Compensated Convexity Methods for Approximations and Interpolations of Sampled Functions in Euclidean Spaces: Applications to Contour Lines, Sparse Data and Inpainting

TL;DR

This paper applies compensated convexity methods to surface reconstruction, scattered data approximation, and image inpainting, demonstrating their effectiveness through explicit examples and numerical experiments in various practical scenarios.

Contribution

The paper extends the theory of compensated convex transforms to practical applications like surface reconstruction, scattered data interpolation, and image inpainting with explicit computational examples.

Findings

Effective surface reconstruction from level sets.

Accurate scattered data approximation with natural triangulation.

Successful noise reduction and inpainting in image processing.

Abstract

This paper is concerned with applications of the theory of approximation and interpolation based on compensated convex transforms developed in [K. Zhang, E. Crooks, A. Orlando, Compensated convexity methods for approximations and interpolations of sampled functions in Euclidean spaces: Theoretical Foundations. SIAM Journal on Mathematical Analysis 48 (2016) 4126-4154]. 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 and 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.