Skip to main content

Research Repository

Advanced Search

Compensated convexity and Hausdorff stable geometric singularity extractions

Zhang, Kewei; Orlando, Antonio; Crooks, Elaine

Authors

KEWEI ZHANG Kewei.Zhang@nottingham.ac.uk
Professor of Mathematical Analysis

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.

Journal Article Type Article
Publication Date Nov 18, 2014
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
APA6 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
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
Publisher URL http://www.worldscientific.com/doi/abs/10.1142/S0218202515500189
Copyright Statement Copyright information regarding this work can be found at the following address: http://eprints.nottingh.../end_user_agreement.pdf

Files

ZOC-Paper1-m3as.pdf (1.9 Mb)
PDF

Copyright Statement
Copyright information regarding this work can be found at the following address: http://eprints.nottingham.ac.uk/end_user_agreement.pdf





You might also like



Downloadable Citations

;