The Solution of the Covariogram Problem for Plane $\mathcal{C}^2_+$ Convex Bodies

Abstract

We prove that the geometric covariogram determines (up to translation and reflection), among all convex bodies, any plane convex body which is C 2 and has positive curvature everywhere. This gives a partial answer to a problem posed by G. Matheron.

Sums, Projections, and Sections of Lattice Sets, and the Discrete Covariogram

“…The answer is affirmative when n = 2 and K is a polygon (a result of W. Nagel; see [2] for a complete proof) or when K is C 2 + (see [3] and [4]), and is negative when n ≥ 4 (proved by Bianchi [3]). Applications of the covariogram to stereology, image processing, and mathematical morphology are discussed in [6], [27], and [28].…”

Retrieving convex bodies from restricted covariogram functions

“…Here ∂ + /∂r stands for right derivative. Theorem 6.2 from [BSV02] is another result related to Theorem 6, which states that most convex planar bodies are determined by the covariogram function over its entire support. In Section 3 we prove Theorem 5.…”

“…Thus, since K is strictly convex, we get that H is strictly convex, as well. It can be seen that H belongs to C 1 by examining the asymptotic behaviour of g H (x) (restricted to DH) at boundary points of H (see [BSV02,p.190]). Further on, in order to get that H is from the class C 2 + we can argue in the same way as in the proof of Lemma 6.2 from [Bia05a], where the equality of covariograms is used only at points close to their support.…”

“…The following lemma is a strengthening of Proposition 5.1 from [BSV02]. It states that C 2 + regularity of a planar convex body K can be recognized from the covariogram of K over every neighbourhood of bd DK.…”

On Set Covariance and Three-Point Test Sets

“…This yields a partial answer in the planar smooth case to the question of determination of a convex body by its set covariance up to shifts and central reflection, see also Bianchi et al [1].…”