On the reconstruction of convex sets from random normal measurements

TL;DR

This paper investigates how to reconstruct convex bodies from a limited number of random normal measurements, providing bounds on the number of measurements needed for a desired accuracy with high probability.

Contribution

It introduces a method to estimate the number of random normal measurements required for accurate convex set reconstruction using stability theory related to Minkowski's theorem.

Findings

Derived upper bounds on measurement count for eta-approximation

Established probabilistic guarantees for reconstruction accuracy

Utilized stability theory to connect measurements with shape approximation

Abstract

We study the problem of reconstructing a convex body using only a finite number of measurements of outer normal vectors. More precisely, we suppose that the normal vectors are measured at independent random locations uniformly distributed along the boundary of our convex set. Given a desired Hausdorff error eta, we provide an upper bounds on the number of probes that one has to perform in order to obtain an eta-approximation of this convex set with high probability. Our result rely on the stability theory related to Minkowski's theorem.