Mahalanobis Distance for Class Averaging of Cryo-EM Images

Single particle reconstruction (SPR) from cryo-electron microscopy (EM) is a technique in which the 3D structure of a molecule needs to be determined from its contrast transfer function (CTF) affected, noisy 2D projection images taken at unknown viewing directions. One of the main challenges in cryo-EM is the typically low signal to noise ratio (SNR) of the acquired images. 2D classification of images, followed by class averaging, improves the SNR of the resulting averages, and is used for selecting particles from micrographs and for inspecting the particle images. We introduce a new affinity measure, akin to the Mahalanobis distance, to compare cryo-EM images belonging to different defocus groups. The new similarity measure is employed to detect similar images, thereby leading to an improved algorithm for class averaging. We evaluate the performance of the proposed class averaging procedure on synthetic datasets, obtaining state of the art classification.Comment: Final version accepted to the 14th IEEE International Symposium on Biomedical Imaging (ISBI 2017

Orthogonal Matrix Retrieval in Cryo-Electron Microscopy

In single particle reconstruction (SPR) from cryo-electron microscopy (cryo-EM), the 3D structure of a molecule needs to be determined from its 2D projection images taken at unknown viewing directions. Zvi Kam showed already in 1980 that the autocorrelation function of the 3D molecule over the rotation group SO(3) can be estimated from 2D projection images whose viewing directions are uniformly distributed over the sphere. The autocorrelation function determines the expansion coefficients of the 3D molecule in spherical harmonics up to an orthogonal matrix of size $(2l+1)\times (2l+1)$ for each $l=0,1,2,...$. In this paper we show how techniques for solving the phase retrieval problem in X-ray crystallography can be modified for the cryo-EM setup for retrieving the missing orthogonal matrices. Specifically, we present two new approaches that we term Orthogonal Extension and Orthogonal Replacement, in which the main algorithmic components are the singular value decomposition and semidefinite programming. We demonstrate the utility of these approaches through numerical experiments on simulated data.Comment: Modified introduction and summary. Accepted to the IEEE International Symposium on Biomedical Imagin

Shape from Sound: Toward New Tools for Quantum Gravity

To unify general relativity and quantum theory is hard in part because they are formulated in two very different mathematical languages, differential geometry and functional analysis. A natural candidate for bridging this language gap, at least in the case of the Euclidean signature, is the discipline of spectral geometry. It aims at describing curved manifolds in terms of the spectra of their canonical differential operators. As an immediate benefit, this would offer a clean gauge-independent identification of the metric’s degrees of freedom in terms of invariants that should be ready to quantize. However, spectral geometry is itself hard and has been plagued by ambiguities. Here, we regularize and break up spectral geometry into small, finite-dimensional and therefore manageable steps. We constructively demonstrate that this strategy works at least in two dimensions. We can now calculate the shapes of two-dimensional objects from their vibrational spectra

