Compensated convexity and Hausdorff stable geometric singularity extractions
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...
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| Format: | Article |
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World Scientific Publishing
2014
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| Online Access: | https://eprints.nottingham.ac.uk/40936/ |
| _version_ | 1848796166756499456 |
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| author | Zhang, Kewei Orlando, Antonio Crooks, Elaine |
| author_facet | Zhang, Kewei Orlando, Antonio Crooks, Elaine |
| author_sort | Zhang, Kewei |
| building | Nottingham Research Data Repository |
| collection | Online Access |
| description | 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. |
| first_indexed | 2025-11-14T19:43:40Z |
| format | Article |
| id | nottingham-40936 |
| institution | University of Nottingham Malaysia Campus |
| institution_category | Local University |
| last_indexed | 2025-11-14T19:43:40Z |
| publishDate | 2014 |
| publisher | World Scientific Publishing |
| recordtype | eprints |
| repository_type | Digital Repository |
| spelling | nottingham-409362020-05-04T16:57:19Z https://eprints.nottingham.ac.uk/40936/ Compensated convexity and Hausdorff stable geometric singularity extractions Zhang, Kewei Orlando, Antonio Crooks, Elaine 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. World Scientific Publishing 2014-11-18 Article PeerReviewed Zhang, Kewei, Orlando, Antonio and Crooks, Elaine (2014) Compensated convexity and Hausdorff stable geometric singularity extractions. Mathematical Models and Methods in Applied Sciences, 25 (04). pp. 747-801. ISSN 1793-6314 Compensated convex transforms; mathematical morphology; non-flat morphological operators; Moreau envelopes; ridges; valleys; edges; exterior corners; top-hat transform; locality property; Hausdorff stability http://www.worldscientific.com/doi/abs/10.1142/S0218202515500189 doi:10.1142/S0218202515500189 doi:10.1142/S0218202515500189 |
| spellingShingle | Compensated convex transforms; mathematical morphology; non-flat morphological operators; Moreau envelopes; ridges; valleys; edges; exterior corners; top-hat transform; locality property; Hausdorff stability Zhang, Kewei Orlando, Antonio Crooks, Elaine Compensated convexity and Hausdorff stable geometric singularity extractions |
| title | Compensated convexity and Hausdorff stable geometric singularity extractions |
| title_full | Compensated convexity and Hausdorff stable geometric singularity extractions |
| title_fullStr | Compensated convexity and Hausdorff stable geometric singularity extractions |
| title_full_unstemmed | Compensated convexity and Hausdorff stable geometric singularity extractions |
| title_short | Compensated convexity and Hausdorff stable geometric singularity extractions |
| title_sort | compensated convexity and hausdorff stable geometric singularity extractions |
| topic | Compensated convex transforms; mathematical morphology; non-flat morphological operators; Moreau envelopes; ridges; valleys; edges; exterior corners; top-hat transform; locality property; Hausdorff stability |
| url | https://eprints.nottingham.ac.uk/40936/ https://eprints.nottingham.ac.uk/40936/ https://eprints.nottingham.ac.uk/40936/ |