56 research outputs found
Prolate spheroidal wave functions on a disc—Integration and approximation of two-dimensional bandlimited functions
AbstractWe consider the problem of integrating and approximating 2D bandlimited functions restricted to a disc by using 2D prolate spheroidal wave functions (PSWFs). We derive a numerical scheme for the evaluation of the 2D PSWFs on a disc, which is the basis for the numerical implementation of the presented quadrature and approximation schemes. Next, we derive a quadrature formula for bandlimited functions restricted to a disc and give a bound on the integration error. We apply this quadrature to derive an approximation scheme for such functions. We prove a bound on the approximation error and present numerical results that demonstrate the effectiveness of the quadrature and approximation schemes
Common lines ab-initio reconstruction of -symmetric molecules
Cryo-electron microscopy is a state-of-the-art method for determining
high-resolution three-dimensional models of molecules, from their
two-dimensional projection images taken by an electron microscope. A crucial
step in this method is to determine a low-resolution model of the molecule
using only the given projection images, without using any three-dimensional
information, such as an assumed reference model. For molecules without
symmetry, this is often done by exploiting common lines between pairs of
images. Common lines algorithms have been recently devised for molecules with
cyclic symmetry, but no such algorithms exist for molecules with dihedral
symmetry. In this work, we present a common lines algorithm for determining the
structure of molecules with symmetry. The algorithm exploits the common
lines between all pairs of images simultaneously, as well as common lines
within each image. We demonstrate the applicability of our algorithm using
experimental cryo-electron microscopy data
Sampling and Approximation of Bandlimited Volumetric Data
We present an approximation scheme for functions in three dimensions, that
requires only their samples on the Cartesian grid, under the assumption that
the functions are sufficiently concentrated in both space and frequency. The
scheme is based on expanding the given function in the basis of generalized
prolate spheroidal wavefunctions, with the expansion coefficients given by
weighted dot products between the samples of the function and the samples of
the basis functions. As numerical implementations require all expansions to be
finite, we present a truncation rule for the expansions. Finally, we derive a
bound on the overall approximation error in terms of the assumed
space/frequency concentration
A perturbation based out-of-sample extension framework
Out-of-sample extension is an important task in various kernel based
non-linear dimensionality reduction algorithms. In this paper, we derive a
perturbation based extension framework by extending results from classical
perturbation theory. We prove that our extension framework generalizes the
well-known Nystr{\"o}m method as well as some of its variants. We provide an
error analysis for our extension framework, and suggest new forms of extension
under this framework that take advantage of the structure of the kernel matrix.
We support our theoretical results numerically and demonstrate the advantages
of our extension framework both on synthetic and real data.Comment: 22 pages, 9 figure
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