As plans move forward toward upgrading synchrotron light sources worldwide, a parallel effort is proceeding without fanfare to ensure that data processing methods are capable of exploiting the new groundbreaking experimental opportunities soon to arrive. As important as designing lightning-fast electronics is, new concepts of data recovery will need to be invented and deployed as well. For example, after the upgrade of the U.S. Department of Energy’s Advanced Photon Source (the APS Upgrade Project) at Argonne National Laboratory is complete, coherent x-ray beams produced by the APS will be as much as 1,000 times brighter, leading to skyrocketing increases in the acquisition rates of high-quality data, which in operando experiments will need to be processed in real time. The impact of these improved beams will be significant in the area of coherent diffractive imaging, where current data processing algorithms can be slow and will likely be overwhelmed by the anticipated flood of new results. Now, theoretical work by two Argonne scientists has defined a new conceptual framework for understanding the informational content in coherent diffraction data that may provide a valuable perspective on recent advances in phase retrieval methodologies, opening the potential for real-time data processing that will be needed to enable operando coherent diffraction microscopy (CDM) even with complex samples and at the upgraded APS’s enhanced data-producing capabilities.
Coherent diffractive imaging involves shining coherent x-rays on a sample, such as a crystal or nanoparticle, and measuring the positions and intensities of the resulting diffracted pattern using area detectors placed in the far field. The intent is to use this data to produce three-dimensional images of the sample, as represented by an electron density map (including irregular morphologies of three-dimensional crystals, defects in the arrangement of atoms in the lattice). Data processing can be slow because the diffraction pattern data is missing needed information about the phases of the diffracted x-rays. Since these phases cannot be directly measured, they are currently “pulled from the air” by applying complex algorithms. At present, these algorithms are the gold standard for coherent data analysis and have enabled dramatic images of nanoscale structures. But they present multiple challenges: they can be slow (requiring thousands of iterations to converge); they can be noisy (i.e., they do not always give exactly the same answer); they can be sensitive to initial assumptions about the unknown sample structure; and they usually require some a priori information about the sample (i.e., the sample size). This horse-and-buggy approach will likely become overloaded when the data-producing capabilities of the upgraded APS enter the picture.
But what if there were a different approach to analyzing the coherent scattering data? What if it were possible to directly image complicated nanoscale samples, using deterministic algorithms capable of yielding structures of assured validity? The two Argonne physicists explored these momentous questions and showed that deterministic direct imaging at the APS is possible for a particular type of sample.
Finding a basis for proving all this initially was a daunting challenge. But there is a saying among theoretical physicists that if the solution of a problem seems easy, it will turn out to be hard. And if it seems hard at first, it will reveal itself to be either darn hard or, at some point, intuitively obvious. The solution became obvious to the researchers as they developed a new perspective on the internal structure of the so-called Patterson function, which is a mathematically transformed (Fourier transform) version of the scattered intensities and a function of the inter-atom separations of the sample under study.
Their analysis of the informational content in coherent x-ray scattering intensities for the case of point-wise structures revealed connections between coherent scattering as described by the Patterson function (which is used to solve the phase problem in x-ray crystallography) and two other areas of study: Fourier holography and graph theory. From Fourier holography, they took the concept of the “hologram,” which is obtained when an object and a point-like reference are illuminated by a coherent x-ray beam. A direct image of the structure then can be obtained by performing a Fourier transform of the hologram. The researchers generalized the concept of the hologram and demonstrated that the Patterson function obtained from coherent illumination of an N-atom structure can be described as an “auto-holographic image” of the unknown structure. That is, it consists of N superimposed images of the unknown structure (“cliques,” in the language of graph theory) that are displaced such that each image has a different atom located at the origin acting as a reference; i.e., the set of coherent scattering intensities is itself a superposition of holograms. This insight was sufficient to uniquely determine unknown structures directly from the Patterson function without the need for error minimization (in the ideal noiseless case).
The researchers illustrated this new understanding by developing a proof-of-principle algorithm that could retrieve an arbitrary unknown structure using only information that could be obtained directly from the measured intensities in the coherent diffraction pattern (Fig. 1), specifically, the Patterson function. The researchers showed that any unknown structure with N-atoms can be revealed uniquely and directly solely from the experimentally measured coherent scattering intensities as long as the features in the Patterson function were sufficiently well-resolved. Or put another way, they demonstrated that there is no conceptual barrier to using coherent scattering as a routine microscopy (i.e., through a direct and deterministic inversion process).
This evaluation suggests that the ability to invert sparse structures may be a direct manifestation of the internal structure of the Patterson function itself, since the Patterson function is not only an auto-hologram but also a graph whose maximal clique corresponds to the unknown structure. The structural information contained in the Patterson function is highly redundant, which suggests that the unknown positions are likely to be significantly overdetermined. This provides physical insight into the success of algorithms like “super resolution phase retrieval,” which are able to retrieve an atomistic structure even in the presence of noise. More broadly, the identification of the Patterson function as a graph consisting with a set of N copies of identical cliques (each with N atoms) may enable the design of machine learning algorithms that can recognize the hidden patterns (e.g., cliques), directly revealing the unknown structure (either by itself, or as an added support constraint to iterative algorithms).
This new framework may also be useful in providing insights into the information content in various standard CDM modalities, such as Bragg coherent diffraction imaging and ptychography. ― Vic Comello
See: Irene Calvo-Almazán and Paul Fenter*, “The Patterson function as auto-hologram and graph enables the direct solution to the phase problem for coherently illuminated atomistic structures,” New J. Phys. 23, 073018 (2021). DOI: 10.1088/1367-2630/ac0d2d
Author affiliation: Argonne National Laboratory
Correspondence: * fenter@anl.gov
This work was supported by U.S. Department of Energy Office of Science-Basic Energy Sciences, Chemical Sciences, Geosciences, and Biosciences Division (Geoscience Research Program) under Contracts DE-AC02-06CH11d357 to UChicago Argonne, LLC as operator of Argonne National Laboratory. Argonne, a U.S. Department of Energy Office of Science laboratory, is operated under Contract No. DE-AC02-06CH11357.
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