An algorithm for reconstructing a convex polygon from its covariogram

The covariogram $g_{K}(x)$ of a convex body $K$ gives the volume of the intersections of $K$ with its translates $K+x$. Matheron conjectured in 1986 that the covariogram determines, up to translations and reflections, a convex body. Recently, Averkov and Bianchi proved Matheron's conjecture for arbitrary planar convex bodies. In this work, the authors give a new algorithm for reconstructing a convex polygon given its covariogram. This algorithm simplifies another one given in [M. Schmitt, in Mathematical morphology in image processing, 151--169, Dekker, New York, 1993].