8 research outputs found
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 for each
. 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 metricâs 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 moleculeâs 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 CTFâs 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 âphase flippingâ, 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 âCovariance Wiener Filteringâ (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
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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 âphase flippingâ, 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 âCovariance Wiener Filteringâ (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