8,446 research outputs found
Stability of the Recovery of Missing Samples in Derivative Oversampling
This paper deals with the problem of reconstructing a band-limited signal
when a finite subset of its samples and of its derivative are missing. The
technique used, due to P.J.S.G. Ferreira, is based on the use of a particular
frame for band-limited functions and the relative oversampling formulas. We
study the eigenvalues of the matrices arising in the procedure of recovering
the lost samples, finding estimates of their eigenvalues and their dependence
on the oversampling parameter and on the number of missing samples.
When the missing samples are consecutive, the problem may become very
ill-conditioned. We present a numerical procedure to overcome this difficulty,
also in presence of noisy data, by using Tikhonov regularization techniques.Comment: 17 pages,8 figure
OPED reconstruction algorithm for limited angle problem
The structure of the reconstruction algorithm OPED permits a natural way to
generate additional data, while still preserving the essential feature of the
algorithm. This provides a method for image reconstruction for limited angel
problems. In stead of completing the set of data, the set of discrete sine
transforms of the data is completed. This is achieved by solving systems of
linear equations that have, upon choosing appropriate parameters, positive
definite coefficient matrices. Numerical examples are presented.Comment: 17 page
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
Subspace procrustes analysis
Postprint (author's final draft
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