Regularized Newton methods for x-ray phase contrast and general imaging problems

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.

Phase retrieval for near-field X-ray imaging beyond linearisation or compact support

“…34 However, the reconstruction obtained by RAAR did not yield a better result at the same number of iterations. We have also tested an iteratively regularized Gauss Newton method 35 using the same set of constraints, which also yielded good reconstructions, but at same or higher cost of computations, as detailed in the supplementary material. The challenge for the future improvement thus lies in finding an optimal preconditioner for iterative schemes, i.e., an appropriate starting guess.…”

Three-dimensional single-cell imaging with X-ray waveguides in the holographic regime

“…As a rather new approach for iterative phase retrieval, we also use the iteratively regularized Gauss-Newton (IRGN) method in this work. The IRGN approach differs from the widespread alternating-projection-type algorithms in that it exploits differentiability and simultaneously processes constraints and observed data, resulting in improved convergence (Maretzke et al, 2016). Mathematically, IRGN is a Tikhonov regularized version of a Newton-type iterative solution,…”

“…Note that the linearization is local with respect to the current iterate f k and thereby better justified than static linearization as in the CTF approach. In contrast to Maretzke et al (2016), where IRGN was used for single-distance recordings, we have implemented it here also for multiple-distance data sets, in order to provide a valid comparison with CTF phase retrieval and holo-TIE (Krenkel et al, 2013). Similar to CTF, holo-TIE is a deterministic inversion based on a multiple-distance data set, treated by Fourier methods, but without linearization of the object's optical constants (Krenkel et al, 2013).…”

“…The demonstrated progress in image quality has been obtained via improvements on different levels, starting from the waveguide optics, the alignment and imaging processing procedure, the recording and detection scheme, and finally the reconstruction. To this end, we first provide a thorough study of phaseretrieval techniques for the holographic regime, including a recent approach based on an iterative regularized GaussNewton method (Maretzke et al, 2016), and also a novel variant of the so-called holo-TIE reconstruction (Krenkel et al, 2013). Importantly, this holo-TIE approach put forward here for tomography of single cells does not rely on assumptions on material composition nor on linearization of the specimen's optical constants, as is typically the case in conventional non-iterative reconstruction schemes.…”

Fast three-dimensional phase retrieval in propagation-based X-ray tomography

“…Multi-material phase retrieval has also been attempted by the application of a 3D correction filter operation applied to the reconstructed volume in addition to conventional 2D TIE-Hom (Ullherr & Zabler, 2015). Also, other non-TIE based methods of 3D phase retrieval have been developed (Vassholz et al, 2016, Ruhlandt et al, 2014, Maretzke et al, 2016. For clarity our proposed method will be referred to as post-reconstruction 3D TIE-Hom (PostTIE-Hom3D) for which the derivation is given in section 2.…”