316 research outputs found
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
Autocorrelation analysis for cryo-EM with sparsity constraints: Improved sample complexity and projection-based algorithms
The number of noisy images required for molecular reconstruction in
single-particle cryo-electron microscopy (cryo-EM) is governed by the
autocorrelations of the observed, randomly-oriented, noisy projection images.
In this work, we consider the effect of imposing sparsity priors on the
molecule. We use techniques from signal processing, optimization, and applied
algebraic geometry to obtain new theoretical and computational contributions
for this challenging non-linear inverse problem with sparsity constraints. We
prove that molecular structures modeled as sums of Gaussians are uniquely
determined by the second-order autocorrelation of their projection images,
implying that the sample complexity is proportional to the square of the
variance of the noise. This theory improves upon the non-sparse case, where the
third-order autocorrelation is required for uniformly-oriented particle images
and the sample complexity scales with the cube of the noise variance.
Furthermore, we build a computational framework to reconstruct molecular
structures which are sparse in the wavelet basis. This method combines the
sparse representation for the molecule with projection-based techniques used
for phase retrieval in X-ray crystallography.Comment: 31 pages, 5 figures, 1 movi
Compressed Representations of Macromolecular Structures and Properties
SummaryWe introduce a new and unified, compressed volumetric representation for macromolecular structures at varying feature resolutions, as well as for many computed associated properties. Important caveats of this compressed representation are fast random data access and decompression operations. Many computational tasks for manipulating large structures, including those requiring interactivity such as real-time visualization, are greatly enhanced by utilizing this compact representation. The compression scheme is obtained by using a custom designed hierarchical wavelet basis construction. Due to the continuity offered by these wavelets, we retain very good accuracy of molecular surfaces, at very high compression ratios, for macromolecular structures at multiple resolutions
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