Signal sampling and reconstruction is a fundamental engineering task at the heart of signal processing. The celebrated Shannon-Nyquist theorem guarantees perfect signal reconstruction from uniform samples, obtained at a rate twice the maximum frequency present in the signal. Unfortunately a large number of signals of interest are far from being band-limited. This motivated research on reconstruction from sub-Nyquist samples, which mainly hinges on the use of random/incoherent sampling procedures. However, uniform or regular sampling is more appealing in practice and from the system design point of view, as it is far simpler to implement, and often necessary due to system constraints. In this work, we study regular sampling and reconstruction of three- or higher-dimensional signals (tensors). We show that reconstructing a tensor signal from regular samples is feasible. Under the proposed framework, the sample complexity is determined by the tensor rank - rather than the signal bandwidth. This result offers new perspectives for designing practical regular sampling patterns and systems for signals that are naturally tensors, e.g., images and video. For a concrete application, we show that functional magnetic resonance imaging (fMRI) acceleration is a tensor sampling problem, and design practical sampling schemes and an algorithmic framework to handle it. Numerical results show that our tensor sampling strategy accelerates the fMRI sampling process significantly without sacrificing reconstruction accuracy.
Bibliographical noteFunding Information:
Manuscript received February 23, 2019; revised August 2, 2019 and September 25, 2019; accepted October 5, 2019. Date of publication November 6, 2019; date of current version December 13, 2019. The associate editor coordinating the review of this manuscript and approving it for publication was Dr. Abd-Krim Seghouane. The work of C. I. Kanatsoulis was supported by the National Science Foundation under Grant IIS-1447788 and Grant IIS-1704074. The work of X. Fu was supported in part by the National Science Foundation under Project ECCS 1808159 and by Army Research Office under grant ARO W911NF-19-1-0247. The work of N. D. Sidiropoulos was supported in part by the National Science Foundation under Grant ECCS-1807660. The work of M. Akçakaya was supported in part by the National Science Foundation under Grant CCF-1651825 and in part by the National Institutes of Health under Grant P41EB015894 and Grant P41EB027061. (Corresponding author: Nicholas D. Sidiropoulos.) C. I. Kanatsoulis and M. Akçakaya are with the Department of Electrical and Computer Engineering, University of Minnesota, Minneapolis, MN 55455 USA (e-mail: email@example.com; firstname.lastname@example.org).
- MRI acceleration
- functional MRI
- tensor completion