2,882 research outputs found
Left-Invariant Diffusion on the Motion Group in terms of the Irreducible Representations of SO(3)
In this work we study the formulation of convection/diffusion equations on
the 3D motion group SE(3) in terms of the irreducible representations of SO(3).
Therefore, the left-invariant vector-fields on SE(3) are expressed as linear
operators, that are differential forms in the translation coordinate and
algebraic in the rotation. In the context of 3D image processing this approach
avoids the explicit discretization of SO(3) or , respectively. This is
particular important for SO(3), where a direct discretization is infeasible due
to the enormous memory consumption. We show two applications of the framework:
one in the context of diffusion-weighted magnetic resonance imaging and one in
the context of object detection
Diffusive spin transport
Information to be stored and transported requires physical carriers. The
quantum bit of information (qubit) can for instance be realised as the spin 1/2
degree of freedom of a massive particle like an electron or as the spin 1
polarisation of a massless photon. In this lecture, I first use irreducible
representations of the rotation group to characterise the spin dynamics in a
least redundant manner. Specifically, I describe the decoherence dynamics of an
arbitrary spin S coupled to a randomly fluctuating magnetic field in the
Liouville space formalism. Secondly, I discuss the diffusive dynamics of the
particle's position in space due to the presence of randomly placed impurities.
Combining these two dynamics yields a coherent, unified picture of diffusive
spin transport, as applicable to mesoscopic electronic devices or photons
propagating in cold atomic clouds.Comment: Lecture notes, published in A. Buchleitner, C. Viviescas, and M.
Tiersch (Eds.), "Entanglement and Decoherence. Foundations and Modern
Trends", Lecture Notes in Physics 768, Springer, Berlin (2009
Numerical Approaches for Linear Left-invariant Diffusions on SE(2), their Comparison to Exact Solutions, and their Applications in Retinal Imaging
Left-invariant PDE-evolutions on the roto-translation group (and
their resolvent equations) have been widely studied in the fields of cortical
modeling and image analysis. They include hypo-elliptic diffusion (for contour
enhancement) proposed by Citti & Sarti, and Petitot, and they include the
direction process (for contour completion) proposed by Mumford. This paper
presents a thorough study and comparison of the many numerical approaches,
which, remarkably, is missing in the literature. Existing numerical approaches
can be classified into 3 categories: Finite difference methods, Fourier based
methods (equivalent to -Fourier methods), and stochastic methods (Monte
Carlo simulations). There are also 3 types of exact solutions to the
PDE-evolutions that were derived explicitly (in the spatial Fourier domain) in
previous works by Duits and van Almsick in 2005. Here we provide an overview of
these 3 types of exact solutions and explain how they relate to each of the 3
numerical approaches. We compute relative errors of all numerical approaches to
the exact solutions, and the Fourier based methods show us the best performance
with smallest relative errors. We also provide an improvement of Mathematica
algorithms for evaluating Mathieu-functions, crucial in implementations of the
exact solutions. Furthermore, we include an asymptotical analysis of the
singularities within the kernels and we propose a probabilistic extension of
underlying stochastic processes that overcomes the singular behavior in the
origin of time-integrated kernels. Finally, we show retinal imaging
applications of combining left-invariant PDE-evolutions with invertible
orientation scores.Comment: A final and corrected version of the manuscript is Published in
Numerical Mathematics: Theory, Methods and Applications (NM-TMA), vol. (9),
p.1-50, 201
Discrepancy convergence for the drunkard's walk on the sphere
We analyze the drunkard's walk on the unit sphere with step size theta and
show that the walk converges in order constant/sin^2(theta) steps in the
discrepancy metric. This is an application of techniques we develop for
bounding the discrepancy of random walks on Gelfand pairs generated by
bi-invariant measures. In such cases, Fourier analysis on the acting group
admits tractable computations involving spherical functions. We advocate the
use of discrepancy as a metric on probabilities for state spaces with isometric
group actions.Comment: 20 pages; to appear in Electron. J. Probab.; related work at
http://www.math.hmc.edu/~su/papers.htm
New Exact and Numerical Solutions of the (Convection-)Diffusion Kernels on SE(3)
We consider hypo-elliptic diffusion and convection-diffusion on , the quotient of the Lie group of rigid body motions SE(3) in
which group elements are equivalent if they are equal up to a rotation around
the reference axis. We show that we can derive expressions for the convolution
kernels in terms of eigenfunctions of the PDE, by extending the approach for
the SE(2) case. This goes via application of the Fourier transform of the PDE
in the spatial variables, yielding a second order differential operator. We
show that the eigenfunctions of this operator can be expressed as (generalized)
spheroidal wave functions. The same exact formulas are derived via the Fourier
transform on SE(3). We solve both the evolution itself, as well as the
time-integrated process that corresponds to the resolvent operator.
Furthermore, we have extended a standard numerical procedure from SE(2) to
SE(3) for the computation of the solution kernels that is directly related to
the exact solutions. Finally, we provide a novel analytic approximation of the
kernels that we briefly compare to the exact kernels.Comment: Revised and restructure
The Eyring-Kramers law for Markovian jump processes with symmetries
We prove an Eyring-Kramers law for the small eigenvalues and mean
first-passage times of a metastable Markovian jump process which is invariant
under a group of symmetries. Our results show that the usual Eyring-Kramers law
for asymmetric processes has to be corrected by a factor computable in terms of
stabilisers of group orbits. Furthermore, the symmetry can produce additional
Arrhenius exponents and modify the spectral gap. The results are based on
representation theory of finite groups.Comment: 39 pages, 9 figure
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