Fast algorithms for nonlinear and constrained phase retrieval in near-field X-ray holography based on Tikhonov regularization

Abstract

Based on phase retrieval, lensless coherent imaging and in particular holography offers quantitative phase and amplitude images. This is of particular importance for spectral ranges where suitable lenses are challenging, such as for hard x-rays. Here, we propose a phase retrieval approach for inline x-ray holography based on Tikhonov regularization applied to the full nonlinear forward model of image formation. The approach can be seen as a nonlinear generalization of the well-established contrast transfer function (CTF) reconstruction method. While similar methods have been proposed before, the current work achieves nonlinear, constrained phase retrieval at competitive computation times. We thus enable high-throughput imaging of optically strong objects beyond the scope of CTF. Using different examples of inline holograms obtained from illumination by a x-ray waveguide-source, we demonstrate superior image quality even for samples which do not obey the assumption of a weakly varying phase. Since the presented approach does not rely on linearization, we expect it to be well suited also for other probes such as visible light or electrons, which often exhibit strong phase interaction.

3D imaging of neuronal inclusions and protein aggregates in human neurodegeneration by multiscale X-ray phase-contrast tomography

“…Projections were acquired and saved using the .tiff or .raw format. After flat field and dark image correction, operating in the holographic regime, phase retrieval was performed using either the contrast transfer function (CTF, [53, 54]), the nonlinear Tikhonov (NLT, [55]) or a Paganin-based iterative scheme [56] (see supplementary information). On the phase retrieved projections, ring removal techniques were conducted (only random displacement at SR2-setup) as well as an automatic rotation axis correction.…”

Hierarchically guided in situ nanolaminography for the visualisation of damage nucleation in alloy sheets

“…This algorithm is not applicable to holographic data obtained by X-ray projection microscopy. On the other hand, all published phase-retrieval approaches suited for projection microscopy require time-consuming multi-distance acquisitions 1,25 or iterative procedures 10,26 , which allow compensating for non-measured frequencies and overcoming certain limitations of the underlying physical models. The multidistance approaches 1,25 require time-consuming measurements, the computation of several Fourier transforms and prior image registration.…”

“…The multidistance approaches 1,25 require time-consuming measurements, the computation of several Fourier transforms and prior image registration. Depending on the number of iterations necessary 10,26,27 , the iterative approaches require the computation of ten to several thousands Fourier transforms, even with state-of-the art hardware resulting in typical reconstruction times of not less than 2 s per projection 10 .…”

Micropipette aspiration as a tool for single-particle X-ray imaging and diffraction

“…They are recorded with a focus-to-sample distance of z 01 = 190 mm with a summed illumination time of 40 s. In this case, the droplet was rather large with a diameter of 7 mm, resulting in low transmission of about T = 0.26. The hologram is reconstructed using a Tikhonov reconstruction (Huhn et al, 2022;Lohse et al, 2020) with the parameters = = 0.037, lim1 = 9 Â 10 À3 , lim2 = 10 À1 in Fig. 7(b).…”