Researchers use ‘complexity’ as a guiding tool to aid phase recovery for coherent X-ray imaging

Reconstruction from noisy diffraction data using RAAR-ER and CG-RAAR algorithms. (a) Ground-truth amplitude for the red blood cell (RBC) object (phase not shown). Simulated noisy oversampled Fourier intensity data of RBC in (a) corresponding to average light levels of (b) 1000 photons/pixel and (c) 500 photons/pixel. The data is displayed as |G|^0.1 for display purposes. Solutions recovered after (d,e) 1000 iterations of RAAR alone and (f,g) after concluding the solution in (d) and (e) by 100 ER steps and (h,i) 1000 CG-RAAR iterations from the noisy data shown in (b) and (c), respectively. For both cases of noisy data, the CG-RAAR solution can be clearly considered to be artifact-free. All solutions are displayed in the colorbar range [0,1]. Credit: Intelligent Computing (2022). DOI: 10.34133/2022/9819716

Since their discovery by Roentgen in 1895, X-rays have become an effective tool for doctors to take images of their patients’ bones, organs and vessels and make better diagnoses. Scientists continued this quest to acquire high-resolution images of interesting objects. Only this time, with highly advanced imaging equipment at their fingertips, they are examining micro and nano-sized objects in the fields of structural biology, materials science, chemistry and medical sciences while trying determine their structures.

Coherent X-ray imaging (CXI), a technique widely enabled by X-ray free-electron lasers (XFEL), applies to such structural analysis. But reconstructing an image from incomplete and noisy Fourier intensity data obtained by a typical CXI experiment, which consists of recovering the undetectable phase information, is not trivial. It is processed by phase recovery algorithms – preferably, iteratively.

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

To break this impasse, researchers from the Indian Institute of Technology in Delhi have been working since 2018 and have proposed a new approach involving a complexity parameter to guide phase recovery algorithms. On October 10, they published their latest research on this topic in Intelligent Computing.

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

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

“In this paper, our effort is to understand the nature of the phase recovery solution from a new perspective of the complexity parameter,” the researchers said. For the first time, the researchers put their idea of ​​complexity orientation to the test against experimental data available in the Coherent X-ray Imaging Data Bank (CXIDB), used with the phase recovery algorithm Relaxed Mean Alternate Reflection (RAAR), another well-known algorithm popular in the CXI community.

CXIDB is an excellent initiative that provides access to consistent raw X-ray diffraction data that can currently be recorded at a few synchrotron facilities around the world. The immediate availability of this data allows researchers around the world to design and test new phase recovery algorithms.

The research team began by observing the complexity behavior of iterative solutions obtained using the popular RAAR-ER methodology, which is a combination of a larger number of RAAR iterations followed by a smaller number error reduction (ER) iterations. The quality of the recovered solutions and their resolution were assessed by evaluating the phase recovery transfer function (PRTF).

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

And both types of solutions, as they discovered, consisted of unwanted grainy artifacts that had a smaller feature size compared to the resolution estimated by the PRTF and therefore considered “false”. It was this inconsistency that prompted researchers to add the complexity-guiding component to the RAAR algorithm and introduce the so-called complexity-guided RAAR (CG-RAAR) algorithm.

CG-RAAR was first tested with simulated noisy data (with two levels of noise) that had no missing pixels, and then applied to real cyanobacteria diffraction data (noisy, with missing pixels) from the base CXIDB data for further validation.

“It should be emphasized that the single run of CG-RAAR produces solutions with greatly reduced artifacts and therefore the number of trial solutions required for the averaging process with this methodology is lower than the half the number needed for the traditional RAAR-ER method,” the researchers observed. Meanwhile, the CG-RAAR solution had the smallest features compatible with the resolution estimated by the PRTF.

According to the researchers, the main idea behind complexity orientation is to match the complexity of the RAAR solution with the complexity of the desired ground truth. “CG-RAAR essentially provides a regularized solution that does not contain false grainy features. Regularization is controlled in this methodology by means of the complexity parameter, thus making the solution consistent with the data,” they added.

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

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