Subspace method of moments for ab initio 3-D single-particle cryo-EM reconstruction

TL;DR

This work tackles ab initio 3-D cryo-EM reconstruction under unknown nonuniform viewing distributions by leveraging the method of moments (MoM). It introduces SubspaceMoM, a scalable framework that compresses the first three moments into low-dimensional subspace moments via basis expansions and randomized linear algebra (range-finding and CUR), coupled with quadrature over $SO(3)$. The approach enables joint recovery of the volume and the viewing-direction distribution with reduced memory and computational demands, and it demonstrates that including the third-order moment substantially improves reconstruction quality, yielding meaningful ab initio models even at low SNRs and in the presence of CTFs. The method is modular, supports symmetry exploitation, and offers a Python implementation, with extensions proposed for translations and higher-order moments to broaden applicability in practical cryo-EM pipelines.

Abstract

Cryo-electron microscopy (cryo-EM) is a widely used technique for recovering the 3-D structure of biological molecules from a large number of experimentally generated noisy 2-D tomographic projection images of the 3-D structure, taken from unknown viewing angles. Through computationally intensive algorithms, these observed images are processed to reconstruct the 3-D structures. Many popular computational methods rely on estimating the unknown angles as part of the reconstruction process, which becomes particularly challenging at low signal-to-noise ratios. The method of moments (MoM) offers an alternative approach that circumvents the estimation of viewing orientations of individual projection images by instead estimating the underlying distribution of the viewing angles, and is robust to noise given sufficiently many images. However, the method of moments typically entails computing higher-order moments of the projection images, incurring significant computational and memory costs. To mitigate this, we propose a new approach called the subspace method of moments (SubspaceMoM), which compresses the first three moments using data-driven low-rank tensor techniques as well as expansion into a suitable function basis. The compressed moments can be efficiently computed from the set of projection images using numerical quadrature and can be employed to jointly reconstruct the 3-D structure and the distribution of viewing orientations. We illustrate the practical applicability of SubspaceMoM through numerical experiments using up to the third-order moment on synthetic datasets with a simplified cryo-EM image formation model, which significantly improves the reconstruction resolution compared to previous MoM approaches.

Figures (16)

• Figure 1: (A) A cryo-EM micrograph from the Beta-galactosidase Falcon-II dataset SCHERES2015114. (B) The result of the particle picking procedure using the APPLE picker HEIMOWITZ2018215 implemented in the ASPIRE software aspire.
• Figure 2: (A) The dimension parameter $r_2$ for the second subspace moment as a function of the threshold value $\tau^{(2)}$, obtained in Section \ref{['sec:thirdmomenteffect']}. Specifically, we obtain $r_2=20, 47, 87, 123, 147, 157$. (B) The relative errors evaluated on a $60\times60$ sub-matrix of the second moment, as a function of the dimension parameter of the second subspace moment $r_2$. (C) The running time (in hours) for completing the optimization \ref{['eqn:secondstageoptm']} that matches the first two subspace moments in the second stage as a function of the dimension parameter of the second subspace moment $r_2$. (D) The resolutions (in angstrom$\textup{~\AA}$) compared to the expanded ground truth, obtained from the first two (subspace) moments as a function of the dimension parameter of the second subspace moment $r_2$.
• Figure 3: (A) The dimension parameter $r_3$ for the third subspace moment as a function of the threshold value $\tau^{(2)}$ obtained in Section \ref{['sec:thirdmomenteffect']}. Specifically, we obtain $r_3=22, 37, 55, 77, 97, 111, 118$. (B) The relative errors evaluated on a $60\times60\times 60$ sub-tensor of the third moment, as a function of the dimension parameter of the second subspace moment $r_3$. (C) The running time (in hours) for completing the optimization \ref{['eqn:subspaceMoM']} in the last stage as a function of the dimension parameter of the third subspace moment $r_3$. (D) The reconstructed resolutions (in angstrom $\textup{~\AA}$) compared to the expanded ground truth, obtained from the first three subspace moments as a function of the dimension parameter of the third subspace moment $r_3$.
• Figure 4: The gray volume represents the ground truth obtained by expanding the downsampled EMD-34948 into the spherical Bessel basis with $L=5$ used in Section \ref{['sec:thirdmomenteffect']}. The others are the reconstructions of the sequential moment matching at different stages. Specifically, the yellow volume represents the reconstructed volume obtained from the first subspace moment via solving \ref{['eqn:firststageoptm']}, achieving a resolution of $54.94 \textup{~\AA}$ compared to the expanded ground truth. The blue volume represents the reconstructed volume obtained from the first two subspace moments via solving \ref{['eqn:secondstageoptm']}, achieving a resolution of $45.03 \textup{~\AA}$. The purple volume represents the reconstructed volume obtained from the first three subspace moments via solving \ref{['eqn:subspaceMoM']}, achieving the Nyquist-limited resolution $6.37 \textup{~\AA}$. The dimension parameters for the subspace moments are $r_1=400$, $r_2=157$ and $r_3=118$.
• Figure 5: The viewing direction densities plotted as functions of first two Euler angles $\alpha\in [0,2\pi]$ and $\beta\in [0,\pi]$) that are invariant to in-plane reflection obtained in Section \ref{['sec:thirdmomenteffect']}. The leftmost figure shows the ground truth density. The second figure shows the estimated density from the first subspace moment via solving \ref{['eqn:firststageoptm']}, which has a relative error $0.4086$ compared to the ground truth. The third figure shows the estimated density obtained from the first two subspace moments via solving \ref{['eqn:secondstageoptm']}, which has a relative $L^2$ error $0.4080$. The last one shows the estimated density obtained from all three subspace moments via solving \ref{['eqn:subspaceMoM']}, which has a relative $L^2$ error $0.0811$. The dimension parameters for the subspace moments are $r_1=400$, $r_2=157$ and $r_3=118$.

Theorems & Definitions (2)

• Remark
• Remark