Projection methods for high numerical aperture phase retrieval
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We develop for the first time a mathematical framework in which the class of projection algorithms can be applied to high numerical aperture (NA) phase retrieval. Within this framework, we first analyze the basic steps of solving the high-NA phase retrieval problem by projection algorithms and establish the closed forms of all the relevant prox-operators. We then study the geometry of the high-NA phase retrieval problem and the obtained results are subsequently used to establish convergence criteria of projection algorithms in the presence of noise. Making use of the vectorial point-spread-function (PSF) is, on the one hand, the key difference between this paper and the literature of phase retrieval mathematics which deals with the scalar PSF. The results of this paper, on the other hand, can be viewed as extensions of those concerning projection methods for low-NA phase retrieval. Importantly, the improved performance of projection methods over the other classes of phase retrieval algorithms in the low-NA setting now also becomes applicable to the high-NA case. This is demonstrated by the accompanying numerical results which show that available solution approaches for high-NA phase retrieval are outperformed by projection methods.Keywords:
Projection method
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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 algorithms3 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.
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We develop for the first time a mathematical framework in which the class of projection algorithms can be applied to high numerical aperture (NA) phase retrieval. Within this framework, we first analyze the basic steps of solving the high-NA phase retrieval problem by projection algorithms and establish the closed forms of all the relevant prox-operators. We then study the geometry of the high-NA phase retrieval problem and the obtained results are subsequently used to establish convergence criteria of projection algorithms in the presence of noise. Making use of the vectorial point-spread-function (PSF) is, on the one hand, the key difference between this paper and the literature of phase retrieval mathematics which deals with the scalar PSF. The results of this paper, on the other hand, can be viewed as extensions of those concerning projection methods for low-NA phase retrieval. Importantly, the improved performance of projection methods over the other classes of phase retrieval algorithms in the low-NA setting now also becomes applicable to the high-NA case. This is demonstrated by the accompanying numerical results which show that available solution approaches for high-NA phase retrieval are outperformed by projection methods.
Projection method
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In this article we introduce line Smoothness-Increasing Accuracy-Conserving Multi-Resolution Analysis. This is a procedure for exploiting convolution kernel post-processors for obtaining more accurate multidimensional multiresolution analysis in terms of error reduction. This filtering-projection tool allows for the transition of data between different resolutions while simultaneously decreasing errors in the fine grid approximation. It specifically allows for defining detail multiwavelet coefficients when translating coarse data onto finer meshes. These coefficients are usually not defined in such cases. We show how to analytically evaluate the resulting convolutions and express the filtered approximation in a new basis. This is done by combining the filtering procedure with projection operators that allow for computational implementation of this scale transition procedure. Further, this procedure can be applied to piecewise constant approximations to functions, as it provides error reduction. We demonstrate the effectiveness of this technique in two and three dimensions.
Multiresolution analysis
Smoothness
Convolution (computer science)
Kernel (algebra)
Basis function
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