiomatisation des analyses multiéchelles d’images et de films. PhD thesis, Université Paris Dauphine, 1993.
F. Guichard and J.M. Morel. Partial Differential Equations and Image Iterative Filtering. Tutorial of ICIP95, Washington D.C., 1995.
H. Ishii. A Generalization of the Bence, Merriman and Osher Algorithm for Motion By Mean Curvature. In GAKUTO, editor, Curvature flows and related topics, volume 5, pages 111–127. Levico, 1994.
J.J. Koenderink. The structure of images. Biol. Cybern., 50:363–370, 1984.
D. Marr. Vision. N.York, W.H. and Co, 1982.
G. Matheron. Random Sets and Integral Geometry. John Wiley N.Y., 1975.
L. Moisan. Affine plane curve evolution: a fully consistent scheme. Technical Report 9628, Cahiers du Ceremade, 1996.
D. Pasquignon. Approximation of viscosity solutions by morphological filters. Preprint, 1997.
J. Serra. Image Analysis and Mathematical Morphology. Academic Press, 1982.
A.P. Witkin. Scale space filtering. In Proc. of IJCAI, Karlsruhe, pages 1019–1021, 1983.
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Cao, F. (1999). Morphological Scale Space and Mathematical Morphology. In: Nielsen, M., Johansen, P., Olsen, O.F., Weickert, J. (eds) Scale-Space Theories in Computer Vision. Scale-Space 1999. Lecture Notes in Computer Science, vol 1682. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-48236-9_15
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DOI: https://doi.org/10.1007/3-540-48236-9_15
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