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

Abstract:

We propose strongly consistent algorithms for reconstructing the characteristic function $1_K$ of an unknown convex body $K$ in $\mathbb {R}^n$ from possibly noisy measurements of the modulus of its Fourier transform $\widehat {1_K}$. 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 $\pm K$ tends to zero as $k$ tends to infinity.