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Phase retrieval theoretical development

Phase retrieval theoretical development

Ralf Hofmann
Julian Moosmann
Martin Köhl
Andrey Minkevich

Phase retrieval theoretical development

Phase MapPhase Map
Figure 1: Left: simulated phase map with moderately large phase shifts
ranging up to unity. Right: computed intensity map due to free-space
propagation of the pure-phase map (left) at an energy of 12 keV and a
distance of 0.5 m. The intensity is also subject to Poisson noise.

Algorithms were developed for coherent X-ray imaging based on the free-space Fresnel propagation to extend the two standard approaches for the phase-retrieval problem of  ure-phase object from a single-distance intensity map beyond the regime of their validity. The first conventional approach, based on a linearized version of the transport-of-intensity equation (TIE), is valid in the edge-enhancement regime. In this regime (Figure 1), the recorded intensity pattern is proportional to the Laplacian of the phase shift exiting the object. For large distances and/or large phase shifts, where the algorithm starts to fail, we devised a method to go beyond this limitation. The latter is an expansion of the TIE in powers of the object-detector distance z, where nonlinear corrections to the linearized TIE are evaluated perturbatively in terms of the linear-TIE estimate. The second conventional approach, based on the contrast transfer- function (CTF), is valid only for small relative phase shifts (compare Figure 2, left). If one likes to scan entire biological samples, the CTF approach is not applicable anymore due to the occurrence of large phase variations. The TIE approach still works but exhibits less resolution since high frequencies are suppressed in this method. For moderately large phase variations this limitation can be circumvented via a nonperturbative projection onto the CTF model employing an effective phase in Fourier space (compare Figure 2, right). This effective phase obeys a modified CTF relation (denoted projected CTF) between intensity contrast at z>0 and phase contrast at z=0: Unphysical singularities are cut off to yield “quasiparticles” in analogy to the theory of a Fermi liquid (which includes nonlocal effects that arise beyond linearity). Projected CTF was theoretically investigated and justified in terms of its statistical-model behavior, and it was successfully applied to real data.