Compensated convexity and Hausdorff stable extraction of intersections for smooth manifolds

Zhang, Kewei; Orlando, Antonio; Crooks, Elaine

Abstract

We apply compensated convex transforms to define a multiscale Hausdorff stable method to extract intersections between smooth compact manifolds represented by their characteristic functions or as point clouds embedded in Rn. We prove extraction results on intersections of smooth compact manifolds and for points of high curvature. As a result of the Hausdorff–Lipschitz continuity of our transforms, we show that our method is stable against dense sampling of smooth manifolds with noise. Examples of explicitly calculated prototype models for some simple cases are presented, which are also used in the proofs of our main results. Numerical experiments in two- and three-dimensional space, and applications to geometric objects are also shown.

Citation

Zhang, K., Orlando, A., & Crooks, E. (2015). Compensated convexity and Hausdorff stable extraction of intersections for smooth manifolds. Mathematical Models and Methods in Applied Sciences, 25(5), https://doi.org/10.1142/S0218202515500207

• Journal Article Type: Article
• Acceptance Date: Sep 15, 2014
• Online Publication Date: Dec 16, 2014
• Publication Date: May 31, 2015
• Deposit Date: Feb 28, 2017
• Publicly Available Date: Feb 28, 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: 5
• DOI: https://doi.org/10.1142/S0218202515500207
• Keywords: Compensated convex transforms; mathematical morphology; non-flat morphological operators; characteristic function; point clouds; Hausdorff–Lipschitz continuity; locality property; surface-to-surface intersection; transversal intersections; random sample
• Public URL: https://nottingham-repository.worktribe.com/output/750988
• Publisher URL: http://www.worldscientific.com/doi/abs/10.1142/S0218202515500207
• Additional Information: Electronic version of an article published as Mathematical Models and Methods in Applied Sciences, v. 25, no. 5, 2015, p. 839-873, doi:10.1142/S0218202515500207, © copyright World Scientific Publishing Company. http://www.worldscientific.com/doi/abs/10.1142/S0218202515500207