Autocorrelation analysis for cryo-EM with sparsity constraints: Improved sample complexity and projection-based algorithms

Abstract

The number of noisy images required for molecular reconstruction in single-particle cryoelectron microscopy (cryo-EM) is governed by the autocorrelations of the observed, randomly oriented, noisy projection images. In this work, we consider the effect of imposing sparsity priors on the molecule. We use techniques from signal processing, optimization, and applied algebraic geometry to obtain theoretical and computational contributions for this challenging nonlinear inverse problem with sparsity constraints. We prove that molecular structures modeled as sums of Gaussians are uniquely determined by the second-order autocorrelation of their projection images, implying that the sample complexity is proportional to the square of the variance of the noise. This theory improves upon the nonsparse case, where the third-order autocorrelation is required for uniformly oriented particle images and the sample complexity scales with the cube of the noise variance. Furthermore, we build a computational framework to reconstruct molecular structures which are sparse in the wavelet basis. This method combines the sparse representation for the molecule with projection-based techniques used for phase retrieval in X-ray crystallography.

Moment-based metrics for molecules computable from cryogenic electron microscopy images

“…Beyond its application to experiments, d iKam demonstrates that Kam's method is a feasible strategy for high-resolution reconstruction. Recent works have improved the viability of Kam's method by using sparsity (7) or neural network (36) priors; likewise, the search over the PDB using Kam's metric can be interpreted as simply running Kam's method under a very strong prior, where only a finite number of structures appear with nonzero probability. Our results suggest that, if one could formulate a tractable prior over the manifold of proteins, Kam's method could yield high-resolution reconstructions.…”

“…We detail the procedure for computing m 1 , m 2 and therefore d vKam in Appendix A.1. Under certain conditions, it has been demonstrated that the second moment of the image collection identifies the 3D structure uniquely (2)(3)(4)6,7) or up to a finite list of candidate structures. (8) In Section 4.2, we show that our metric is alike other similarity scores but remarkably does not rely on alignment.…”

“…Kam's method has helped push the theoretical understanding of the reconstruction processunder certain conditions, it is a provable algorithm and provides bounds for the estimated structure's quality in terms of the noise level and the number of images. (2)(3)(4)(5)(6)(7)(8) Kam's method also enjoys other remarkable properties:…”

“…(9) While theoretically attractive, Kam's method has not been able to yield high-resolution reconstructions as yet. One direction that is currently being explored to resolve this issue is the development of better priors, for instance, based on the sparsity of the solution as in the study by Bendory et al (7) Separately, there has been considerable, continued interest in utilizing the vast amount of solved structures stored in the Protein Data Bank (PDB) (10) to improve cryo-EM reconstructions.…”