Abstract
The partial differential equations describing the propagation of (wave) fronts in space are closely connected with the morphological erosion and dilation. Strangely enough this connection has not been ex- plored in the derivation of numerical schemes to solve the differential equations. In this paper the morphological facet model is introduced in which an analytical function is locally fitted to the data. This function is then dilated analytically with an infinitesimal small structuring element. These sub-pixel dilationsform the core of the numerical solution schemes presented in this paper. One of the simpler morphological facet models leads to a numerical scheme that is identical with a well known classical upwind finite difference scheme. Experiments show that the morpholog- ical facet model provides stable numerical solution schemes for these partial differential equations.
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References
L. Alvarez, F. Guichard, P.L. Lions, and J.M. Morel. Axioms and fundamental equations of image processing. Archives for rational mechanics, 123(3):199–257, 1993.
P. Maragos. Max-min difference equations and recursive morphological systems. In J. Serra and P. Salembier, editors, Mathematical Morphology and its Applications to Signal Processing, pages 128–133. Kluwer academic publishers, 1993.
J. Mattioli. Differential equations of morphological operators. In J. Serra and P. Salembier, editors, Mathematical Morphology and its Applications to Signal Processing, pages 162–167, Amsterdam, 1993. Kluwer academic publishers.
R. v.d. Boomgaard and A.W.M. Smeulders. The morphological structure of images, the differential equations of morphological scale-space. IEEE transactions PAMI, 16(11):1101–1113, 1994.
S. Osher and J.A. Sethian. Fronts propagating with curvature dependent speed: algorithms based on hamilton-jacobi formulations. Journal of Computational Physics, 79:12–49, 1988.
G. Sapiro and A. Tannenbaum. Affine invariant scale-space. International journal of computer vision, 11:25–44, 1994.
B.B. Kimia, A. Tannenbaum, and S.W. Zucker. Shape, shocks and deformations, i. International journal of computer vision, 15(3):189–224, 1994.
R. v.d. Boomgaard. Numerical solutions of morphological partial differential equations, based on the morphological facet model. In H.J.A.M. Heijmans and J. Roerdink, editors, Mathematical Morphology and its Applications to Image and Signal Processing, pages 175–182, 1998.
L. Dorst and R. v.d. Boomgaard. The slope transform and morphological signal processing. Signal Processing, 38:79–98, 1994.
H.J.A.M. Heijmans. Morphological Image Operators. Academic Press, 1994.
L. van Vliet. Grey-scale measurements in multidimensional digitized images. PhD thesis, Delft University of Technology, 1993.
J.A. Sethian. A fast marching level set method for monotonically advancing fronts. Proc. Nat. Acad. Sci., 93(4):1591–1595, 1996.
J.M. Burgers. The non-linear diffusion equation: asymptotic solutions and statistical problems. D. Reidel Publishers, 1974.
R. Haralick and L. Shapiro. Computer and robot vision. Addison and Wesley, 1992.
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© 1999 Springer-Verlag Berlin Heidelberg
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van den Boomgaard, R. (1999). Numerical Solution Schemes for Continuous-Scale 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_18
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DOI: https://doi.org/10.1007/3-540-48236-9_18
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