Data preparation and evaluation techniques for x-ray diffraction microscopy
Jan SteinbrenerJohanna Nelson WekerXiaojing HuangStefano MarchesiniDavid A. ShapiroJoshua J. TurnerChris Jacobsen
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Abstract:
The post-experiment processing of X-ray Diffraction Microscopy data is often time-consuming and difficult. This is mostly due to the fact that even if a preliminary result has been reconstructed, there is no definitive answer as to whether or not a better result with more consistently retrieved phases can still be obtained. We show here that the first step in data analysis, the assembly of two-dimensional diffraction patterns from a large set of raw diffraction data, is crucial to obtaining reconstructions of highest possible consistency. We have developed software that automates this process and results in consistently accurate diffraction patterns. We have furthermore derived some criteria of validity for a tool commonly used to assess the consistency of reconstructions, the phase retrieval transfer function, and suggest a modified version that has improved utility for judging reconstruction quality.Keywords:
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This paper expands the linear iterative near-field phase retrieval (LIPR) formalism to achieve quantitative material thickness decomposition. Propagation-based phase contrast x-ray imaging with subsequent phase retrieval has been shown to improve the signal-to-noise ratio (SNR) by factors of up to hundreds compared to conventional x-ray imaging. This is a key step in biomedical imaging, where radiation exposure must be kept low without compromising the SNR. However, for a satisfactory phase retrieval from a single measurement, assumptions must be made about the object investigated. To avoid such assumptions, we use two measurements collected at the same propagation distance but at different x-ray energies. Phase retrieval is then performed by incorporating the Alvarez-Macovski (AM) model, which models the x-ray interactions as being comprised of distinct photoelectric and Compton scattering components. We present the first application of dual-energy phase retrieval with the AM model to monochromatic experimental x-ray projections at two different energies for obtaining split x-ray interactions. Our phase retrieval method allows us to separate the object investigated into the projected thicknesses of two known materials. Our phase retrieval output leads to no visible loss in spatial resolution while the SNR improves by factors of 2 to 10. This corresponds to a possible x-ray dose reduction by a factor of 4 to 100, under the Poisson noise assumption.
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X-Ray Phase-Contrast Imaging
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In 2006, Balan/Casazza/Edidin \cite{BCE} introduced the frame theoretic study of phaseless reconstruction. Since then, this has turned into a very active area of research. Over the years, many people have replaced the term {\it phaseless reconstruction} with {\it phase retrieval}. Casazza then asked: {\it Are these really the same?} In this paper, we will show that phase retrieval is equivalent to phaseless reconstruction. We then show, more generally, that phase retrieval by projections is equivalent to phaseless reconstruction by projections. Finally, we study {\it weak phase retrieval} and discover that it is very different from phaseless reconstruction.
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In 2006, Balan/Casazza/Edidin \cite{BCE} introduced the frame theoretic study of phaseless reconstruction. Since then, this has turned into a very active area of research. Over the years, many people have replaced the term {\it phaseless reconstruction} with {\it phase retrieval}. Casazza then asked: {\it Are these really the same?} In this paper, we will show that phase retrieval is equivalent to phaseless reconstruction. We then show, more generally, that phase retrieval by projections is equivalent to phaseless reconstruction by projections. Finally, we study {\it weak phase retrieval} and discover that it is very different from phaseless reconstruction.
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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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Intensity
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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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X-Ray Phase-Contrast Imaging
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An improved iterative phase retrieval algorithm is put forward,which makes use of intensity information in several planes,and divides the retrieval process into two steps: profile retrieval and detail retrieval.First the input plane and other three far diffraction planes are needed to retrieve the profile of phase distribution,then the phase detail is available in a similar way.The phase distribution of any one dimensional optical field,including very complicated ones,can be successfully retrieved with this algorithm,and the accuracy of retrieval is greatly enhanced.This algorithm is also very robust.It has convergence independent on the initial phase,and high stability towards additive noise,which are both confirmed by simulations.
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A novel two-dimensional phase unwrapping method is proposed.In this idea,the phase wrapped is propagated through an optical diffraction calculation as an input optical field and transformed into the intensity distribution.Then the phase retrieval algorithm is engaged in order to find the best fitting Zernike polynomials for the unwrapped phase based on the diffraction intensity.Because of the maturity of phase retrieval theory,the uniqueness and accuracy of phase unwrapping can be guaranteed.As a result,the phase noise can be remarkably reduced by intensity filtering.The validity of this algorithm is verified by experiment.
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Phase Unwrapping
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