222 research outputs found
Covariance estimation using conjugate gradient for 3D classification in Cryo-EM
Classifying structural variability in noisy projections of biological
macromolecules is a central problem in Cryo-EM. In this work, we build on a
previous method for estimating the covariance matrix of the three-dimensional
structure present in the molecules being imaged. Our proposed method allows for
incorporation of contrast transfer function and non-uniform distribution of
viewing angles, making it more suitable for real-world data. We evaluate its
performance on a synthetic dataset and an experimental dataset obtained by
imaging a 70S ribosome complex
Structural Variability from Noisy Tomographic Projections
In cryo-electron microscopy, the 3D electric potentials of an ensemble of
molecules are projected along arbitrary viewing directions to yield noisy 2D
images. The volume maps representing these potentials typically exhibit a great
deal of structural variability, which is described by their 3D covariance
matrix. Typically, this covariance matrix is approximately low-rank and can be
used to cluster the volumes or estimate the intrinsic geometry of the
conformation space. We formulate the estimation of this covariance matrix as a
linear inverse problem, yielding a consistent least-squares estimator. For
images of size -by- pixels, we propose an algorithm for calculating this
covariance estimator with computational complexity
, where the condition number
is empirically in the range --. Its efficiency relies on the
observation that the normal equations are equivalent to a deconvolution problem
in 6D. This is then solved by the conjugate gradient method with an appropriate
circulant preconditioner. The result is the first computationally efficient
algorithm for consistent estimation of 3D covariance from noisy projections. It
also compares favorably in runtime with respect to previously proposed
non-consistent estimators. Motivated by the recent success of eigenvalue
shrinkage procedures for high-dimensional covariance matrices, we introduce a
shrinkage procedure that improves accuracy at lower signal-to-noise ratios. We
evaluate our methods on simulated datasets and achieve classification results
comparable to state-of-the-art methods in shorter running time. We also present
results on clustering volumes in an experimental dataset, illustrating the
power of the proposed algorithm for practical determination of structural
variability.Comment: 52 pages, 11 figure
Heterogeneous multireference alignment: a single pass approach
Multireference alignment (MRA) is the problem of estimating a signal from
many noisy and cyclically shifted copies of itself. In this paper, we consider
an extension called heterogeneous MRA, where signals must be estimated, and
each observation comes from one of those signals, unknown to us. This is a
simplified model for the heterogeneity problem notably arising in cryo-electron
microscopy. We propose an algorithm which estimates the signals without
estimating either the shifts or the classes of the observations. It requires
only one pass over the data and is based on low-order moments that are
invariant under cyclic shifts. Given sufficiently many measurements, one can
estimate these invariant features averaged over the signals. We then design
a smooth, non-convex optimization problem to compute a set of signals which are
consistent with the estimated averaged features. We find that, in many cases,
the proposed approach estimates the set of signals accurately despite
non-convexity, and conjecture the number of signals that can be resolved as
a function of the signal length is on the order of .Comment: 6 pages, 3 figure
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
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Covariance Matrix Estimation for the Cryo-EM Heterogeneity Problem
In cryo-electron microscopy (cryo-EM), a microscope generates a top view of a sample of randomly oriented copies of a molecule. The problem of single particle reconstruction (SPR) from cryo-EM is to use the resulting set of noisy two-dimensional projection images taken at unknown directions to reconstruct the three-dimensional (3D) structure of the molecule. In some situations, the molecule under examination exhibits structural variability, which poses a fundamental challenge in SPR. The heterogeneity problem is the task of mapping the space of conformational states of a molecule. It has been previously suggested that the leading eigenvectors of the covariance matrix of the 3D molecules can be used to solve the heterogeneity problem. Estimating the covariance matrix is challenging, since only projections of the molecules are observed, but not the molecules themselves. In this paper, we formulate a general problem of covariance estimation from noisy projections of samples. This problem has intimate connections with matrix completion problems and high-dimensional principal component analysis. We propose an estimator and prove its consistency. When there are finitely many heterogeneity classes, the spectrum of the estimated covariance matrix reveals the number of classes. The estimator can be found as the solution to a certain linear system. In the cryo-EM case, the linear operator to be inverted, which we term the projection covariance transform, is an important object in covariance estimation for tomographic problems involving structural variation. Inverting it involves applying a filter akin to the ramp filter in tomography. We design a basis in which this linear operator is sparse and thus can be tractably inverted despite its large size. We demonstrate via numerical experiments on synthetic datasets the robustness of our algorithm to high levels of noise
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