Sampling trajectories for mobile sensing

Unnikrishnan, Jayakrishnan; Vetterli, Martin

doi:10.1109/allerton.2011.6120308

Cited by 6 publications

(12 citation statements)

References 15 publications

(32 reference statements)

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“…For a field , we define its Fourier transform as (1) where denotes the imaginary unit, and denotes the scalar product between vectors and in . We use to denote the collection of fields with finite energy such that the Fourier transform of is supported on a set , i.e., …”

Section: Notations and Conventionsmentioning

confidence: 99%

“…We say that a set of points is uniformly discrete if we have , i.e., there exists such that for any two distinct points , we have . 1 For a set , let denote the class of fields bandlimited to as defined in (2). Let denote the collection of all uniformly discrete sets which have the property that any field can be reconstructed exactly from its values on , i.e., is uniquely determined from .…”

Section: A Preliminariesmentioning

confidence: 99%

“…In a practical deployment, it is not possible to take measurements of the field at all points along a continuous path because 1 For example, lattices in are uniformly discrete, but a sequence in converging to a point in is not. a continuous path has an infinite number of points.…”

Section: Definition 22mentioning

confidence: 99%

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

Unnikrishnan

Vetterli

2012

2012 IEEE International Symposium on Information Theory Proceedings

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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.

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Section: Notations and Conventionsmentioning

confidence: 99%

Section: A Preliminariesmentioning

confidence: 99%

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

Unnikrishnan

Vetterli

2012

2012 IEEE International Symposium on Information Theory Proceedings

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“…for all ), then the expressions for the variance in (16), (18) and (21) tend to infinity. This is because unfiltered white noise samples have infinite variance.…”

Section: Mobile Sampling With Ideal Low-pass Filtermentioning

confidence: 99%

“…However, to the best of our knowledge, this is the first work to illustrate the possibility of implementing spatial smoothing by using a mobile sensing scheme together with time-domain anti-aliasing filtering, and to quantify the improvements of such a scheme over static sensing. In earlier work [15], [16], we studied the problem of designing trajectories for mobile sensing which minimize the total distance required to be traveled by the sensors per unit area of the field being sampled.…”

Section: Introductionmentioning

confidence: 99%

Sampling and Reconstruction of Spatial Fields Using Mobile Sensors

Unnikrishnan

Vetterli

2013

IEEE Trans. Signal Process.

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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.

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