Reconstruction of convex bodies from moments

TL;DR

This paper explores how convex bodies can be reconstructed from a finite set of geometric or Legendre moments, providing conditions for uniqueness, stability results, and a reconstruction algorithm that handles noisy data.

Contribution

It introduces a sufficient condition for unique determination of convex bodies from finite moments, and develops a stable reconstruction algorithm using Legendre moments.

Findings

Convex bodies can be uniquely determined by finite geometric moments under certain conditions.

Using Legendre moments improves stability in reconstructing convex bodies.

The proposed algorithm is consistent even with noisy moment measurements.

Abstract

We investigate how much information about a convex body can be retrieved from a finite number of its geometric moments. We give a sufficient condition for a convex body to be uniquely determined by a finite number of its geometric moments, and we show that among all convex bodies, those which are uniquely determined by a finite number of moments form a dense set. Further, we derive a stability result for convex bodies based on geometric moments. It turns out that the stability result is improved considerably by using another set of moments, namely Legendre moments. We present a reconstruction algorithm that approximates a convex body using a finite number of its Legendre moments. The consistency of the algorithm is established using the stability result for Legendre moments. When only noisy measurements of Legendre moments are available, the consistency of the algorithm is established under certain assumptions on the variance of the noise variables.