12,199 research outputs found
Sampling of operators
Sampling and reconstruction of functions is a central tool in science. A key
result is given by the sampling theorem for bandlimited functions attributed to
Whittaker, Shannon, Nyquist, and Kotelnikov. We develop an analogous sampling
theory for operators which we call bandlimited if their Kohn-Nirenberg symbols
are bandlimited. We prove sampling theorems for such operators and show that
they are extensions of the classical sampling theorem
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
Irregular and multi--channel sampling of operators
The classical sampling theorem for bandlimited functions has recently been
generalized to apply to so-called bandlimited operators, that is, to operators
with band-limited Kohn-Nirenberg symbols. Here, we discuss operator sampling
versions of two of the most central extensions to the classical sampling
theorem. In irregular operator sampling, the sampling set is not periodic with
uniform distance. In multi-channel operator sampling, we obtain complete
information on an operator by multiple operator sampling outputs
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