Compensated convexity and Hausdorff stable geometric singularity extractions

Zhang, Kewei; Orlando, Antonio; Crooks, Elaine

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 1, 2017
• 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
• Contract Date: Mar 1, 2017