Compensated Convexity on Bounded Domains, Mixed Moreau Envelopes and Computational Methods

Abstract: Compensated convex transforms have been introduced for extended real-valued functions defined over $\mathbb{R}^n$. In their application to image processing, interpolation, and shape interrogation, where one deals with functions defined over a bounded domain, one was making the implicit assumption that the function coincides with its transform at the boundary of the data domain. In this paper, we introduce local compensated convex transforms for functions defined in bounded open convex subsets $\Omega$ of $\mathbb{R}^n$ by making specific extensions of the function to the whole space, and establish their relations to globally defined compensated convex transforms via the mixed critical Moreau envelopes. We find that the compensated convex transforms of such extensions coincide with the local compensated convex transforms in the closure of $\Omega$. We also propose a numerical scheme for computing Moreau envelopes, establishing convergence of the scheme with the rate of convergence depending on the regularity of the original function. We give an estimate of the number of iterations needed for computing the discrete Moreau envelope. We then apply the local compensated convex transforms to image processing and shape interrogation. Our results are compared with those obtained by using schemes based on computing the convex envelope from the original definition of compensated convex transforms.