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

Advances in image processing for single-particle analysis by electron cryomicroscopy and challenges ahead

Electron cryomicroscopy (cryo-EM) is essential for the study and functional understanding of non-crystalline macromolecules such as proteins. These molecules cannot be imaged using X-ray crystallography or other popular methods. CryoEM has been successfully used to visualize molecules such as ribosomes, viruses, and ion channels, for example. Obtaining structural models of these at various conformational states leads to insight on how these molecules function. Recent advances in imaging technology have given cryo-EM a scientific rebirth. Because of imaging improvements, image processing and analysis of the resultant images have increased the resolution such that molecular structures can be resolved at the atomic level. Cryo-EM is ripe with stimulating image processing challenges. In this article, we will touch on the most essential in order to build an accurate structural three-dimensional model from noisy projection images. Traditional approaches, such as k-means clustering for class averaging, will be provided as background. With this review, however, we will highlight fresh approaches from new and varied angles for each image processing sub-problem, including a 3D reconstruction method for asymmetric molecules using just two projection images and deep learning algorithms for automated particle picking. Keywords: Cryo-electron microscopy, Single Particle Analysis, Image processing algorithms

Bispectrum Inversion with Application to Multireference Alignment

We consider the problem of estimating a signal from noisy circularly-translated versions of itself, called multireference alignment (MRA). One natural approach to MRA could be to estimate the shifts of the observations first, and infer the signal by aligning and averaging the data. In contrast, we consider a method based on estimating the signal directly, using features of the signal that are invariant under translations. Specifically, we estimate the power spectrum and the bispectrum of the signal from the observations. Under mild assumptions, these invariant features contain enough information to infer the signal. In particular, the bispectrum can be used to estimate the Fourier phases. To this end, we propose and analyze a few algorithms. Our main methods consist of non-convex optimization over the smooth manifold of phases. Empirically, in the absence of noise, these non-convex algorithms appear to converge to the target signal with random initialization. The algorithms are also robust to noise. We then suggest three additional methods. These methods are based on frequency marching, semidefinite relaxation and integer programming. The first two methods provably recover the phases exactly in the absence of noise. In the high noise level regime, the invariant features approach for MRA results in stable estimation if the number of measurements scales like the cube of the noise variance, which is the information-theoretic rate. Additionally, it requires only one pass over the data which is important at low signal-to-noise ratio when the number of observations must be large

Cram\'er-Rao bounds for synchronization of rotations

Synchronization of rotations is the problem of estimating a set of rotations R_i in SO(n), i = 1, ..., N, based on noisy measurements of relative rotations R_i R_j^T. This fundamental problem has found many recent applications, most importantly in structural biology. We provide a framework to study synchronization as estimation on Riemannian manifolds for arbitrary n under a large family of noise models. The noise models we address encompass zero-mean isotropic noise, and we develop tools for Gaussian-like as well as heavy-tail types of noise in particular. As a main contribution, we derive the Cram\'er-Rao bounds of synchronization, that is, lower-bounds on the variance of unbiased estimators. We find that these bounds are structured by the pseudoinverse of the measurement graph Laplacian, where edge weights are proportional to measurement quality. We leverage this to provide interpretation in terms of random walks and visualization tools for these bounds in both the anchored and anchor-free scenarios. Similar bounds previously established were limited to rotations in the plane and Gaussian-like noise

Lifting-based Global Optimisation over Riemannian Manifolds: Theory and Applications in Cryo-EM

We propose ellipsoidal support lifting (ESL), a lifting-based optimisation scheme for approximating the global minimiser of a smooth function over a Riemannian manifold. Under a uniqueness assumption on the minimiser we show several theoretical results, in particular an error bound with respect to the global minimiser. Additionally, we use the developed theory to integrate the lifting-based optimisation scheme into an alternating minimisation method for joint homogeneous volume reconstruction and rotation estimation in single particle cryogenic-electron microscopy (Cryo-EM), where typically tens of thousands of manifold-valued minimisation problems have to be solved. The joint recovery method arguably overcomes the typical trade-off between noise-robustness and data-consistency -- while remaining computationally feasible --, and is used to test both the theoretical predictions and algorithmic performance through numerical experiments with Cryo-EM data