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A Novel Sampling Theorem on the Rotation Group

McEwen, JD; Buettner, M; Leistedt, B; Peiris, HV; Wiaux, Y; (2015) A Novel Sampling Theorem on the Rotation Group. IEEE Signal Processing Letters , 22 (12) pp. 2425-2429. 10.1109/LSP.2015.2490676. Green open access

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Abstract

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 4L 3 samples to capture all of the information content of a signal band-limited at L, 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 O(L 4 ), compared to the naive scaling of O(L 6 ). For the common case of a low directional band-limit N, complexity is reduced to O(NL3 ). 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.

Type: Article
Title: A Novel Sampling Theorem on the Rotation Group
Open access status: An open access version is available from UCL Discovery
DOI: 10.1109/LSP.2015.2490676
Publisher version: http://doi.org/10.1109/LSP.2015.2490676
Language: English
Additional information: © 2015 IEEE. Personal use of this material is permitted. Permission from IEEE must be obtained for all other users, including reprinting/ republishing this material for advertising or promotional purposes, creating new collective works for resale or redistribution to servers or lists, or reuse of any copyrighted components of this work in other works.
Keywords: Science & Technology, Technology, Engineering, Electrical & Electronic, Engineering, Harmonic analysis, rotation group, sampling, spheres, wigner transform, SPHERICAL HARMONIC TRANSFORMS, WAVELETS, GRAVITY, SERIES
UCL classification: UCL
UCL > Provost and Vice Provost Offices > UCL BEAMS
UCL > Provost and Vice Provost Offices > UCL BEAMS > Faculty of Maths and Physical Sciences
UCL > Provost and Vice Provost Offices > UCL BEAMS > Faculty of Maths and Physical Sciences > Dept of Physics and Astronomy
UCL > Provost and Vice Provost Offices > UCL BEAMS > Faculty of Maths and Physical Sciences > Dept of Space and Climate Physics
URI: https://discovery.ucl.ac.uk/id/eprint/1508537
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