Abstract
Like many other advanced imaging methods, x-ray phase contrast imaging and tomography require mathematical inversion of the observed data to obtain real-space information. While an accurate forward model describing the generally nonlinear image formation from a given object to the observations is often available, explicit inversion formulas are typically not known. Moreover, the measured data might be insufficient for stable image reconstruction, in which case it has to be complemented by suitable a priori information. In this work, regularized Newton methods are presented as a general framework for the solution of such ill-posed nonlinear imaging problems. For a proof of principle, the approach is applied to x-ray phase contrast imaging in the near-field propagation regime. Simultaneous recovery of the phase- and amplitude from a single near-field diffraction pattern without homogeneity constraints is demonstrated for the first time. The presented methods further permit all-at-once phase contrast tomography, i.e. simultaneous phase retrieval and tomographic inversion. We demonstrate the potential of this approach by three-dimensional imaging of a colloidal crystal at 95nm isotropic resolution.
Highlights
Lensless coherent diffractive x-ray imaging (CDI) has opened up a new field of high resolution structure analysis beyond the ensemble averaging of conventional x-ray diffraction [1,2,3]
We have presented iteratively regularized Gauss-Newton methods (IRGNM) as a generic approach to solve nonlinear ill-posed image reconstruction problems
In order to compensate for missing information, the iterates are computed to provide an optimal compromise between the measurements and additional a priori information on the unknown object
Summary
Lensless coherent diffractive x-ray imaging (CDI) has opened up a new field of high resolution structure analysis beyond the ensemble averaging of conventional x-ray diffraction [1,2,3]. We present iteratively regularized Gauss-Newton methods (IRGNM) [21] as an alternative approach to phase retrieval and other imaging problems. In this method, each iterate is computed to provide an optimal compromise between agreement with the measured data and additional constraints on the basis of a local linearization of contrast formation, as we discuss further below. We apply an IRGNM approach to three-dimensional (3d) imaging, i.e. we show how the method can be used to perform phase retrieval and tomographic reconstruction simultaneously This strategy has been argued to enable improved accuracy compared to splitting the reconstruction into phase retrieval problems for each angle to recover the fields in the object plane and a subsequent inversion of the Radon transform [24,25,26,27]. A Kaczmarz-type IRGNM suitable for tomographic imaging is presented in §4 and applied to resolve the structure of a colloidal micro-crystal
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