2,649 research outputs found
A semidiscrete version of the Citti-Petitot-Sarti model as a plausible model for anthropomorphic image reconstruction and pattern recognition
In his beautiful book [66], Jean Petitot proposes a sub-Riemannian model for
the primary visual cortex of mammals. This model is neurophysiologically
justified. Further developments of this theory lead to efficient algorithms for
image reconstruction, based upon the consideration of an associated
hypoelliptic diffusion. The sub-Riemannian model of Petitot and Citti-Sarti (or
certain of its improvements) is a left-invariant structure over the group
of rototranslations of the plane. Here, we propose a semi-discrete
version of this theory, leading to a left-invariant structure over the group
, restricting to a finite number of rotations. This apparently very
simple group is in fact quite atypical: it is maximally almost periodic, which
leads to much simpler harmonic analysis compared to Based upon this
semi-discrete model, we improve on previous image-reconstruction algorithms and
we develop a pattern-recognition theory that leads also to very efficient
algorithms in practice.Comment: 123 pages, revised versio
Fast Ewald summation for electrostatic potentials with arbitrary periodicity
A unified treatment for fast and spectrally accurate evaluation of
electrostatic potentials subject to periodic boundary conditions in any or none
of the three space dimensions is presented. Ewald decomposition is used to
split the problem into a real space and a Fourier space part, and the FFT based
Spectral Ewald (SE) method is used to accelerate the computation of the latter.
A key component in the unified treatment is an FFT based solution technique for
the free-space Poisson problem in three, two or one dimensions, depending on
the number of non-periodic directions. The cost of calculations is furthermore
reduced by employing an adaptive FFT for the doubly and singly periodic cases,
allowing for different local upsampling rates. The SE method will always be
most efficient for the triply periodic case as the cost for computing FFTs will
be the smallest, whereas the computational cost for the rest of the algorithm
is essentially independent of the periodicity. We show that the cost of
removing periodic boundary conditions from one or two directions out of three
will only marginally increase the total run time. Our comparisons also show
that the computational cost of the SE method for the free-space case is
typically about four times more expensive as compared to the triply periodic
case. The Gaussian window function previously used in the SE method, is here
compared to an approximation of the Kaiser-Bessel window function, recently
introduced. With a carefully tuned shape parameter that is selected based on an
error estimate for this new window function, runtimes for the SE method can be
further reduced. Keywords: Fast Ewald summation, Fast Fourier transform,
Arbitrary periodicity, Coulomb potentials, Adaptive FFT, Fourier integral,
Spectral accuracy.Comment: 21 pages, 11 figure
The exponentially convergent trapezoidal rule
It is well known that the trapezoidal rule converges geometrically when applied to analytic functions on periodic intervals or the real line. The mathematics and history of this phenomenon are reviewed and it is shown that far from being a curiosity, it is linked with computational methods all across scientific computing, including algorithms related to inverse Laplace transforms, special functions, complex analysis, rational approximation, integral equations, and the computation of functions and eigenvalues of matrices and operators
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