Optimally space-localized band-limited wavelets on <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si15.gif" overflow="scroll"><mml:msup><mml:mrow><mml:mi mathvariant="double-struck">S</mml:mi></mml:mrow><mml:mrow><mml:mi>q</mml:mi><mml:mo>-</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msup></mml:math>

Fernández, Noemí Laín

doi:10.1016/j.cam.2005.03.096

Cited by 21 publications

(1 citation statement)

References 7 publications

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“…In the late 1990s, Slepian functions on the sphere were derived for use in geodesy and planetary science [2, 4, 34-36, 44, 45]. In parallel, a few alternative approaches, using different measures of optimality, have been developed for constructing approximating structures on the sphere, see, for example, [20,21,24].…”

Section: Introductionmentioning

confidence: 99%

A general approach to regularizing inverse problems with regional data using Slepian wavelets

Michel

Simons

2017

Inverse Problems

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Slepian functions are orthogonal function systems that live on subdomains (for example, geographical regions on the Earth's surface, or bandlimited portions of the entire spectrum). They have been firmly established as a useful tool for the synthesis and analysis of localized (concentrated or confined) signals, and for the modeling and inversion of noise-contaminated data that are only regionally available or only of regional interest. In this paper, we consider a general abstract setup for inverse problems represented by a linear and compact operator between Hilbert spaces with a known singular-value decomposition (svd). In practice, such an svd is often only given for the case of a global expansion of the data (e.g. on the whole sphere) but not for regional data distributions. We show that, in either case, Slepian functions (associated to an arbitrarily prescribed region and the given compact operator) can be determined and applied to construct a regularization for the ill-posed regional inverse problem. Moreover, we describe an algorithm for constructing the Slepian basis via an algebraic eigenvalue problem. The obtained Slepian functions can be used to derive an svd for the combination of the regionalizing projection and the compact operator. As a result, standard regularization techniques relying on a known svd become applicable also to those inverse problems where the data are regionally given only. In particular, wavelet-based multiscale techniques can be used. An example for the latter case is elaborated theoretically and tested on two synthetic numerical examples.

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Section: Introductionmentioning

confidence: 99%