The cross covariogram of a pair of polygons determines both polygons, with a few exceptions

Abstract

The cross covariogram g K,L of two convex sets K and L in R n is the function which associates to each x ∈ R n the volume of K ∩ (L + x). Very recently Averkov and Bianchi [AB] have confirmed Matheron's conjecture on the covariogram problem, that asserts that any planar convex body K is determined by the knowledge of g K,K . The problem of determining the sets from their covariogram is relevant in probability, in statistical shape recognition and in the determination of the atomic structure of a quasicrystal from X-ray diffraction images. We prove that when K and L are convex polygons (and also when K and L are planar convex cones) g K,L determines both K and L, up to a described family of exceptions. These results imply that, when K and L are in these classes, the information provided by the cross covariogram is so rich as to determine not only one unknown body, as required by Matheron's conjecture, but two bodies, with a few classified exceptions. These results are also used by Bianchi [Bia] to prove that any convex polytope P in R 3 is determined by g P,P .

Confirmation of Matheron's conjecture on the covariogram of a planar convex body

“…The cross covariogram of two convex bodies K and L in E 2 is the function defined for each x ∈ E 2 by g K,L (x) := V 2 (K ∩ (L + x)). Bianchi [Bia08b] proves that if K and L are convex polygons, then g K,L determines both K and L, with exclusion of a completely described family of exceptions. The family of exceptions is composed of pairs of parallelograms.…”

The covariogram determines three-dimensional convex polytopes

“…Problem 1.3 is completely solved by Bianchi [9], which proves that, up to an affine transformation, the previous counterexamples are the only ones. Theorem 1.4.…”

“…Theorem 1.4. (See [9].) Let K, L be convex polygons and K , L be planar convex bodies with g K,L = g K ,L .…”

On the Reconstruction of Planar Lattice-Convex Sets from the Covariogram

“…Let P be the convex hull of (s + T 1 ) ∪ (s + T 1 ). The point s := w 2 = (k, 1) = s + w 2 lies in (10), (11) and (12), (13), (14) are written in a pictographic style, respectively. At the bottom, the lattice points in the gray area indicate nodes of G ; if a lattice point on the boundary of the hatched area was contained in G S , then G could be extended and would not be maximal.…”

“…It is not hard to show that g K is the autocorrelation of the characteristic function of K. By this, the covariogram problem is a special case of the phase retrieval problem. Recently there has been much progress on the retrieval of K from g K within the class of convex bodies (this version of the covariogram problem is usually called Matheron's problem); see [1,3,7,8,11,12,26].…”