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Can x-ray pictures that are almost empty be used to reconstruct physical models?

Many forward-thinking scientists at synchrotron and newer free-electron laser x-ray facilities are hoping that someday scientists will be able to determine the structure of a single molecule using x-ray techniques. Lacking millions of similar molecules periodically spaced on a crystalline lattice, the x-ray signal from a single molecule will be extraordinarily weak. Still, with the advent of imaging detectors that have very low background noise, it still seems feasible that even a few photons scattered from a single molecule should be sufficient. Imaging single molecules is only one extreme example; there are also many more common situations where indexing or analyzing a single image is not possible either because of the nature of the sample (e.g. a non-crystalline specimen that produces no diffraction peaks) or because the number of scattered photons detected in a single frame simply does not provide enough information.

Experimentally, in the case of studying single molecules, a full data set will involve recording and subsequently processing many frames of data obtained by taking short snapshots of many identical molecules, each with a random and unknown orientation. Due to the small size of the molecules and short exposure times, average signal levels of much less than 1 photon/pixel/frame are expected, much too low to be processed using standard methods.

Tomography

(Right) Plastic figure used to generate sparse data set. (Left) X-ray radiographic photographs of the target. Shown are typical frames of data containing 96 photons in a 396x266 pixel detector. (Center) Angle-averaged pattern produced by summing over all 15,650,615 frames in the 99 photons/frame data set. The numbers in the legend refer to photon counts.

Simulating such an experiment, this paper discusses a new approach to process the data using statistical methods called an expand-maximize-compress (EMC) algorithm, which processes a larger data set as a whole [1]. The research groups of Elser and Gruner (Physics, Cornell University) demonstrate this new method by attempting a real-space tomographic reconstruction using sparse frames of data [2]. They create the data set by performing x-ray transmission measurements on a low-contrast, randomly-oriented three-dimensional object using a lab x-ray source (see figure). To test the method for 3D tomographic reconstructions, they collected over 15 million sparse data images from a 5 cm sized object, where each data frame is from a random and unknown orientation. The sparse data frames used for reconstruction have signal levels of 0.001 to 0.01 photons/pixel on a fast pixel-array detector.

The analysis process can be thought of as unknown-angle tomography. Starting with an initial random model for the object density, three operations are applied to generate an updated model for each iteration of the EMC process. Each cycle improves the tomograms from the current model (called the Expand phase), updates the tomograms (Maximize), and combines them to generate the new model (Compress).

The paper discusses systematically how reconstruction quality improves as the number of photons per image grows, but also shows how discernible specimen features can be found even at the lowest photon densities. By using a 3D object with random orientations, this paper extends previous work [3] to three dimensions and is one step closer to solving single molecule structures.

References:

[1] N. T. D. Loh and V. Elser, "Reconstruction algorithm for single-particle diffraction imaging experiments", Phys Rev E 80(2) (2009).
[2] K. Ayyer, H. T. Philipp, M. W. Tate, V. Elser and S. M. Gruner, "Real-Space x-ray tomographic reconstruction of randomly oriented objects with sparse data frames", Opt Express 22(3), 2403-2413 (2014).
[3] H. T. Philipp, K. Ayyer, M. W. Tate, V. Elser and S. M. Gruner, "Solving structure with sparse, randomly-oriented x-ray data", Opt Express 20(12), 13129-13137 (2012).

 

 

Submitted by: Margaret Koker and Ernest Fontes
CHESS, Cornell University
06/13/2014