Compensated convexity methods for approximations and interpolations of sampled functions in Euclidean spaces: applications to contour lines, sparse data and inpainting

Zhang, Kewei; Crooks, Elaine; Orlando, Antonio

Abstract

This paper is concerned with applications of the theory of approximation and interpolation based on compensated convex transforms developed in [55]. We apply our methods to (i) surface reconstruction starting from the knowledge of finitely many level sets (or ‘contour lines’); (ii) scattered data approximation; (iii) image inpainting. For (i) and (ii) our methods give interpolations. For the case of finite sets (scattered data), in particular, our approximations provide a natural triangulation and piecewise affine interpolation. Prototype examples of explicitly calculated approximations and inpainting results are presented for both finite and compact sets. We also show numerical experiments for applications of our methods to high density salt & pepper noise reduction in image processing, for image inpainting and for approximation and interpolations of continuous functions sampled on finitely many level sets and on scattered points.

Citation

Zhang, K., Crooks, E., & Orlando, A. (2018). Compensated convexity methods for approximations and interpolations of sampled functions in Euclidean spaces: applications to contour lines, sparse data and inpainting. SIAM Journal on Imaging Sciences, 11(4), 2368-2428. doi:10.1137/17M116152X

• Journal Article Type: Article
• Acceptance Date: Aug 23, 2018
• Online Publication Date: Oct 23, 2018
• Publication Date: Oct 23, 2018
• Deposit Date: Aug 24, 2018
• Publicly Available Date: Aug 24, 2018
• Electronic ISSN: 1936-4954
• Publisher: Society for Industrial and Applied Mathematics
• Peer Reviewed: Peer Reviewed
• Volume: 11
• Issue: 4
• Pages: 2368-2428
• DOI: https://doi.org/10.1137/17M116152X
• Public URL: https://nottingham-repository.worktribe.com/output/1047226
• Publisher URL: https://epubs.siam.org/doi/10.1137/17M116152X