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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)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)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)Uniqueness for projection tomography—Unique 

Abstract 3D tomographic phase retrieval under the Born approximation for discrete objects supported on a n × n × n grid is analyzed. It is proved that n projections are sufficient and necessary for unique determination by computed tomography 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. 

Phase retrieval, i.e. the problem of recovering a function from the squared magnitude of its Fourier transform, arises in many applications, such as X-ray crystallography, diffraction imaging, optics, quantum mechanics and astronomy. This problem has confounded engineers, physicists, and mathematicians for many decades. Recently, phase retrieval has seen a resurgence in research activity, ignited by new imaging modalities and novel mathematical concepts. As our scientific experiments produce larger and larger datasets and we aim for faster and faster throughput, it is becoming increasingly important to study the involved numerical algorithms in a systematic and principled manner. Indeed, the past decade has witnessed a surge in the systematic study of computational algorithms for phase retrieval. In this paper we will review these recent advances from a numerical viewpoint. 

The problem of imaging point objects can be formulated as estimation of an unknown atomic measure from its M+1 consecutive noisy Fourier coefficients. The standard resolution of this inverse problem is 1/M and super-resolution refers to the capability of resolving atoms at a higher resolution. When any two atoms are less than 1/M apart, this recovery problem is highly challenging and many existing algorithms either cannot deal with this situation or require restrictive assumptions on the sign of the measure. ESPRIT is an efficient method which does not depend on the sign of the measure. This paper provides an explicit error bound on the support matching distance of ESPRIT in terms of the minimum singular value of Vandermonde matrices. When the support consists of multiple well-separated clumps and noise is sufficiently small, the support error by ESPRIT scales like SRF2λ-2×Noise, where the Super-Resolution Factor (SRF) governs the difficulty of the problem and λ is the cardinality of the largest clump. Our error bound matches the min-max rate of a special model with one clump of closely spaced atoms up to a factor of M in the small noise regime, and therefore establishes the near-optimality of ESPRIT. Our theory is validated by numerical experiments. Keywords: Super-resolution, subspace methods, ESPRIT, stability, uncertainty principle.