  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.
