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

Sampling theory is a fundamental area of study in harmonic analysis and signal and image processing. Our talk will 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. In particular, we discuss spherical geometry, hyperbolic geometry, and the geometry of general surfaces.

There are numerous motivations for extending sampling to non-Euclidean geometries. Applications of sampling in spherical and hyperbolic geometries are showing up areas from EIT to cosmology. 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. Irregular sampling of band-limited functions by iteration in hyperbolic space is possible, as shown by Feichtinger and Pensenson. 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 develop an approach to determine Nyquist tiles and sampling groups for spherical and hyperbolic space. We then connect this to arbitrary orientable analytic surfaces using Uniformization. We close with an application to network tomography, and in particular, a model of how one could monitor internet traffic in a computationally efficient manner.

Speaker Bio: Stephen Casey is a Professor of Mathematics and Affiliate Professor of Computer Science at American University. His research is in complex analysis, harmonic analysis, and number theory with applications to signal and image processing. He is a founding member of the Editorial Board for Sampling Theory in Signal and Image Processing and an Associate Editor of The Journal of Signal and Image Processing. He was chair of SampTA 2015, the 11th biennial international conference on Sampling Theory and its Applications, and was named Guest Editor-in-Chief of Sampling Theory in Signal and Image Processing in June 2015. His work includes multichannel deconvolution and multi-rate sampling, sampling in Euclidean and non-Euclidean geometries, sampling for wideband signals, signal adaptive frame theory, and the analysis of pulse train signals. He was just awarded a U.S. Patent for his work on sampling for wideband signals and signal adaptive frame theory.

Presentation Slides: PDF

Note: Visitors from outside NIST must contact Cathy Graham; (301) 975-3800; at least 24 hours in advance.