Rotationally invariant image representation for viewing direction classification in cryo-EM

Abstract

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.

Cross-Validation of Data Compatibility Between Small Angle X-ray Scattering and Cryo-Electron Microscopy

“…The improvement of SNR usually has been done either by class averaging, that is, averaging many different images in a class corresponding to similar orientations (van Heel and Frank, 1981;van Heel, 1984;Schatz and Van Heel, 1990;Penczek et al, 1992;Penczek, 2002;Scheres et al, 2005;Park et al, 2011;Singer et al, 2011;Shkolnisky and Singer, 2012;Park and Chirikjian, 2014;Zhao and Singer, 2014), or by applying denoising techniques, such as bilateral filtering ( Jiang et al, 2003) sinograms (Mielikäinen and Ravantti, 2005) and covariance Wiener filtering (Bhamre et al, 2016) to EM images. Note that the method we propose in this work is to remove the noise on a single image rather than over a class, which makes our work very different from others.…”

Temporal ordering and registration of images in studies of developmental dynamics

“…In contrast to most of the existing registration approaches, which rely on a knowledge of appropriate landmarks in the images (Dryden and Mardia, 1998), such as the eyes in facerecognition applications (Zhao et al, 2003), algorithms based on angular synchronization can register images even in the absence of such information, making them relevant for a wide variety of applications. Angular synchronization and vector diffusion maps have been used to reconstruct molecular shapes from cryo-electron microscopy images Singer and Wu, 2012;Zhao and Singer, 2014). Because of high levels of instrument noise in these data, thousands of images were needed for successful shape reconstruction.…”

“…The approach is based on vector diffusion maps (Singer and Wu, 2012), a manifold learning algorithm that simultaneously addresses the problems of registration and temporal ordering. This algorithm is one of several nonlinear dimensionality reduction techniques that have been developed over the past decade (Roweis and Saul, 2000;Tenenbaum et al, 2000;Belkin and Niyogi, 2003;Coifman et al, 2005;Coifman and Lafon, 2006) for applications ranging from the analysis of cryo-electron microscopy (cryo-EM) images of individual molecules Zhao and Singer, 2014) to face recognition and the classification of CT scans (Fernández et al, 2014).…”

Joint Angular Refinement and Reconstruction for Single-Particle Cryo-EM

“…, P }) is compared against N θ1 N θ2 clean templates. This requires rotation and in-plane translation alignment whose complexity is O(m 2 log m) if done efficiently using polar Fourier transform [37], spherical harmonics [38], or steerable basis functions [39]. The overall complexity of template matching is thus given by O(N θ1 N θ2 (P m 2 log m + C H )).…”