A comparison of ptychographic phase retrieval algorithms

Ptychography, a relatively new form of phase retrieval, can reconstruct both intensity and phase images of a sample from a group of diffraction patterns, which are recorded as the sample is translated through a grid of positions¹. To recover the phase information lost in the recording of these diffraction patterns, iterative algorithms must optimise an objective function full of local minima, in a huge multidimensional space². Many such algorithms have been developed, each aiming to converge rapidly whilst avoiding stagnation. Our research aims to compare some of the more popular algorithms, to determine their advantages and disadvantages under a range of different conditions, and hence to suggest guidelines for choosing suitable algorithms for any given data set. We have implemented and tested several well-known phase retrieval algorithms: the ‘PIE’ family of algorithms^{3 4}, the difference map (DM)^{5 6}, hybrid projection/reflection (HPR)⁷ and relaxed averaged alternating reflections (RAAR)⁸. The PIE-type algorithms are based on the stochastic gradient descent concept⁴, whilst the rests are based on the set projection and reflection concept², and hence named the ‘PR’ algorithms in this paper. We began our tests by tuning algorithm parameters using multiple sets of simulated calibration data. Then, these tuned algorithms were tested on simulated data generated from a range of scenarios using either a randomised illumination function or convergent beam illumination, combined with either a weakly- or a strongly-scattering sample. Because ptychographic reconstructions are subject to certain pathological ambiguities, we then used an ambiguity-invariant error measure to evaluate the differences between the resulting images⁴.