Fast multiscale reconstruction for Cryo-EM

Abstract

We present a multiscale reconstruction framework for single-particle analysis (SPA). The representation of three-dimensional (3D) objects with scaled basis functions permits the reconstruction of volumes at any desired scale in the real-space. This multiscale approach generates interesting opportunities in SPA for the stabilization of the initial volume problem or the 3D iterative refinement procedure. In particular, we show that reconstructions performed at coarse scale are more robust to angular errors and permit gains in computational speed. A key component of the proposed iterative scheme is its fast implementation. The costly step of reconstruction, which was previously hindering the use of advanced iterative methods in SPA, is formulated as a discrete convolution with a cost that does not depend on the number of projection directions. The inclusion of the contrast transfer function inside the imaging matrix is also done at no extra computational cost. By permitting full 3D regularization, the framework is by itself a robust alternative to direct methods for performing reconstruction in adverse imaging conditions (e.g., heavy noise, large angular misassignments, low number of projections). We present reconstructions obtained at different scales from a dataset of the 2015/2016 EMDataBank Map Challenge. The algorithm has been implemented in the Scipion package.

Joint Angular Refinement and Reconstruction for Single-Particle Cryo-EM

“…Currently, state-of-the-art refinement techniques [1][2][3][4] produce a high-resolution density map by alternating between 1) the reconstruction of the 3D density map for a given set of (however inaccurate) projection orientations; 2) the refinement of the projection orientations for all 2D particles based on the previously reconstructed 3D volume. The reconstruction problem can be solved using different approaches such as algebraic methods [5], [6], weighted backprojection (WBP) [7], direct Fourier methods [8][9][10], and iterative regularized approaches [11][12][13].…”

“…When TV regularization is used, the proximal operator at Line 4 admits a closed-form expression that can be computed efficiently [34]. Then, the linear step at Line 6 is solved iteratively using a conjugate-gradient algorithm together with a fast formulation of the H T H(Θ, Γ) term [13]. Finally, Line 7 corresponds to a simple update of the dual variable u, while ρ > 0 is a penalty parameter.…”

“…An option would be to add regularization for the latent variable estimation. The proposed framework could also be combined with the multiscale approach proposed in [13] to perform angle refinements at coarser scales, which demonstrates increased robustness to noise. We expect that those extensions would further improve the performance of the method while keeping an attractive numerical complexity as demonstrated in Section III-D.…”

“…The minimization of L with respect to u We solve it in terms of c using a conjugate-gradient method. Note that, for the x-ray operator H(Θ, Γ), the quantity H T H(Θ, Γ)c can be efficiently computed at the cost of one FFT and one inverse FFT [13], [15], [16]. Indeed, we have that H T H(Θ, Γ)c = w(Θ) * c,…”

“…, g p [m] = g p (Λm)), which can for instance be obtained via some interpolation of the elements of g p . We then have that [13]…”