3D ab initio modeling in cryo-EM by autocorrelation analysis

Single-Particle Reconstruction (SPR) in Cryo-Electron Microscopy (cryo-EM) is the task of estimating the 3D structure of a molecule from a set of noisy 2D projections, taken from unknown viewing directions. Many algorithms for SPR start from an initial reference molecule, and alternate between refining the estimated viewing angles given the molecule, and refining the molecule given the viewing angles. This scheme is called iterative refinement. Reliance on an initial, user-chosen reference introduces model bias, and poor initialization can lead to slow convergence. Furthermore, since no ground truth is available for an unsolved molecule, it is difficult to validate the obtained results. This creates the need for high quality ab initio models that can be quickly obtained from experimental data with minimal priors, and which can also be used for validation. We propose a procedure to obtain such an ab initio model directly from raw data using Kam's autocorrelation method. Kam's method has been known since 1980, but it leads to an underdetermined system, with missing orthogonal matrices. Until now, this system has been solved only for special cases, such as highly symmetric molecules or molecules for which a homologous structure was already available. In this paper, we show that knowledge of just two clean projections is sufficient to guarantee a unique solution to the system. This system is solved by an optimization-based heuristic. For the first time, we are then able to obtain a low-resolution ab initio model of an asymmetric molecule directly from raw data, without 2D class averaging and without tilting. Numerical results are presented on both synthetic and experimental data

Rotationally Invariant Image Representation for Viewing Direction Classification in Cryo-EM

We introduce a new rotationally invariant viewing angle classification method for identifying, among a large number of Cryo-EM projection images, similar views without prior knowledge of the molecule. Our rotationally invariant features are based on the bispectrum. Each image is denoised and compressed using steerable principal component analysis (PCA) such that rotating an image is equivalent to phase shifting the expansion coefficients. Thus we are able to extend the theory of bispectrum of 1D periodic signals to 2D images. The randomized PCA algorithm is then used to efficiently reduce the dimensionality of the bispectrum coefficients, enabling fast computation of the similarity between any pair of images. The nearest neighbors provide an initial classification of similar viewing angles. In this way, rotational alignment is only performed for images with their nearest neighbors. The initial nearest neighbor classification and alignment are further improved by a new classification method called vector diffusion maps. Our pipeline for viewing angle classification and alignment is experimentally shown to be faster and more accurate than reference-free alignment with rotationally invariant K-means clustering, MSA/MRA 2D classification, and their modern approximations

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

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.Comment: 31 pages, 5 figures, 1 movi

3D unknown view tomography via rotation invariants

In this paper, we study the problem of reconstructing a 3D point source model from a set of 2D projections at unknown view angles. Our method obviates the need to recover the projection angles by extracting a set of rotation-invariant features from the noisy projection data. From the features, we reconstruct the density map through a constrained nonconvex optimization. We show that the features have geometric interpretations in the form of radial and pairwise distances of the model. We further perform an ablation study to examine the effect of various parameters on the quality of the estimated features from the projection data. Our results showcase the potential of the proposed method in reconstructing point source models in various noise regimes

Multi-target detection with rotations

We consider the multi-target detection problem of estimating a two-dimensional target image from a large noisy measurement image that contains many randomly rotated and translated copies of the target image. Motivated by single-particle cryo-electron microscopy, we focus on the low signal-to-noise regime, where it is difficult to estimate the locations and orientations of the target images in the measurement. Our approach uses autocorrelation analysis to estimate rotationally and translationally invariant features of the target image. We demonstrate that, regardless of the level of noise, our technique can be used to recover the target image when the measurement is sufficiently large.Comment: 20 pages, 5 figure

High-resolution ab initio three-dimensional X-ray diffraction microscopy

Coherent X-ray diffraction microscopy is a method of imaging non-periodic isolated objects at resolutions only limited, in principle, by the largest scattering angles recorded. We demonstrate X-ray diffraction imaging with high resolution in all three dimensions, as determined by a quantitative analysis of the reconstructed volume images. These images are retrieved from the 3D diffraction data using no a priori knowledge about the shape or composition of the object, which has never before been demonstrated on a non-periodic object. We also construct 2D images of thick objects with infinite depth of focus (without loss of transverse spatial resolution). These methods can be used to image biological and materials science samples at high resolution using X-ray undulator radiation, and establishes the techniques to be used in atomic-resolution ultrafast imaging at X-ray free-electron laser sources.Comment: 22 pages, 11 figures, submitte