When x-rays penetrate soft matter, their phase changes more rapidly than their amplitude. Interference effects
visible with high brightness sources creates higher contrast, edge enhanced images. When the object is piecewise
smooth (made of big blocks of a few components), such higher contrast datasets have a sparse solution. We
apply basis pursuit solvers to improve SNR, remove ring artifacts, reduce the number of views and radiation dose
from phase contrast datasets collected at the Hard X-Ray Micro Tomography Beamline at the Advanced Light
Source. We report a GPU code for the most computationally intensive task, the gridding and inverse gridding
algorithm (non uniform sampled Fourier transform).
The latest development of ultrafast free electron laser makes it now possible to perform single molecule diffraction
imaging. In such an experiment, two-dimensional (2D) diffraction images of randomly oriented molecules of the
same type (single molecules) can be captured within femtosecond exposure time. These images can then be
used to deduce the 3D structure of the molecule. Two of the most challenging problems that must be solved in
order to obtain a high resolution 3D reconstruction are: 1) the determination of the relative orientations of 2D
diffraction images; 2) the retrieval of the phase information of a reconstructed 3D diffraction pattern. In this
paper, we will focus on the first problem and discuss the use of common curve detection techniques to deduce
the relative orientations of 2D diffraction images produced from single-molecule diffraction experiments. Such a
technique is based on the fact that Ewald spheres associated with two diffraction images of the same molecule
intersect along a common curve in the reciprocal space. By detecting these curves on each diffraction image, we
can deduce the relative orientations of diffraction images by solving an eigenvalue problem. When the radius of
the Ewald sphere is sufficiently large relatively to the region of reciprocal space we are interested in, the Ewald
sphere becomes flat near the origin of the reciprocal space, and common curves reduce to common lines. In
this case, the orientation determination problem is similar to the one that arises in single particle cryo-electron
microscopy. The recent work of Singer and Shkolnisky [1] shows that the orientation determination problem
can be solved by computing the largest eigenvalues of a symmetric matrix constructed from the common lines
identified among cryo-EM projection images. In this paper, we will extend their technique to diffraction images
on which common curves can be identified.
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