59,082 research outputs found

    Structural Variability from Noisy Tomographic Projections

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    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 nn images of size NN-by-NN pixels, we propose an algorithm for calculating this covariance estimator with computational complexity O(nN4+κN6logN)\mathcal{O}(nN^4+\sqrt{\kappa}N^6 \log N), where the condition number κ\kappa is empirically in the range 1010--200200. 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

    Theory and implementation of H\mathcal{H}-matrix based iterative and direct solvers for Helmholtz and elastodynamic oscillatory kernels

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    In this work, we study the accuracy and efficiency of hierarchical matrix (H\mathcal{H}-matrix) based fast methods for solving dense linear systems arising from the discretization of the 3D elastodynamic Green's tensors. It is well known in the literature that standard H\mathcal{H}-matrix based methods, although very efficient tools for asymptotically smooth kernels, are not optimal for oscillatory kernels. H2\mathcal{H}^2-matrix and directional approaches have been proposed to overcome this problem. However the implementation of such methods is much more involved than the standard H\mathcal{H}-matrix representation. The central questions we address are twofold. (i) What is the frequency-range in which the H\mathcal{H}-matrix format is an efficient representation for 3D elastodynamic problems? (ii) What can be expected of such an approach to model problems in mechanical engineering? We show that even though the method is not optimal (in the sense that more involved representations can lead to faster algorithms) an efficient solver can be easily developed. The capabilities of the method are illustrated on numerical examples using the Boundary Element Method

    Functional observers for motion control systems

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    This paper presents a novel functional observer for motion control systems to provide higher accuracy and less noise in comparison to existing observers. The observer uses the input current and position information along with the nominal parameters of the plant and can observe the velocity, acceleration and disturbance information of the system. The novelty of the observer is based on its functional structure that can intrinsically estimate and compensate the un-measured inputs (like disturbance acting on the system) using the measured input current. The experimental results of the proposed estimator verifies its success in estimating the velocity, acceleration and disturbance with better precision than other second order observers

    Effects of Destriping Errors on Estimates of the CMB Power Spectrum

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    Destriping methods for constructing maps of the Cosmic Microwave Background (CMB) anisotropies have been investigated extensively in the literature. However, their error properties have been studied in less detail. Here we present an analysis of the effects of destriping errors on CMB power spectrum estimates for Planck-like scanning strategies. Analytic formulae are derived for certain simple scanning geometries that can be rescaled to account for different detector noise. Assuming {Planck-like low-frequency noise, the noise power spectrum is accurately white at high multipoles (l<50). D estriping errors, though dominant at lower multipoles, are small in comparison to the cosmic variance. These results show that simple destriping map-making methods should be perfectly adequate for the analysis of Planck data and support the arguments given in an earlier paper in favour of applying a fast hybrid power spectrum estimator to CMB data with realistic `1/f' noise.Comment: 13 pages, 6 figures, submitted to MNRA
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