On Minimal Trajectories for Mobile Sampling of Bandlimited Fields

We study the design of sampling trajectories for stable sampling and the reconstruction of bandlimited spatial fields using mobile sensors. The spectrum is assumed to be a symmetric convex set. As a performance metric we use the path density of the set of sampling trajectories that is defined as the total distance traveled by the moving sensors per unit spatial volume of the spatial region being monitored. Focussing first on parallel lines, we identify the set of parallel lines with minimal path density that contains a set of stable sampling for fields bandlimited to a known set. We then show that the problem becomes ill-posed when the optimization is performed over all trajectories by demonstrating a feasible trajectory set with arbitrarily low path density. However, the problem becomes well-posed if we explicitly specify the stability margins. We demonstrate this by obtaining a non-trivial lower bound on the path density of an arbitrary set of trajectories that contain a sampling set with explicitly specified stability bounds.Comment: 28 pages, 8 figure

Sampling 2-D Signals on a Union of Lattices that Intersect on a Lattice

This paper presents new sufficient conditions under which a field (or image) can be perfectly reconstructed from its samples on a union of two lattices that share a common coarse lattice. In particular, if samples taken on the first lattice can be used to reconstruct a field bandlimited to some spectral support region, and likewise samples taken on the second lattice can reconstruct a field bandlimited to another spectral support region, then under certain conditions, a field bandlimited to the union of these two spectral regions can be reconstructed from its samples on the union of the two respective lattices. These results generalize a previous perfect reconstruction theorem for Manhattan sampling, where data is taken at high density along evenly spaced rows and columns of a rectangular grid. Additionally, a sufficient condition is given under which the Landau lower bound is achieved

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.Comment: Submitted to IEEE Transactions on Information Theory, Nov 2011; revised July 2012; accepted Oct 201

On sampling a high-dimensional bandlimited field on a union of shifted lattices

We study the problem of sampling a high-dimensional bandlimited field on a union of shifted lattices under certain assumptions motivated by some practical sampling applications. Under these assumptions, we show that simple necessary and sufficient conditions for perfect reconstruction can be identified. We also obtain an explicit scheme for reconstructing the field from its samples on the various shifted lattices. We illustrate our results using examples

Manhattan Cutset Sampling and Sensor Networks.

Cutset sampling is a new approach to acquiring two-dimensional data, i.e., images, where values are recorded densely along straight lines. This type of sampling is motivated by physical scenarios where data must be taken along straight paths, such as a boat taking water samples. Additionally, it may be possible to better reconstruct image edges using the dense amount of data collected on lines. Finally, an advantage of cutset sampling is in the design of wireless sensor networks. If battery-powered sensors are placed densely along straight lines, then the transmission energy required for communication between sensors can be reduced, thereby extending the network lifetime. A special case of cutset sampling is Manhattan sampling, where data is recorded along evenly-spaced rows and columns. This thesis examines Manhattan sampling in three contexts. First, we prove a sampling theorem demonstrating an image can be perfectly reconstructed from Manhattan samples when its spectrum is bandlimited to the union of two Nyquist regions corresponding to the two lattices forming the Manhattan grid. An efficient ``onion peeling'' reconstruction method is provided, and we show that the Landau bound is achieved. This theorem is generalized to dimensions higher than two, where again signals are reconstructable from a Manhattan set if they are bandlimited to a union of Nyquist regions. Second, for non-bandlimited images, we present several algorithms for reconstructing natural images from Manhattan samples. The Locally Orthogonal Orientation Penalization (LOOP) algorithm is the best of the proposed algorithms in both subjective quality and mean-squared error. The LOOP algorithm reconstructs images well in general, and outperforms competing algorithms for reconstruction from non-lattice samples. Finally, we study cutset networks, which are new placement topologies for wireless sensor networks. Assuming a power-law model for communication energy, we show that cutset networks offer reduced communication energy costs over lattice and random topologies. Additionally, when solving centralized and decentralized source localization problems, cutset networks offer reduced energy costs over other topologies for fixed sensor densities and localization accuracies. Finally, with the eventual goal of analyzing different cutset topologies, we analyze the energy per distance required for efficient long-distance communication in lattice networks.PhDElectrical Engineering: SystemsUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttp://deepblue.lib.umich.edu/bitstream/2027.42/120876/1/mprelee_1.pd