Moment-based metrics for molecules computable from cryo-EM images

Single particle cryogenic electron microscopy (cryo-EM) is an imaging technique capable of recovering the high-resolution 3-D structure of biological macromolecules from many noisy and randomly oriented projection images. One notable approach to 3-D reconstruction, known as Kam's method, relies on the moments of the 2-D images. Inspired by Kam's method, we introduce a rotationally invariant metric between two molecular structures, which does not require 3-D alignment. Further, we introduce a metric between a stack of projection images and a molecular structure, which is invariant to rotations and reflections and does not require performing 3-D reconstruction. Additionally, the latter metric does not assume a uniform distribution of viewing angles. We demonstrate uses of the new metrics on synthetic and experimental datasets, highlighting their ability to measure structural similarity.

Alignment of Density Maps in Wasserstein Distance

In this paper we propose an algorithm for aligning three-dimensional objects when represented as density maps, motivated by applications in cryogenic electron microscopy. The algorithm is based on minimizing the 1-Wasserstein distance between the density maps after a rigid transformation. The induced loss function enjoys a more benign landscape than its Euclidean counterpart and Bayesian optimization is employed for computation. Numerical experiments show improved accuracy and efficiency over existing algorithms on the alignment of real protein molecules. In the context of aligning heterogeneous pairs, we illustrate a potential need for new distance functions.

A stochastic approximate expectation-maximization for structure determination directly from cryo-EM micrographs

A single-particle cryo-electron microscopy (cryo-EM) measurement, called a micrograph, consists of multiple two-dimensional tomographic projections of a three-dimensional molecular structure at unknown locations, taken under unknown viewing directions. All existing cryo-EM algorithmic pipelines first locate and extract the projection images, and then reconstruct the structure from the extracted images. However, if the molecular structure is small, the signal-to-noise ratio (SNR) of the data is very low, and thus accurate detection of projection images within the micrograph is challenging. Consequently, all standard techniques fail in low-SNR regimes. To recover molecular structures from measurements of low SNR, and in particular small molecular structures, we devise a stochastic approximate expectation-maximization algorithm to estimate the three-dimensional structure directly from the micrograph, bypassing locating the projection images. We corroborate our computational scheme with numerical experiments, and present successful structure recoveries from simulated noisy measurements.

Fast expansion into harmonics on the disk: a steerable basis with fast radial convolutions

We present a fast and numerically accurate method for expanding digitized $L \times L$ images representing functions on $[-1,1]^2$ supported on the disk $\{x \in \mathbb{R}^2 : |x|<1\}$ in the harmonics (Dirichlet Laplacian eigenfunctions) on the disk. Our method runs in $\mathcal{O}(L^2 \log L)$ operations. This basis is also known as the Fourier-Bessel basis and it has several computational advantages: it is orthogonal, ordered by frequency, and steerable in the sense that images expanded in the basis can be rotated by applying a diagonal transform to the coefficients. Moreover, we show that convolution with radial functions can also be efficiently computed by applying a diagonal transform to the coefficients.

A Molecular Prior Distribution for Bayesian Inference Based on Wilson Statistics

Background and Objective: Wilson statistics describe well the power spectrum of proteins at high frequencies. Therefore, it has found several applications in structural biology, e.g., it is the basis for sharpening steps used in cryogenic electron microscopy (cryo-EM). A recent paper gave the first rigorous proof of Wilson statistics based on a formalism of Wilson's original argument. This new analysis also leads to statistical estimates of the scattering potential of proteins that reveal a correlation between neighboring Fourier coefficients. Here we exploit these estimates to craft a novel prior that can be used for Bayesian inference of molecular structures. Methods: We describe the properties of the prior and the computation of its hyperparameters. We then evaluate the prior on two synthetic linear inverse problems, and compare against a popular prior in cryo-EM reconstruction at a range of SNRs. Results: We show that the new prior effectively suppresses noise and fills-in low SNR regions in the spectral domain. Furthermore, it improves the resolution of estimates on the problems considered for a wide range of SNR and produces Fourier Shell Correlation curves that are insensitive to masking effects. Conclusions: We analyze the assumptions in the model, discuss relations to other regularization strategies, and postulate on potential implications for structure determination in cryo-EM.

