Abstract
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.
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Maragos, P. (2005). A Variational Formulation of PDE’s for Dilations and Levelings. In: Ronse, C., Najman, L., Decencière, E. (eds) Mathematical Morphology: 40 Years On. Computational Imaging and Vision, vol 30. Springer, Dordrecht. https://doi.org/10.1007/1-4020-3443-1_29
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DOI: https://doi.org/10.1007/1-4020-3443-1_29
Publisher Name: Springer, Dordrecht
Print ISBN: 978-1-4020-3442-8
Online ISBN: 978-1-4020-3443-5
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