Sampling and Reconstruction of Spatial Fields Using Mobile Sensors

Abstract

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.

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

“…2) Union of Uniform Sets: For vectors let denote the uniform set defined by where (13) where is a unit vector in orthogonal to and . Thus, is a uniform set oriented parallel to with the spacing between adjacent lines equal to apart.…”

“…As in the example we considered in Section II-B1, if we choose a small enough value of , then it can be shown via classical sampling results [19] that the sampled spectrum comprises spectral repetitions in one direction. We obtain the following spectrum analogous to (11): (15) Now suppose we have such uniform sets of the form (13). Let denote the union of all the sets (16) An example of such a trajectory set for is depicted in Fig.…”

“…Moreover, in some applications [11], moving sensors can sample the fields along their paths at high spatial frequencies thereby reducing the amount of spatial aliasing introduced in the samples. Furthermore, as we point out in [12] and [13], 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. Such spatial filtering helps in reducing the amount of aliasing and the contribution of out-of-band noise in the reconstructed field.…”

Unlabeled sensing: Reconstruction algorithm and theoretical guarantees

“…Therefore, at the end of the measurement phase, a set of sample values with some potential locations are accessible [1][2][3]. The ultimate goal is to recover the underlying spatial field which, at the same time, requires knowledge about the sampling locations.…”

Handheld Reflectance Acquisition of Paintings

“…A rigorous treatment of reconstruction of 2D spatial fields from samples along a trajectory is presented in [20]. However samples are obtained along well-defined curves, hence the approach is unsuitable for handheld sampling.…”