The phase retrieval problem arises in various fields ranging from physics and astronomy to biology and microscopy. Computational reconstruction of the Fourier phase from a single diffraction pattern is typically achieved using iterative alternating projections algorithms imposing a non-convex computational challenge. A different approach is holography, relying on a known reference field. Here we present a conceptually new approach for the reconstruction of two (or more) sufficiently separated objects. In our approach we combine the constraint the objects are finite as well as the information in the interference between them to construct an overdetermined set of linear equations. We show that this set of equations is guaranteed to yield the correct solution almost always and that it can be solved efficiently by standard numerical algebra tools. Essentially, our method combine commonly used constraint (that the object is finite) with a holographic approach (interference information). It differs from holographic methods in the fact that a known reference field is not required, instead the unknown objects serve as reference to one another (hence blind holography). Our method can be applied in a single-shot for two (or more) separated objects or with several measurements with a single object. It can benefit phase imaging techniques such as Fourier phytography microscopy, as well as coherent diffractive X-ray imaging in which the generation of a well-characterized, high resolution reference beam imposes a major challenge. We demonstrate our method experimentally both in the optical domain and in the X-ray domain using XFEL pulses.
Ben Leshem, Rui Xu, Jianwei Miao, Boaz Nadler, Dan Oron, Nirit Dudovich, and Oren Raz, "Direct single-shot phase retrieval for separated objects
(Conference Presentation)," Proc. SPIE 9718, Quantitative Phase Imaging II, 97182S (Presented at SPIE BiOS: February 17, 2016; Published: 27 April 2016); https://doi.org/10.1117/12.2217563.4848767859001.
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