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.

翻译：本文发展了针对强相位物体与弱相位物体的有限离散测量数据下三维相位恢复的唯一性理论，包括：(i) 从衍射图样唯一确定（相位）投影——提出了带编码与未编码孔径的通用测量方案，并证明这些方案能确保在不依赖于相对取向和位置的情况下，分别从每个方向独立地将衍射图样唯一还原为强相位物体的相位投影（或弱相位物体的投影）；(ii) 三维相位解包裹的唯一性——建立了从相位投影数据唯一确定三维强相位物体的一般条件，涵盖但不限于从三个正交方向顶点构成的球面三角形中密集采样的随机倾斜方案及其他确定性倾斜方案；(iii) 投影层析的唯一性——证明了从n个通用投影或n+1个编码衍射图样唯一确定包含n³体素的物体。这种将三维相位恢复简化为（相位）投影层析问题的方法具有实际意义，当相对取向未知时，可基于（相位）投影而非衍射图样实现分类与配准。文中讨论了单轴倾斜、圆锥倾斜、双轴倾斜、随机圆锥倾斜及通用随机倾斜等测量方案的应用。