Inverse Problems in Propagation-based X-ray Phase Contrast Imaging and Tomography: Stability Analysis and Reconstruction Methods

by Simon Maretzke
Cumulative thesis
Date of Examination:2019-03-04
Date of issue:2019-06-12
Advisor:Prof. Dr. Thorsten Hohage
Referee:Prof. Dr. Thorsten Hohage
Referee:Prof. Dr. Tim Salditt
Referee:Prof. Dr. Russell Luke
Referee:Prof. Dr. Gerlind Plonka-Hoch
Referee:Prof. Dr. Claus Ropers
Referee:Dr. Frank Werner
Persistent Address: http://dx.doi.org/10.53846/goediss-7458

 

 

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Abstract

English

Propagation-based X-ray phase contrast imaging (XPCI) and -tomography (XPCT) extend the capabilities of classical X-ray radiography and computed tomography (CT) to imaging of microscopic specimens with nanometer-sized structures, such as single biological cells. Enabled by ongoing progress in the experimental generation and control of highly coherent and intense X-ray beams, XPCI and XPCT use refraction of X-rays as a contrast-mechanism. While this enables greatly enhanced contrast compared to conventional absorption-based imaging, the inability to measure phases of the X-ray wave-field directly (the phase problem of optics) gives rise to an inverse problem: to obtain an interpretable image, the sample-induced refractive phase shifts have to be numerically reconstructed from the measured near-field intensity patterns (holograms). This thesis provides mathematical analysis of the inverse problems in XPCI and XPCT and proposes practical algorithms for their solution. Firstly, stability of image reconstruction with respect to errors in the hologram data is studied. In the linear contrast regime of weakly interacting samples, the inverse problem of XPCI is shown to be Lipschitz-stable for finitely large, i.e. compactly supported, objects and explicit bounds on the governing stability constant are derived in terms of the geometric setup parameters. The stability results are extended to several settings of interest including XPCT and, to some degree, imaging with nonlinear contrast. Moreover, it is analyzed how stability is affected by the real-world constraint that holograms are acquired by detectors of finite size. It is shown that the resulting truncation of the data gives rise to a sharp resolution limit down to which image features may be stably recovered, in agreement with the numerical aperture of the imaging system. The second, algorithmic part of this work considers regularized Newton-type methods for image reconstruction, which may flexibly exploit available a priori knowledge on the imaged object and also account for nonlinear contrast. Special emphasis is laid on making the devised algorithms computationally efficient enough for processing large-scale experimental data as encountered in XPCT at synchrotron facilities. To this end, Kaczmarz-type schemes are proposed as well as efficient formulas, termed generalized SART, for the computation of the individual Kaczmarz iterates, exploiting the specific mathematical structure of tomographic inverse problems. The capabilities of the novel reconstruction methods in terms of image-quality and computational performance are assessed for different experimental data sets. The theoretical results and numerical algorithms of this thesis may be adapted to a wide range of imaging modalities beyond XPCI and XPCT.
Keywords: X-ray imaging; phase contrast; tomography; in-line holography; image reconstruction; a priori constraints; stability; Kaczmarz-type methods
 
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