An Analytic Method of Phase Retrieval for X-Ray Phase Contrast Imaging
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(2020) 26:79
An Analytic Method of Phase Retrieval for X-Ray Phase Contrast Imaging Victor Palamodov1 Received: 10 June 2019 / Revised: 13 April 2020 / Accepted: 14 April 2020 © Springer Science+Business Media, LLC, part of Springer Nature 2020
Abstract New lensless diffractive X-ray technic for micro-scale imaging of biological tissue is based on quantitative information on the phase. This method yields improved contrast compared to purely absorption-based tomography but involves a phase retrieval problem since of physical limitation of detectors. An analytic method is proposed in the paper for reconstruction of the ray projection of complex refraction index from intensity distribution of one hologram. Keywords Refractive index · Phase contrast imaging · Fresnel propagator · Sampling · Interpolation Mathematics Subject Classification 78A45 · 78A46
1 Introduction A sample is illuminated by a parallel beam of coherent X-rays; the intensity of the diffracted pattern (hologram) is registered at a plane detector for determination of distribution of attenuation and refractivity of the object. This type of imaging is implemented at third generation synchrotron radiation sources due to their high degree of coherence. This method guarantees improved contrast compared to purely absorptionbased radiography but involves phase retrieval problem. The linearized version of this problem is known as the contrast transfer function model (CTF [9]). For narrow beams the Helmholtz equation is replaced by the paraxial approximation. Pogany et al. [12], Cloetens et al. [4], Paganin et al. [10], Gureyev et al. [5] applied reconstruction algorithms for medium with zero absorption by replacing the Fresnel propagator by its
Communicated by Todd Quinto.
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Victor Palamodov [email protected] Tel Aviv University, Tel Aviv, Israel 0123456789().: V,-vol
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Journal of Fourier Analysis and Applications
(2020) 26:79
quadratic term in the frequency plane (the dashed parabola on Fig. 1). Bronnikov’s method [1], [2] is based on Radon’s and Lorentz’s inversion formulas. See Burvall et al. [3] for similar algorithms. Methods of complex analysis are applied in for stating uniqueness [6] and stability of inversion [7]. The exact estimate of the norm of the inversion shows exponential growth for large Fresnel numbers. In this paper an analytic reconstruction method of arbitrary refraction index is proposed. Our method is exact in frame of CTF model for optically weak objects and requires measurements of intensity of a single hologram. The key point is interpolation in the Fourier domain from a nonuniform sampling of data. The asymptotic density of the sampling is minimal for any natural Fresnel number which depends on size of the support of the object. The interpolation operator looks as the classical WhitteckerShannon-Kotelnikov interpolation. An earlier version is given in [11].
2 Setup and Linearization A three-dimensional object of interest is subjected by a plane wave beam of perfectly coherent radiation of high frequ
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