3D Unwrapped Phase Retrieval with Coded Aperture is Reducible to Projection Tomography

A discrete framework of 3D tomographic phase retrieval with a coded aperture under the Rytov and Born approximations is analyzed. With the introduction of a beam splitter together with coded and uncoded apertures in the measurement, the dataset of diffraction patterns is reducible to that of projections under various measurement uncertainties such as sample heterogeneities, unknown orientations and positions. Without a beam splitter, this data reducibility holds true for {\em random conical tilt} (RCT) and {\em orthogonal tilt} (OT) schemes widely used in cryo-EM if performed with a coded aperture. This approach has the potential of leveraging highly successful projection-based techniques in cryo-EM to process diffraction data collected under uncertainties. The resulting phase unwrapping problem for 3D projection tomography is solved by the proposed sampling schemes including as special cases (i) the conical tilting of range at least $\pi$ at a conical angle { slightly greater than $\pi/4$}, (ii) the orthogonal dual-axis tilting of a tilt range at least $\pi/2$ for each axis and (iii) a combination of a conical tilting of range at least $\pi/2$ at any conical angle $\tau\in (\pi/4, \pi/2]$ and an orthogonal single-axis tilting of a tilt range at least $\tau$.