A Variational Formulation of PDE’s for Dilations and Levelings
Partial differential equations (PDEs) have become very useful modeling and computational tools for many problems in image processing and computer vision related to multiscale analysis and optimization using variational calculus. In previous works, the basic continuous-scale morphological operators have been modeled by nonlinear geometric evolution PDEs. However, these lacked a variational interpretation. In this paper we contribute such a variational formulation and show that the PDEs generating multiscale dilations and erosions can be derived as gradient flows of variational problems with nonlinear constraints. We also extend the variational approach to more advanced object-oriented morphological filters by showing that levelings and the PDE that generates them result from minimizing a mean absolute error functional with local sup-inf constraints.
Keywordsscale-spaces PDEs variational methods morphology
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- A. Arehart, L. Vincent and B. Kimia, “Mathematical Morphology: The Hamilton-Jacobi Connection,” in Proc. ICCV’93, pp.215–219, 1993.Google Scholar
- R.W. Brockett and P. Maragos, “Evolution Equations for Continuous-Scale Morphology,” in Proc. ICASSP-92, San Francisco, Mar. 1992.Google Scholar
- R. Courant and D. Hilbert, Methods of Mathematical Physics, Wiley-Interscience Publ. 1953 (Wiley Classics Edition 1989).Google Scholar
- F. Guichard and J.-M. Morel, Image Analysis and P.D.E.s, book to appear.Google Scholar
- F. Meyer, “The Levelings”, in Mathematical Morphology and Its Applications to Image and Signal Processing, H. Heijmans and J. Roerdink, editors, KluwerAcad. Publ., 1998, p.199–207.Google Scholar
- F. Meyer and P. Maragos, “Nonlinear Scale-Space Representation with Morphological Levelings”, J. Visual Commun. & Image Representation, 11, p.245–265, 2000.Google Scholar
- J.-M. Morel and S. Solimini, Variational Methods in Image Processing, Birkhauser, 1994.Google Scholar
- G. Sapiro, Geometric Partial Differential Equations and Image Analysis, Cambridge University Press, 2001.Google Scholar
- J. Serra, “Connections for Sets and Functions”, Fundamentae Informatica 41, pp.147–186, 2000.Google Scholar