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

TL;DR

This paper demonstrates that imposing sparsity priors in cryo-EM allows for unique molecular structure determination using second-order autocorrelations, reducing sample complexity compared to non-sparse models, and introduces a new computational reconstruction method.

Contribution

The work provides theoretical guarantees for sparse cryo-EM models using second-order autocorrelations and develops a novel projection-based reconstruction algorithm leveraging wavelet sparsity.

Findings

Molecular structures as sums of Gaussians are uniquely identified by second-order autocorrelations.

Sample complexity scales with the square of noise variance in sparse models, improving over non-sparse cases.

A new computational framework combines sparsity and projection techniques for molecular reconstruction.

Abstract

The number of noisy images required for molecular reconstruction in single-particle cryo-electron 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 new theoretical and computational contributions for this challenging non-linear 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 non-sparse 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.