Researchers use ‘complexity’ as a guiding tool to make phase retrieval easier for coherent X-ray imaging

Since their discovery by Roentgen in 1895, X-rays have become an effective tool for doctors to create images of their patients’ bones, organs, and vessels and to make better diagnoses. Scientists have continued this quest to obtain high-resolution images of objects of interest. Only this time, with advanced imaging hardware at hand, they’re looking at micro- and nano-sized objects in structural biology, materials science, chemistry and medical science as they try to determine their structures.

Coherent X-ray imaging (CXI), a technique largely enabled by X-ray free electron lasers (XFELs), is applicable to such structural analysis. But reconstructing an image from incomplete, noisy Fourier intensity data obtained by a typical CXI experiment, retrieving the undetectable phase information, is not trivial. It is addressed by phase retrieval algorithms – preferably in an iterative manner.

In other words, iterations, running the algorithms hundreds of times, are expected to provide more stable and reliable solutions when a problem cannot be solved immediately. But sometimes the quality of solutions does not improve even after a large number of iterations, resulting in the so-called stagnation problems.

To break this deadlock, researchers at the Indian Institute of Technology Delhi have been working since 2018 on a new approach with a complexity parameter to guide the phase retrieval algorithms. On October 10, they published their latest research on the subject Intelligent computers.

“In recent works, we have proposed a new approach that we call ‘complexity-guided phase retrieval’ (CGPR) that is intended to address the typical stagnation problems with phase retrieval algorithms,” the researchers said. “This methodology uses a complexity parameter calculated directly from the Fourier intensity data and provides a measure of fluctuations in the desired phase retrieval solution.”

In their previous research, the researchers developed the CGPR methodology mainly with simulation noisy data in combination with Fienup’s hybrid input-output (HIO). algorithma well-known phase retrieval algorithm.

“In this paper, we try to understand the nature of the phase retrieval solution from a new perspective of the complexity parameter,” the researchers said. For the first time, the researchers tested their idea of ​​complexity guidance on experimental data available in the Coherent X-ray Imaging Data Bank (CXIDB) database, used in conjunction with the relaxed mean alternating reflection (RAR) phase retrieval algorithm, another well-known algorithm popular in the CXI community.

CXIDB is a great initiative that provides access to raw coherent X-ray diffraction data that can currently be incorporated in only a few synchrotron facilities worldwide. The ready availability of this data allows researchers around the world to design and test newer phase retrieval algorithms.

The research team started by observing the complexity behavior of the iterative solutions obtained using the popular RAAR-ER methodology, which combines a larger number of RAAR iterations followed by a smaller number of error reduction iterations (ER). The quality of the recovered solutions and their resolution was assessed by Phase Retrieval Transfer Function (PRTF) evaluation.

They observed both the single run of the RAAR-ER algorithm and the averaged solutions, as the latter – hundreds of trial solutions obtained first from random initial guesses and then averaged with phase adjustment – are considered more reliable than the former.

And both types of solutions, they found, consisted of unwanted grainy artifacts that had a smaller feature size compared to PRTF’s estimated resolution and were therefore considered “false.” This inconsistency was the reason for the researchers to add the complexity-guided component to the RAAR algorithm and present the so-called complexity-guided RAAR algorithm (CG-RAR).

CG-RAAAR was first tested with simulated noise data (with two noise levels) with no missing pixels, then applied to the real cyanobacterium diffraction data (noise, with missing pixels) from the CXIDB database for further validation.

“It is worth emphasizing that the single run of CG-RAR yields solutions with far fewer artifacts, and as a result the number of trial solutions required for the averaging process using this methodology is less than half the number required for the traditional RAAR-ER method,” the researchers noted. Meanwhile, the CG-RAAAR solution had the smallest features consistent with PRTF estimated resolution.

According to the researchers, the main idea behind complexity guidance is to match the complexity of the RAAR solution with the desired ground-truth complexity. “CG-RAAAR essentially provides a regularized solution that does not contain false, granular features. The regularization is controlled in this methodology through the complexity parameter, which makes the solution consistent with the data,” they added.

In conclusion, the concept of complexity guidance, when combined with traditional phase retrieval algorithms such as HIO and RAAR, can provide better noise-robust estimation of the object. “We believe that complexity guidance as an idea could potentially be integrated into existing software tools and improve the performance of existing phasing algorithms in coherent X-ray imaging,” the researchers conclude.

Provided by Intelligent Computing