Phase retrieval for characteristic functions of convex bodies and reconstruction from covariograms

TL;DR

This paper introduces strongly consistent algorithms for reconstructing convex bodies from noisy Fourier modulus measurements, solving the phase retrieval problem for characteristic functions via covariogram data, applicable in multiple dimensions.

Contribution

The paper presents the first complete theoretical algorithms for reconstructing convex bodies from noisy covariogram measurements, addressing the phase retrieval problem for characteristic functions.

Findings

Algorithms are strongly consistent, with Hausdorff distance tending to zero almost surely.

Reconstruction uses O(k^n) measurements, applicable to various convex bodies.

Methods work in all dimensions for symmetric and many arbitrary convex bodies.

Abstract

We propose strongly consistent algorithms for reconstructing the characteristic function 1_K of an unknown convex body K in R^n from possibly noisy measurements of the modulus of its Fourier transform \hat{1_K}. This represents a complete theoretical solution to the Phase Retrieval Problem for characteristic functions of convex bodies. The approach is via the closely related problem of reconstructing K from noisy measurements of its covariogram, the function giving the volume of the intersection of K with its translates. In the many known situations in which the covariogram determines a convex body, up to reflection in the origin and when the position of the body is fixed, our algorithms use O(k^n) noisy covariogram measurements to construct a convex polytope P_k that approximates K or its reflection -K in the origin. (By recent uniqueness results, this applies to all planar convex bodies, all three-dimensional convex polytopes, and all symmetric and most (in the sense of Baire category) arbitrary convex bodies in all dimensions.) Two methods are provided, and both are shown to be strongly consistent, in the sense that, almost surely, the minimum of the Hausdorff distance between P_k and K or -K tends to zero as k tends to infinity.