Algorithms for Image Restoration and 3D Reconstruction from Cryo-EM Images

Single particle reconstruction (SPR) in cryo-electron microscopy (cryo-EM) has recently emerged as the method of choice to determine the structure of biological macromolecules to near atomic resolution. The typical procedure for obtaining the final high resolution 3D structure is by starting with an initial guess and iteratively refining it using the acquired dataset of the molecule’s 2D projection images. The final estimate from the refinement procedure is known to often depend heavily on the initial model used as the starting point, thereby making a good initial estimate crucial for success. In this thesis, we propose and test two novel approaches, which we call Orthogonal Extension and Orthogonal Replacement, for 3D ab-initio and homology modeling in SPR using cryo-EM and X-ray free electron lasers (XFEL). Our approach is inspired by the molecular replacement technique used in X-ray crystallography. We first test both approaches on noisy synthetic datasets. Motivated by the need for a reliable estimator of the covariance matrix, we de- velop a new image restoration method to perform contrast transfer function (CTF) correction and denoising in a single step. Through results on several experimental datasets, we demonstrate the efficacy of our method as a single, preliminary step to inspect particle images, detect outliers, and estimate the covariance matrix of the un- derlying clean images. Our covariance matrix estimator is asymptotically consistent and successfully corrects for the CTF. An immediate application of improved covariance estimation is an improvement in the 2D classification or class averaging procedure in the cryo-EM pipeline. We digress from 3D homology/ab-initio modeling to focus on this application. Since different cryo-EM images are affected by noise as well as different CTF’s or point spread functions from the microscope, the Euclidean distance between two images is not an optimal metric for their affinity. We derive and test a new affinity measure akin to the Mahalanobis distance to compare cryo-EM images belonging to different defocus groups. We demonstrate that the new metric leads to an improvement in nearest neighbor detection and therefore the obtained class averages. Finally, we revisit the homology modeling procedure of Orthogonal Extension. We incorporate our improved covariance matrix estimator into the Orthogonal Extension algorithm and propose a family of asymptotically unbiased estimators to recover the 3D structure. We demonstrate the advantage of our estimator through numerical experiments on synthetic and experimental datasets. We foresee this method as a good way to provide models to initialize refinement, directly from experimental images without performing class averaging and orientation estimation in cryo-EM and XFEL. Our second algorithm for ab-initio modeling, Orthogonal Replacement, is tested on synthetic datasets. In future work, Orthogonal Replacement would require designing an appropriate experiment to collect datasets that would facilitate its usage

Denoising And Covariance Estimation Of Single Particle Cryo-Em Images

The problem of image restoration in cryo-EM entails correcting for the effects of the Contrast Transfer Function (CTF) and noise. Popular methods for image restoration include ‘phase flipping’, which corrects only for the Fourier phases but not amplitudes, and Wiener filtering, which requires the spectral signal to noise ratio. We propose a new image restoration method which we call ‘Covariance Wiener Filtering’ (CWF). In CWF, the covariance matrix of the projection images is used within the classical Wiener filtering framework for solving the image restoration deconvolution problem. Our estimation procedure for the covariance matrix is new and successfully corrects for the CTF. We demonstrate the efficacy of CWF by applying it to restore both simulated and experimental cryo-EM images. Results with experimental datasets demonstrate that CWF provides a good way to evaluate the particle images and to see what the dataset contains even without 2D classification and averaging

MAHALANOBIS DISTANCE FOR CLASS AVERAGING OF CRYO-EM IMAGES

Single particle reconstruction (SPR) from cryo-electron microscopy (EM) is a technique in which the 3D structure of a molecule needs to be determined from its contrast transfer function (CTF) affected, noisy 2D projection images taken at unknown viewing directions. One of the main challenges in cryo-EM is the typically low signal to noise ratio (SNR) of the acquired images. 2D classification of images, followed by class averaging, improves the SNR of the resulting averages, and is used for selecting particles from micrographs and for inspecting the particle images. We introduce a new affinity measure, akin to the Mahalanobis distance, to compare cryo-EM images belonging to different defocus groups. The new similarity measure is employed to detect similar images, thereby leading to an improved algorithm for class averaging. We evaluate the performance of the proposed class averaging procedure on synthetic datasets, obtaining state of the art classification

Denoising and covariance estimation of single particle cryo-EM images

The problem of image restoration in cryo-EM entails correcting for the effects of the Contrast Transfer Function (CTF) and noise. Popular methods for image restoration include ‘phase flipping’, which corrects only for the Fourier phases but not amplitudes, and Wiener filtering, which requires the spectral signal to noise ratio. We propose a new image restoration method which we call ‘Covariance Wiener Filtering’ (CWF). In CWF, the covariance matrix of the projection images is used within the classical Wiener filtering framework for solving the image restoration deconvolution problem. Our estimation procedure for the covariance matrix is new and successfully corrects for the CTF. We demonstrate the efficacy of CWF by applying it to restore both simulated and experimental cryo-EM images. Results with experimental datasets demonstrate that CWF provides a good way to evaluate the particle images and to see what the dataset contains even without 2D classification and averaging. (C) 2016 Elsevier Inc. All rights reserved