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Sampling in Euclidean and Non-Euclidean Domains: A Unified Approach

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Sampling Theory, a Renaissance

Part of the book series: Applied and Numerical Harmonic Analysis ((ANHA))

Abstract

Sampling theory is a fundamental area of study in harmonic analysis and signal and image processing. The purpose of this paper is to connect sampling theory with the geometry of the signal and its domain. It is relatively easy to demonstrate this connection in Euclidean spaces, but one quickly gets into open problems when the underlying space is not Euclidean. We focus primarily on Euclidean and hyperbolic geometries.There are numerous motivations for extending sampling to non-Euclidean geometries. Applications of sampling in non-Euclidean geometries are showing up areas from EIT to cosmology. Irregular sampling of bandlimited functions by iteration in hyperbolic space is possible, as shown by Feichtinger and Pesenson. Sampling in spherical geometry has been analyzed by many authors, e.g., Driscoll, Healy, Keiner, Kunis, McEwen, Potts, and Wiaux, and brings up questions about tiling the sphere. In Euclidean space, the minimal sampling rate for Paley-Wiener functions on \(\mathbb{R}^{d}\), the Nyquist rate, is a function of the bandwidth. No such rate has yet been determined for hyperbolic or spherical spaces. We look to develop a structure for the tiling of frequency spaces in both Euclidean and non-Euclidean domains. In particular, we establish Nyquist tiles and sampling groups in Euclidean geometry, and discuss the extension of these concepts to hyperbolic and spherical geometry and general orientable surfaces.

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Acknowledgements

The authors would like to thank the referees for their valuable input. First author’s research was partially supported by US Army Research Office Scientific Services program, administered by Battelle (TCN 06150, Contract DAAD19-02-D-0001) and US Air Force Office of Scientific Research Grant Number FA9550-12-1-0430.

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Authors and Affiliations

  1. Department of Mathematics and Statistics, American University, Washington, DC, 20016-8050, USA

    Stephen D. Casey & Jens Gerlach Christensen

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  1. Stephen D. Casey
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  2. Jens Gerlach Christensen
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Correspondence to Stephen D. Casey .

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Editors and Affiliations

  1. School of Engineering and Science, Jacobs University Bremen, Bremen, Bremen, Germany

    Götz E. Pfander

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Casey, S.D., Christensen, J.G. (2015). Sampling in Euclidean and Non-Euclidean Domains: A Unified Approach. In: Pfander, G. (eds) Sampling Theory, a Renaissance. Applied and Numerical Harmonic Analysis. Birkhäuser, Cham. https://doi.org/10.1007/978-3-319-19749-4_9

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