Novel spherical informatics techniques for studying cosmic evolution
Lead Research Organisation:
UNIVERSITY COLLEGE LONDON
Department Name: Physics and Astronomy
Abstract
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Publications
Roddy P
(2021)
Sifting Convolution on the Sphere
in IEEE Signal Processing Letters
Roddy PJ
(2022)
https://discovery.ucl.ac.uk/id/eprint/10156089/
Studentship Projects
| Project Reference | Relationship | Related To | Start | End | Student Name |
|---|---|---|---|---|---|
| ST/P006736/1 | 30/09/2017 | 30/03/2026 | |||
| 1966434 | Studentship | ST/P006736/1 | 30/09/2017 | 29/09/2021 | Patrick Roddy |
| Description | This thesis is primarily split into two parts: the development of a spherical convolution, and the construction of a wavelet basis built on this convolution. The sifting convolution is designed as a Euclidean analogue; although initially designed for the sphere, it is indeed general. Slepian wavelets were again considered for the sphere, but may be generalised further to graph and manifold data. These wavelets are intended for the analysis of incomplete manifolds. Convolutions are a fundamental technique in signal processing that are used widely in scientific and engineering applications. Often the convolution is part of a wider technique, although they may be used directly in their own right. Existing convolutions in the spherical domain impose certain restrictions on one or both inputs. The sifting convolution therefore may find use in various applications where one seeks to accept directional inputs whilst remaining on the sphere. Moreover, the convolution is general and, as such, may find success in general manifold settings. With the recent interest in geometric deep learning, the sifting convolution may find application in the development of convolutional neural networks on manifolds. Wavelets are also widely used in various disciplines to analyse signals both in space and scale. Whilst many fields measure data on manifolds (i.e. the sphere), often data are only observed on a partial region of the manifold. Wavelets are a typical approach to data of this form, but the wavelet coefficients which overlap with the boundary become contaminated and must be removed for accurate analysis. Another approach is to estimate the region of missing data and to use existing whole-manifold methods for analysis. However, both approaches introduce uncertainty into any analysis. Slepian wavelets enable one to work directly with only the data present, thus avoiding the problems discussed above. Possible applications of Slepian wavelets to areas of research measuring data on the partial sphere include: gravitational/magnetic fields in geodesy; ground-based measurements in astronomy; measurements of whole-planet properties in planetary science; geomagnetism of the Earth; and in analyses of the cosmic microwave background. |
| Exploitation Route | Slepian wavelets have many potential applications in analyses of manifolds where data are only observed over a partial region. One such application is in CMB analyses, where observations are inherently made on the celestial sphere, and foreground emissions mask the centre of the data. In fields such as astrophysics and cosmology, datasets are increasingly large and thus require analysis at high resolutions for accurate predictions. Whilst Slepian wavelets may be trivially computed from a set of Slepian functions, the computation of the spherical Slepian functions themselves are computationally complex, where the matrix scales as O(L^4). Although symmetries of this matrix and the spherical harmonics have been exploited, filling in this matrix is inherently slow due to the many integrals performed. The matrix is filled in parallel in Python using multiprocessing, and the spherical harmonic transforms are computed in C using SSHT. This may be sped up further by utilising the new ducc0 backend for SSHT, which may allow for a multithreaded solution. Ultimately, the eigenproblem must be solved to compute the Slepian functions, requiring sophisticated algorithms to balance speed and accuracy. Therefore, to work with high-resolution data such as these, one requires HPC methods on supercomputers with massive memory and storage. To this end, Slepian wavelets may be exploited at present at low resolutions, but further work is required for them to be fully scalable. |
| Sectors | Digital/Communication/Information Technologies (including Software) Other |