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 conversion of diffraction patterns into 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 determination of an object of $n^3$ voxels from generic $n$ projections or $n+1$ coded diffraction patterns is proved. This approach 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：本文针对有限离散测量数据下的三维相位重建问题，提出了强相位物体与弱相位物体的唯一性理论，包括：(i) 从衍射图样中唯一确定相位投影——提出了带编码与无编码孔径的通用测量方案，并证明在该方案下，无需已知相对取向和位置信息，即可确保每个方向上独立地将衍射图样唯一转换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通用条件，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) 投影断层成像的唯一性——证明从通用n个投影或n+1个编码衍射图样中可唯一确定包含$n^3$体素的物体。该方法具有实际意义，即在相对取向未知时，可基于相位投影而非衍射图样进行分类与对齐。本文还讨论了单轴倾斜、锥形倾斜、双轴倾斜、随机锥形倾斜及通用随机倾斜等测量方案的应用。