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This paper is concerned with applications of the theory of approximation andinterpolation based on compensated convex transforms developed in [K. Zhang, E.Crooks, A. Orlando, Compensated convexity methods for approximations andinterpolations of sampled functions in Euclidean spaces: TheoreticalFoundations. SIAM Journal on Mathematical Analysis 48 (2016) 4126-4154]. Weapply our methods to $(i)$ surface reconstruction starting from the knowledgeof finitely many level sets (or `contour lines'); $(ii)$ scattered dataapproximation; $(iii)$ image inpainting. For $(i)$ and $(ii)$ our methods giveinterpolations. For the case of finite sets (scattered data), in particular,our approximations provide a natural triangulation and piecewise affineinterpolation. Prototype examples of explicitly calculated approximations andinpainting results are presented for both finite and compact sets. We also shownumerical experiments for applications of our methods to high density salt andpepper noise reduction in image processing, for image inpainting and forapproximation and interpolations of continuous functions sampled on finitelymany level sets and on scattered points.

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