Abstract
Based on a new, general formulation of the geometric method of moving frames, invariantization of numerical schemes has been established during the last years as a powerful tool to guarantee symmetries for numerical solutions while simultaneously reducing the numerical errors. In this paper, we make the first step to apply this framework to the differential equations of image processing. We focus on the Hamilton–Jacobi equation governing dilation and erosion processes which displays morphological symmetry, i.e. is invariant under strictly monotonically increasing transformations of gray-values. Results demonstrate that invariantization is able to handle the specific needs of differential equations applied in image processing, and thus encourage further research in this direction.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Alvarez, L., et al.: Axioms and fundamental equations in image processing. Archive for Rational Mechanics and Analysis 123, 199–257 (1993)
Alvarez, L., Lions, P.-L., Morel, J.-M.: Image selective smoothing and edge detection by nonlinear diffusion. II. SIAM Journal on Numerical Analysis 29, 845–866 (1992)
Cao, F.: Geometric Curve Evolution and Image Processing. Lecture Notes in Mathematics, vol. 1805. Springer, Berlin (2003)
Cartan, É.: La méthode du repère mobile, la théorie des groupes continus, et les espaces généralisés. Exposés de Géométrie, vol. 5. Hermann, Paris (1935)
Breuß, M., Weickert, J.: A shock-capturing algorithm for the differential equations of dilation and erosion. Journal of Mathematical Imaging and Vision, in press.
Brockett, R.W., Maragos, P.: Evolution equations for continuous-scale morphological filtering. IEEE Transactions on Signal Processing 42, 3377–3386 (1994)
Fels, M., Olver, P.J.: Moving coframes. II. Acta Appl. Math. 55, 127–208 (1999)
Gage, M., Hamilton, R.S.: The heat equation shrinking convex plane curves. Journal of Differential Geometry 23, 69–96 (1986)
Grayson, M.: The heat equation shrinks embedded plane curves to round points. Journal of Differential Geometry 26, 285–314 (1987)
Kim, P.: Invariantization of Numerical Schemes for Differential Equations Using Moving Frames. Ph.D. Thesis, University of Minnesota, Minneapolis (2006)
Kim, P.: Invariantization of the Crank-Nicholson Method for Burgers’ Equation. Preprint, University of Minnesota, Minneapolis.
Kim, P., Olver, P.J.: Geometric integration via multi-space. Regular and Chaotic Dynamics 9(3), 213–226 (2004)
Kimmel, R.: Numerical Geometry of Images. Springer, Berlin (2004)
Lax, P.D.: Gibbs Phenomena. Journal of Scientific Computing 28, 445–449 (2006)
Mathéron, G.: Random Sets and Integral Geometry. Wiley, New York (1975)
Olver, P.J.: Applications of Lie Groups to Differential Equations. Springer, New York (1986)
Olver, P.J.: Geometric foundations of numerical algorithms and symmetry. Applicable Algebra in Engineering, Communication and Computing 11, 417–436 (2001)
Olver, P.J.: A survey of moving frames. In: Li, H., Olver, P.J., Sommer, G. (eds.) IWMM-GIAE 2004. LNCS, vol. 3519, pp. 105–138. Springer, Heidelberg (2005)
Sapiro, G.: Geometric Partial Differential Equations and Image Analysis. Cambridge University Press, Cambridge (2001)
Sapiro, G., Tannenbaum, A.: Affine invariant scale-space. International Journal of Computer Vision 11, 25–44 (1993)
Serra, J.: Image Analysis and Mathematical Morphology, vol. 1. Academic Press, London (1982)
Sethian, J.A.: Level Set Methods. Cambridge University Press, Cambridge (1996)
Author information
Authors and Affiliations
Editor information
Rights and permissions
Copyright information
© 2007 Springer Berlin Heidelberg
About this paper
Cite this paper
Welk, M., Kim, P., Olver, P.J. (2007). Numerical Invariantization for Morphological PDE Schemes. In: Sgallari, F., Murli, A., Paragios, N. (eds) Scale Space and Variational Methods in Computer Vision. SSVM 2007. Lecture Notes in Computer Science, vol 4485. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-72823-8_44
Download citation
DOI: https://doi.org/10.1007/978-3-540-72823-8_44
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-72822-1
Online ISBN: 978-3-540-72823-8
eBook Packages: Computer ScienceComputer Science (R0)