Sampling and reconstructing spatial fields using mobile sensors

2012 IEEE International Symposium on Information Theory Proceedings

2012

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Abstract-Consider the task of sampling and reconstructing a bandlimited spatial field in 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 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-D manifolds, we extend some of our ideas to higher dimensional sampling manifolds. We formulate the problem of designing -dimensional sampling manifolds for -dimensional spatial fields that are minimal in manifold density, a natural generalization of the path density. We show that our results on sampling trajectories for fields in can be generalized to analogous results on -dimensional sampling manifolds for -dimensional spatial fields.

“…The path density of a uniform set of the form in (7) is given by (12) Furthermore, satisfies condition (C2).…”

Section: Lemma 22mentioning

confidence: 99%

mentioning

confidence: 99%

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

2012 IEEE International Symposium on Information Theory Proceedings

2012

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“…This has been noted to lead to a significant reduction in the amount of spatial aliasing in some applications [3]. Furthermore, as we point out in [4], a moving sensor admits filtering over space in the direction of motion of the sensor whereas no such spatial filtering is possible in the case of static sampling.…”

Section: Introductionmentioning

confidence: 99%

“…Let ℓ i denote the length of the intersection of the disc with the trajectory p i defined in (4). Let N ≈ a α denote the highest index of the trajectories with non-zero intersection with the disc.…”

Section: Proof Of Lemma 23mentioning

confidence: 99%

Sampling trajectories for mobile sensing

2011 49th Annual Allerton Conference on Communication, Control, and Computing (Allerton)

2011

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Abstract-Classical sampling theory for sampling and reconstructing bandlimited fields in R 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 R 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 path density, that quantifies the total length of these trajectories per unit volume in R 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.

“…Suppose further that the field is slowly varying in space and can hence be modeled as a spatially bandlimited field. Let represent the observed field that is a noisy version of the field of interest expressed as (1) where denotes non-bandlimited spatial noise, which we refer to as environmental noise. The objective in a typical sampling and reconstructing scheme is to use the samples of the observed field to obtain a reconstruction of the field such that the mean-square error (MSE) in the reconstruction is minimal.…”

Section: Introductionmentioning

confidence: 99%

Sampling and Reconstruction of Spatial Fields Using Mobile Sensors

IEEE Trans. Signal Process.

2013

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Abstract-Spatial sampling is traditionally studied in a static setting where static sensors scattered around space take measurements of the spatial field at their locations. In this paper, we study the emerging paradigm of sampling and reconstructing spatial fields using sensors that move through space. We show that mobile sensing offers some unique advantages over static sensing in sensing bandlimited spatial fields. Since a moving sensor encounters such a spatial field along its path as a time-domain signal, a time-domain anti-aliasing filter can be employed prior to sampling the signal received at the sensor. Such a filtering procedure, when used by a configuration of sensors moving at constant speeds along equispaced parallel lines, leads to a complete suppression of spatial aliasing in the direction of motion of the sensors. We analytically quantify the advantage of using such a sampling scheme over a static sampling scheme by computing the reduction in sampling noise due to the filter. We also analyze the effects of nonuniform sensor speeds on the reconstruction accuracy. Using simulation examples, we demonstrate the advantages of mobile sampling over static sampling in practical problems. We extend our analysis to sampling and reconstruction schemes for monitoring time-varying bandlimited fields using mobile sensors. We demonstrate that in some situations we require a lower density of sensors when using a mobile sensing scheme instead of the conventional static sensing scheme. The exact advantage is quantified for a problem of sampling and reconstructing an audio field.Index Terms-Mobile sensors, spatial anti-aliasing, spatial field sampling and reconstruction.