Orientation Determination of Cryo-EM Images Using Least Unsquared Deviations

Abstract

A major challenge in single particle reconstruction from cryo-electron microscopy is to establish a reliable ab initio three-dimensional model using two-dimensional projection images with unknown orientations. Common-lines–based methods estimate the orientations without additional geometric information. However, such methods fail when the detection rate of common-lines is too low due to the high level of noise in the images. An approximation to the least squares global self-consistency error was obtained in [A. Singer and Y. Shkolnisky, SIAM J. Imaging Sci., 4 (2011), pp. 543–572] using convex relaxation by semidefinite programming. In this paper we introduce a more robust global self-consistency error and show that the corresponding optimization problem can be solved via semidefinite relaxation. In order to prevent artificial clustering of the estimated viewing directions, we further introduce a spectral norm term that is added as a constraint or as a regularization term to the relaxed minimization problem. The resulting problems are solved using either the alternating direction method of multipliers or an iteratively reweighted least squares procedure. Numerical experiments with both simulated and real images demonstrate that the proposed methods significantly reduce the orientation estimation error when the detection rate of common-lines is low.

Fast and Robust Orientation of Cryo-Electron Microscopy Images

“…Recently, a series of common line based orientation methods were proposed in [22][23][24][25]. In [22] the orientations of all the discrete radial Fourier lines of projections were calculated from first three eigenvectors of a sparse adjacency matrix.…”

“…[24] proposed two algorithms which both used the computed common lines to determine the orientations, one computing the three largest eigenvectors of a symmetric matrix and another solving a semidefinite programming. The method was improved in [25] using least unsquared deviations via semidefinite relaxation. Different from algorithms in [22,24,25] which were based on common lines between pairs of projections, the method in [23] used triplets of projections to construct a synchronization matrix using triplets of projections which improved the noise robustness.…”

“…The method was improved in [25] using least unsquared deviations via semidefinite relaxation. Different from algorithms in [22,24,25] which were based on common lines between pairs of projections, the method in [23] used triplets of projections to construct a synchronization matrix using triplets of projections which improved the noise robustness. Orientation algorithms have been implemented in the softwares.…”

“…Different from the method in [22] which uses all the radial Fourier lines of the projections, we only use the common lines to determine the orientations. Our algorithm also differs from [10,[23][24][25] in that instead of determining the Euler angles or the rotation matrices, our orientation method calculates the coordinates of the common line vectors, then determines the projection directions and the inplane rotation angles from these coordinates. In addition, in order to increase the noise robustness, we use the weighted cross-correlation to compute the common lines in the real space.…”

“…Since q ω,k are computed using theẽ 1ẽ2 coordinate system (2.1) and (2.2), then the in-plane rotation angle θω is determined by minimizing the following energy function 25) where R(θω) is the rotation matrix defined by θω: In this section, we use the simulated cryo-EM images to validate our algorithm. In order to produce the simulated cryo-EM images, a 3D density map (a volume data) with size 140 3 is synthesized using the crystal structure file 1FFK.pdb¹, which is the largest ribosomal subunit.…”