3D Tomographic Phase Retrieval and Unwrapping

TL;DR

This paper establishes theoretical uniqueness results for 3D phase retrieval and unwrapping using various measurement schemes, enabling object reconstruction without prior orientation knowledge.

Contribution

It introduces new uniqueness theorems for 3D phase retrieval and unwrapping under diverse measurement schemes, including coded apertures and random tilts.

Findings

Unique determination of phase projections from diffraction patterns.

Conditions for 3D phase unwrapping from phase projection data.

Proof of unique object reconstruction from limited projections or diffraction patterns.

Abstract

This paper develops uniqueness theory for 3D phase retrieval with finite, discrete measurement data for strong phase objects and weak phase objects, including: (i) {\em Unique determination of (phase) projections from diffraction patterns} -- General measurement schemes with coded and uncoded apertures are proposed and shown to ensure unique reduction of diffraction patterns to the phase projection for a strong phase object (respectively, the projection for a weak phase object) in each direction separately without the knowledge of relative orientations and locations. (ii) {\em Uniqueness for 3D phase unwrapping} -- General conditions for unique determination of a 3D strong phase object from its phase projection data are established, including, but not limited to, random tilt schemes densely sampled from a spherical triangle of vertexes in three orthogonal directions and other deterministic tilt schemes. (iii) {\em Uniqueness for projection tomography} -- Unique determination of an object of $n^3$ voxels from generic $n$ projections or $n+1$ coded diffraction patterns is proved. This approach of reducing 3D phase retrieval to the problem of (phase) projection tomography has the practical implication of enabling classification and alignment, when relative orientations are unknown, to be carried out in terms of (phase) projections, instead of diffraction patterns. The applications with the measurement schemes such as single-axis tilt, conical tilt, dual-axis tilt, random conical tilt and general random tilt are discussed.