Expansion-maximization-compression algorithm with spherical harmonics for single particle imaging with x-ray lasers

Abstract

In 3D single particle imaging with X-ray free-electron lasers, particle orientation is not recorded during measurement but is instead recovered as a necessary step in the reconstruction of a 3D image from the diffraction data. Here we use harmonic analysis on the sphere to cleanly separate the angular and radial degrees of freedom of this problem, providing new opportunities to efficiently use data and computational resources. We develop the Expansion-Maximization-Compression algorithm into a shell-by-shell approach and implement an angular bandwidth limit that can be gradually raised during the reconstruction. We study the minimum number of patterns and minimum rotation sampling required for a desired angular and radial resolution. These extensions provide new avenues to improve computational efficiency and speed of convergence, which are critically important considering the very large datasets expected from experiment.

Reconstruction from limited single-particle diffraction data via simultaneous determination of state, orientation, intensity, and phase

“…Classification work includes manifold mapping (6), spectral clustering (7), principal component analysis, and support vector machines (8). Orientation methods include common curve approaches (9)(10)(11)(12), expectation maximization (13)(14)(15), and manifold embedding (16)(17)(18)(19). Once images are classified, oriented, and assembled into a 3D intensity function, iterative phasing (20) is often used to determine molecular structure.…”

Comparison of EMC and CM methods for orienting diffraction images in single-particle imaging experiments

“…One group of methods to orient the diffraction patterns in SPI relies on the information in the common intersection curves of the patterns (Huldt et al, 2003;Shneerson et al, 2008;Bortel & Tegze, 2011;Yefanov & Vartanyants, 2013;Zhou et al, 2014). Other methods find the possible orientations of the patterns by comparing them with a 3D intensity model updated by every iteration (Loh & Elser, 2009;Tegze & Bortel, 2012;Flamant et al, 2016;Nakano et al, 2017Nakano et al, , 2018. Another group of methods uses the manifold embedding technique to find the similarities between diffraction patterns and order them in the orientation space (Fung et al, 2009;Giannakis et al, 2012;Kassemeyer et al, 2013;Winter et al, 2016).…”

Deep neural networks for classifying complex features in diffraction images

“…Moreover, the ability to analyze such data sets is sought after by the community in general [22]. For example, for the successful determination of 3D-structures from a CDI data set using the expansion-maximization-compression algorithm [22][23][24], it is necessary to sample the 3D Fourier space up to the Nyquist rate for the desired resolution and this for all sub-species contained in the target under study. The achievable resolution, as well as the chance for successful convergence of the algorithm, correlates directly with the number of diffraction patterns with a high signal-tonoise ratio [23].…”

“…For example, for the successful determination of 3D-structures from a CDI data set using the expansion-maximization-compression algorithm [22][23][24], it is necessary to sample the 3D Fourier space up to the Nyquist rate for the desired resolution and this for all sub-species contained in the target under study. The achievable resolution, as well as the chance for successful convergence of the algorithm, correlates directly with the number of diffraction patterns with a high signal-tonoise ratio [23]. Thus, huge data sets are taken and as a consequence of the sheer amount of data, it is getting increasingly complicated to distill the high-quality data subsets that are suitable for subsequent analysis steps.…”