Ab-initio Contrast Estimation and Denoising of Cryo-EM Images

Background and Objective: The contrast of cryo-EM images vary from one to another, primarily due to the uneven thickness of ice layers. The variation of contrast can affect the quality of 2-D class averaging, 3-D ab-initio modeling, and 3-D heterogeneity analysis. Contrast estimation is currently performed during 3-D iterative refinement. As a result, the estimates are not available for class averaging and ab-initio modeling. However, these methods require good initial estimates of 3-D volumes and 3-D rotations of molecules. This paper aims to solve the contrast estimation problem in the ab-initio stage, without estimating the 3-D volume. Methods: The key observation underlying our analysis is that the 2-D covariance matrix of the raw images is related to the covariance of the underlying clean images, the noise variance, and the contrast variability between images. We show that the contrast variability can be derived from the 2-D covariance matrix and use the existing Covariance Wiener Filtering (CWF) framework to estimate it. We also demonstrate a modification of CWF to estimate the contrast of individual images. Results: Our method improves the contrast estimation by a large margin, compared to the previous CWF method. Its estimation accuracy is often comparable to that of an oracle that knows the ground truth covariance of the clean images. The more accurate contrast estimation also improves the quality of image denoising as demonstrated in both synthetic and experimental datasets. Conclusions: This paper proposes an effective method for contrast estimation directly from noisy images without using any 3-D volume information. It enables contrast correction in the earlier stage of single particle analysis, and may improve the accuracy of downstream processing.

An approximate expectation-maximization for two-dimensional multi-target detection

We consider the two-dimensional multi-target detection (MTD) problem of estimating a target image from a noisy measurement that contains multiple copies of the image, each randomly rotated and translated. The MTD model serves as a mathematical abstraction of the structure reconstruction problem in single-particle cryo-electron microscopy, the chief motivation of this study. We focus on high noise regimes, where accurate detection of image occurrences within a measurement is impossible. To estimate the image, we develop an expectation-maximization framework that aims to maximize an approximation of the likelihood function. We demonstrate image recovery in highly noisy environments, and show that our framework outperforms the previously studied autocorrelation analysis in a wide range of parameters. The code to reproduce all numerical experiments is publicly available at https://github.com/krshay/MTD-2D-EM.

Sparse multi-reference alignment: sample complexity and computational hardness

Motivated by the problem of determining the atomic structure of macromolecules using single-particle cryo-electron microscopy (cryo-EM), we study the sample and computational complexities of the sparse multi-reference alignment (MRA) model: the problem of estimating a sparse signal from its noisy, circularly shifted copies. Based on its tight connection to the crystallographic phase retrieval problem, we establish that if the number of observations is proportional to the square of the variance of the noise, then the sparse MRA problem is statistically feasible for sufficiently sparse signals. To investigate its computational hardness, we consider three types of computational frameworks: projection-based algorithms, bispectrum inversion, and convex relaxations. We show that a state-of-the-art projection-based algorithm achieves the optimal estimation rate, but its computational complexity is exponential in the sparsity level. The bispectrum framework provides a statistical-computational trade-off: it requires more observations (so its estimation rate is suboptimal), but its computational load is provably polynomial in the signal's length. The convex relaxation approach provides polynomial time algorithms (with a large exponent) that recover sufficiently sparse signals at the optimal estimation rate. We conclude the paper by discussing potential statistical and algorithmic implications for cryo-EM.

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.

Manifold learning with arbitrary norms

Manifold learning methods play a prominent role in nonlinear dimensionality reduction and other tasks involving high-dimensional data sets with low intrinsic dimensionality. Many of these methods are graph-based: they associate a vertex with each data point and a weighted edge between each pair of close points. Existing theory shows, under certain conditions, that the Laplacian matrix of the constructed graph converges to the Laplace-Beltrami operator of the data manifold. However, this result assumes the Euclidean norm is used for measuring distances. In this paper, we determine the limiting differential operator for graph Laplacians constructed using $\textit{any}$ norm. The proof involves a subtle interplay between the second fundamental form of the underlying manifold and the convex geometry of the norm's unit ball. To motivate the use of non-Euclidean norms, we show in a numerical simulation that manifold learning based on Earthmover's distances outperforms the standard Euclidean variant for learning molecular shape spaces, in terms of both sample complexity and computational complexity.