6,163 research outputs found
A novel sampling theorem on the rotation group
We develop a novel sampling theorem for functions defined on the
three-dimensional rotation group SO(3) by connecting the rotation group to the
three-torus through a periodic extension. Our sampling theorem requires
samples to capture all of the information content of a signal band-limited at
, reducing the number of required samples by a factor of two compared to
other equiangular sampling theorems. We present fast algorithms to compute the
associated Fourier transform on the rotation group, the so-called Wigner
transform, which scale as , compared to the naive scaling of .
For the common case of a low directional band-limit , complexity is reduced
to . Our fast algorithms will be of direct use in speeding up the
computation of directional wavelet transforms on the sphere. We make our SO3
code implementing these algorithms publicly available.Comment: 5 pages, 2 figures, minor changes to match version accepted for
publication. Code available at http://www.sothree.or
Density estimation on the rotation group using diffusive wavelets
This paper considers the problem of estimating probability density functions
on the rotation group . Two distinct approaches are proposed, one based
on characteristic functions and the other on wavelets using the heat kernel.
Expressions are derived for their Mean Integrated Squared Errors. The
performance of the estimators is studied numerically and compared with the
performance of an existing technique using the De La Vall\'ee Poussin kernel
estimator. The heat-kernel wavelet approach appears to offer the best
convergence, with faster convergence to the optimal bound and guaranteed
positivity of the estimated probability density function
On representations of the rotation group and magnetic monopoles
Recently (Phys. Lett. A302 (2002) 253, hep-th/0208210; hep-th/0403146)
employing bounded infinite-dimensional representations of the rotation group we
have argued that one can obtain the consistent monopole theory with generalized
Dirac quantization condition, , where is the
weight of the Dirac string. Here we extend this proof to the unbounded
infinite-dimensional representations.Comment: References adde
Infinite-dimensional representations of the rotation group and Dirac's monopole problem
Within the context of infinite-dimensional representations of the rotation
group the Dirac monopole problem is studied in details. Irreducible
infinite-dimensional representations, being realized in the indefinite metric
Hilbert space, are given by linear unbounded operators in infinite-dimensional
topological spaces, supplied with a weak topology and associated weak
convergence. We argue that an arbitrary magnetic charge is allowed, and the
Dirac quantization condition can be replaced by a generalized quantization rule
yielding a new quantum number, the so-called topological spin, which is related
to the weight of the Dirac string.Comment: JHEP style. Extended version of hep-th/0403146. Revised version,
title and some notations are changed. References and Appendix B are adde
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