Sampling of Structured Signals: Techniques and Imaging Applications
The celebrated Shannon sampling theorem is a key mathematical tool that allows one to seamlessly switch between the continuous-time and discrete-time representations of bandlimited signals. Sampling and reconstruction of signals that are not bandlimited has been addressed within several sampling frameworks, each suitably designed to accommodate a particular class of signals. The design of these sampling frameworks stems from the careful observation of the implicit structure present in the signals. My thesis focuses on the sampling of a class of signals called finite-rate-of-innovation (FRI) signals --- these signals are not necessarily bandlimited, but are completely specified by a finite number of parameters per unit interval of time. In the case of FRI sampling, we consider signals that are a sum-of-weighted and time-shifted (SWTS) pulses, asymmetric pulse trains, and modulated signals. We also consider sampling of FRI signals that are 2-D counterparts of the 1-D FRI signals of the SWTS form. Further, we address two alternatives to the uniform sampling mechanism: (i) time-encoding of FRI signals, which is a neuromorphic sampling scheme that results in nonuniformly spaced samples; and (ii) unlimited sampling of signals, which involves reconstruction of signal from its modulo measurements. We also demonstrate super-resolution reconstruction in imaging applications such as ultrasound, sonar, and ground penetrating radar.