Uniqueness Theorems for Tomographic Phase Retrieval with Few Diffraction Patterns

TL;DR

This paper establishes the minimum number of diffraction patterns needed for unique 3D tomographic phase retrieval under the Born approximation, demonstrating that $n+1$ patterns suffice for objects on an $n imes n imes n$ grid.

Contribution

It provides the first rigorous proof of the minimal number of diffraction patterns required for unique 3D tomographic phase retrieval under the Born approximation.

Findings

$n$ projections are necessary and sufficient for CT with full measurements

$n+1$ coded diffraction patterns are sufficient for unique phase retrieval

The $n+1$ pattern requirement is nearly optimal for 3D phase retrieval

Abstract

3D tomographic phase retrieval under the Born approximation for discrete objects supported on a $n\times n\times n$ grid is analyzed. It is proved that $n$ projections are sufficient and necessary for unique determination by computed tomography (CT) with full projected field measurements and that $n+1$ coded projected diffraction patterns are sufficient for unique determination, up to a global phase factor, in tomographic phase retrieval. Hence $n+1$ is nearly, if not exactly, the minimum number of diffractions patterns needed for 3D tomographic phase retrieval under the Born approximation.