Sampling High-Dimensional Bandlimited Fields on Low-Dimensional Manifolds

Consider the task of sampling and reconstructing a bandlimited spatial field in $\Re^2$ using moving sensors that take measurements along their path. It is inexpensive to increase the sampling rate along the paths of the sensors but more expensive to increase the total distance traveled by the sensors per unit area, which we call the \emph{path density}. In this paper we introduce the problem of designing sensor trajectories that are minimal in path density subject to the condition that the measurements of the field on these trajectories admit perfect reconstruction of bandlimited fields. We study various possible designs of sampling trajectories. Generalizing some ideas from the classical theory of sampling on lattices, we obtain necessary and sufficient conditions on the trajectories for perfect reconstruction. We show that a single set of equispaced parallel lines has the lowest path density from certain restricted classes of trajectories that admit perfect reconstruction. We then generalize some of our results to higher dimensions. We first obtain results on designing sampling trajectories in higher dimensional fields. Further, interpreting trajectories as $1$-dimensional manifolds, we extend some of our ideas to higher dimensional sampling manifolds. We formulate the problem of designing $\kappa$-dimensional sampling manifolds for $d$-dimensional spatial fields that are minimal in \emph{manifold density}, a natural generalization of the path density. We show that our results on sampling trajectories for fields in $\Re^2$ can be generalized to analogous results on $d-1$-dimensional sampling manifolds for $d$-dimensional spatial fields.