The last two decades have seen a renewed interest in sampling theory, which is concerned with the conversion of continuous-domain signals into discrete sequences. Traditionally, this conversion is achieved by recording the amplitude of the signal at specified time instants.

In this thesis, we explore novel sensing and imaging strategies. The first alternative sampling method we investigate is based on timing rather than amplitude information. Sampling based on timing also seems to appear in nature and may explain how neurons encode information, moreover, sampling based on timing information often leads to energy-efficient sampling devices. Significant progress has taken place in this area recently. Nevertheless, the state-of-the-art research has mostly concentrated on methods for sampling and reconstructing bandlimited functions, which were further extended to signals that belong to shift-invariant spaces.

In our work, we investigate the problem of timing-based sampling and reconstruction of non-bandlimited signals, within the Finite Rate of Innovation (FRI) setting. These signals have a finite number of degrees of freedom per unit of time, called rate of innovation, such as, for example, trains of impulses or piecewise constant polynomials. We show here that these functions can be non-uniformly sampled using a compact-support kernel that satisfies the generalised Strang-Fix conditions, and a comparator with a threshold detector, which outputs the instants where the filtered input crosses the values of some reference signal. In this context, we prove that perfect estimation is possible using a novel local reconstruction method, if the frequency of the comparator's reference signal is large enough, and whenever the input satisfies some density property. We also describe the integrate-and-fire time encoding system, which consists of a filter, followed by an integrator and a threshold detector with a reset. We show that also in this case, we can reconstruct signals with finite rate of innovation from the time-encoded information. We also extend the proposed framework to time encoding and reconstruction of multidimensional signals, such as video scenes.

The second problem we address is that of reconstructing physical phenomena from samples taken along unknown trajectories. In particular we consider diffusion fields induced by multiple localised and instantaneous sources and assume a mobile sensor samples the field, uniformly along a trajectory, which is unknown. The problem we address is the estimation of the amplitudes and locations of the diffusion sources, as well as of the trajectory of the sensor. We first propose a method for diffusion source localisation and trajectory mapping (D-SLAM) in 2D, where we assume the activation times of the sources are known, that the evolution of the diffusion field over time is negligible, and that the trajectory of the sensor is piecewise linear. The reconstruction method we propose maps the measurements obtained using the mobile sensor to a sequence of generalised field samples. From these generalised samples, we can then retrieve the locations of the sources as well as the trajectory of the sensor (up to a 2D orthogonal geometric transformation). We then relax these assumptions and show that we can perform D-SLAM also in the case of unknown activation times, from samples of a time-varying field, from samples along an arbitrary parametric trajectory, as well as in 3D spaces. Finally, simulation results on both synthetic and real data further validate the proposed framework.