Autocorrelation analysis for cryo-EM with sparsity constraints: Improved sample complexity and projection-based algorithms - Bendory, Tamir; Khoo, Yuehaw; Kileel, Joe; Mickelin, Oscar; Singer, Amit (Proceedings of the National Academy of Sciences - PNAS, 2023)
SignificanceCryoelectron microscopy is an imaging technique for determining molecular structures from randomly oriented projection images, with important applications in basic science and drug design. A large number of images is needed for accurate reconstruction due to low signal-to-noise ratio and unknown image orientations. We prove that asymptotically fewer images are required for reconstruction if the structure admits a sparse representation. This may reduce the experimental cost and allow reconstruction from limited datasets. This might also help in the related technology of X-ray free-electron lasers, where throughput is a bottleneck. We introduce sparse representations to the computational pipeline and build projection-based algorithms for low-resolution ab initio modeling that would be used to initialize existing iterative refinement procedures and for model validation. 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.