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Compensated convexity and Hausdorff stable geometric singularity extractions

Zhang, Kewei; Orlando, Antonio; Crooks, Elaine

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Authors

Kewei Zhang

Antonio Orlando

Elaine Crooks



Abstract

We develop and apply the theory of lower and upper compensated convex transforms introduced in [K. Zhang, Compensated convexity and its applications, Ann. Inst. H. Poincaré Anal. Non Linéaire 25 (2008) 743–771] to define multiscale, parametrized, geometric singularity extraction transforms of ridges, valleys and edges of function graphs and sets in ℝn. These transforms can be interpreted as "tight" opening and closing operators, respectively, with quadratic structuring functions. We show that these geometric morphological operators are invariant with respect to translation, and stable under curvature perturbations, and establish precise locality and tight approximation properties for compensated convex transforms applied to bounded functions and continuous functions. Furthermore, we establish multiscale and Hausdorff stable versions of such transforms. Specifically, the stable ridge transforms can be used to extract exterior corners of domains defined by their characteristic functions. Examples of explicitly calculated prototype mathematical models are given, as well as some numerical experiments illustrating the application of these transforms to 2d and 3d objects.

Citation

Zhang, K., Orlando, A., & Crooks, E. (2014). Compensated convexity and Hausdorff stable geometric singularity extractions. Mathematical Models and Methods in Applied Sciences, 25(4), https://doi.org/10.1142/S0218202515500189

Journal Article Type Article
Acceptance Date Sep 15, 2014
Publication Date Nov 18, 2014
Deposit Date Mar 1, 2017
Publicly Available Date Mar 28, 2024
Journal Mathematical Models and Methods in Applied Sciences
Print ISSN 0218-2025
Electronic ISSN 1793-6314
Publisher World Scientific
Peer Reviewed Peer Reviewed
Volume 25
Issue 4
DOI https://doi.org/10.1142/S0218202515500189
Keywords Compensated convex transforms; mathematical morphology; non-flat morphological operators; Moreau envelopes; ridges; valleys; edges; exterior corners; top-hat transform; locality property; Hausdorff stability
Public URL https://nottingham-repository.worktribe.com/output/739527
Publisher URL http://www.worldscientific.com/doi/abs/10.1142/S0218202515500189

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