Real-Space x-ray tomographic reconstruction of randomly oriented objects with sparse data frames

Abstract

Schemes for X-ray imaging single protein molecules using new x-ray sources, like x-ray free electron lasers (XFELs), require processing many frames of data that are obtained by taking temporally short snapshots of 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. One approach to process the data is to use statistical methods developed in the EMC algorithm (Loh & Elser, Phys. Rev. E, 2009) which processes the data set as a whole. In this paper we apply this method to a real-space tomographic reconstruction using sparse frames of data (below 10(-2) photons/pixel/frame) obtained by performing x-ray transmission measurements of a low-contrast, randomly-oriented object. This extends the work by Philipp et al. (Optics Express, 2012) to three dimensions and is one step closer to the single molecule reconstruction problem.

Fig. 1

(a) Photographs of the target object, a plastic toy figure about 50 mm tall. (b) Typical frame of data containing 96 photons in a 396×266 pixel detector. This translates to 9.1 × 10⁻⁴ photons/pixel. (c) Data frame with 1025 photons obtained by combining 10 consecutive frames from the previous data set. The sizes of the pixels recording photons have been enlarged to improve visibility. Pixels with two photons are shown in red.

Fig. 2

(a) 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. (b) Mask representing relevant (green), irrelevant (blue) and ignored (red) pixels (details in Section 3) (c) Histogram of photon counts with a cutoff value for relevant pixels at 17,800 photons.

Fig. 3

Flowchart of EMC reconstruction algorithm applied to this system including the transformations performed in each step.

Fig. 4

The Expand step generates tomograms, W_rz(θ), for many different discrete orientations θ from the 3D Fourier space model M̃_uvw.

Fig. 5

The first four rows show the projected x-ray transmission intensities through the reconstructed object for π/6 rotation intervals in [0, π]. Reconstructions of the object were obtained with data sets of 99, 198, 397, and 992 photons per frame, respectively. The total number of photons in each data set was 1.56 × 10⁹. Details about the data sets are given in Table 1. The bottom row shows for comparison static radiographs of the object which were acquired at the same angles at high signal levels. Some fine, low-contrast features are circled. All images are scaled such that white and black colors represent no and complete attenuation respectively.