Sampling Trajectories for Mobile Sensing

Classical sampling theory for sampling and reconstructing bandlimited fields in $\Re^d$ addresses the problem of sampling on lattice points. We consider a generalization of this problem, in which one samples the field along $1$-dimensional spatial trajectories in $\Re^d$ rather than at points. The process of sampling records the value of the field at all points on the sampling trajectories. Such a sampling setup is relevant in the problem of spatial sampling using mobile sensors. We study various possible designs of sampling trajectories and discuss necessary and sufficient conditions for perfect reconstruction. We introduce a density metric for trajectories which we call \emph{path density}, that quantifies the total length of these trajectories per unit volume in $\Re^d$. We formulate the problem of identifying optimal sampling trajectories that admit perfect reconstruction of bandlimited fields and are minimal in terms of the path density metric. We identify optimal sampling trajectories from a restricted class of trajectories.