text: We focus on a multidimensional field with uncorrelated spectrum and study the quality of the reconstructed signal when the field samples are irregularly spaced and affected by independent and identically distributed noise. More specifically, we apply linear reconstruction techniques and take the mean-square error (MSE) of the field estimate as a metric to evaluate the signal reconstruction quality. We find that the MSE analysis could be carried out by using the closed-form expression of the eigenvalue distribution of the matrix representing the sampling system. Unfortunately, such distribution is still unknown. Thus, we first derive a closed-form expression of the distribution moments, and we find that the eigenvalue distribution tends to the Marcenko-Pastur distribution as the field dimension goes to infinity. Finally, by using our approach, we derive a tight approximation to the MSE of the reconstructed field.
  @cite_4: We bound the number of sensors required to achieve a desired level of sensing accuracy in a discrete sensor network application (e.g. distributed detection). We model the state of nature being sensed as a discrete vector, and the sensor network as an encoder. Our model assumes that each sensor observes only a subset of the state of nature, that sensor observations are localized and dependent, and that sensor network output across different states of nature is neither identical nor independently distributed. Using a random coding argument we prove a lower bound on the 'sensing capacity' of a sensor network, which characterizes the ability of a sensor network to distinguish among all states of nature. We compute this lower bound for sensors of varying range, noise models, and sensing functions. We compare this lower bound to the empirical performance of a belief propagation based sensor network decoder for a simple seismic sensor network scenario. The key contribution of this paper is to introduce the idea of a sharp cut-off function in the number of required sensors, to the sensor network community.
  @cite_7: Wireless Sensor Networks (WSN) are characterized by the dense deployment of sensor nodes that continuously observe physical phenomenon. Due to high density in the network topology, sensor observations are highly correlated in the space domain. Furthermore, the nature of the physical phenomenon constitutes the temporal correlation between each consecutive observation of a sensor node. These spatial and temporal correlations along with the collaborative nature of the WSN bring significant potential advantages for the development of efficient communication protocols well-suited for the WSN paradigm. In this paper, several key elements are investigated to capture and exploit the correlation in the WSN for the realization of advanced efficient communication protocols. A theoretical framework is developed to model the spatial and temporal correlations in WSN. The objective of this framework is to enable the development of efficient communication protocols which exploit these advantageous intrinsic features of the WSN paradigm. Based on this framework, possible approaches are discussed to exploit spatial and temporal correlation for efficient medium access and reliable event transport in WSN, respectively.
  @cite_19: While high resolution, regularly gridded observations are generally preferred in remote sensing, actual observations are often not evenly sampled and have lower-than-desired resolution. Hence, there is an interest in resolution enhancement and image reconstruction. This paper discusses a general theory and techniques for image reconstruction and creating enhanced resolution images from irregularly sampled data. Using irregular sampling theory, we consider how the frequency content in aperture function-attenuated sidelobes can be recovered from oversampled data using reconstruction techniques, thus taking advantage of the high frequency content of measurements made with nonideal aperture filters. We show that with minor modification, the algebraic reconstruction technique (ART) is functionally equivalent to Grochenig's (1992) irregular sampling reconstruction algorithm. Using simple Monte Carlo simulations, we compare and contrast the performance of additive ART, multiplicative ART, and the scatterometer image reconstruction (SIR) (a derivative of multiplicative ART) algorithms with and without noise. The reconstruction theory and techniques have applications with a variety of sensors and can enable enhanced resolution image production from many nonimaging sensors. The technique is illustrated with ERS-2 and SeaWinds scatterometer data.
  @cite_23: The problems of sensor configuration and activation for the detection of correlated random fields using large sensor arrays are considered. Using results that characterize the large-array performance of sensor networks in this application, the detection capabilities of different sensor configurations are analyzed and compared. The dependence of the optimal choice of configuration on parameters such as sensor signal-to-noise ratio (SNR), field correlation, etc., is examined, yielding insights into the most effective choices for sensor selection and activation in various operating regimes.
  @cite_34: Perturbation theorems for regular sampling in shift-invariant spaces are derived using a generalized perturbation theorem for frames in a Hilbert space, which generalizes the irregular sampling theorem established by Chen Using this generalized irregular sampling theorem, estimates for the maximum perturbation are obtained. Some typical examples illustrate the result
  @cite_20: We consider sensor networks that measure spatio-temporal correlated processes. An important task in such settings is the reconstruction at a certain node, called the sink, of the data at all points of the field. We consider scenarios where data is time critical, so delay results in distortion, or suboptimal estimation and control. For the reconstruction, the only data available to the sink are the values measured at the nodes of the sensor network, and knowledge of the correlation structure: this results in spatial distortion of reconstruction. Also, for the sake of power efficiency, sensor nodes need to transmit their data by relaying through the other network nodes: this results in delay, and thus temporal distortion of reconstruction if time critical data is concerned. We study data gathering for the case of Gaussian processes in one- and two-dimensional grid scenarios, where we are able to write explicit expressions for the spatial and time distortion, and combine them into a single total distortion measure. We prove that, for various standard correlation structures, there is an optimal finite density of the sensor network for which the total distortion is minimized. Thus, when power efficiency and delay are both considered in data gathering, it is useless from the point of view of accuracy of the reconstruction to increase the number of sensors above a certain threshold that depends on the correlation structure characteristics.
  @cite_17: Wireless sensor networks have attracted attention from a diverse set of researchers, due to the unique combination of distributed, resource and data processing constraints. However, until now, the lack of real sensor network deployments have resulted in ad-hoc assumptions on a wide range of issues including topology characteristics and data distribution. As deployments of sensor networks become more widespread [1, 2], many of these assumptions need to be revisited.This paper deals with the fundamental issue of spatio-temporal irregularity in sensor networks We make the case for the existence of such irregular spatio-temporal sampling, and show that it impacts many performance issues in sensor networks. For instance, data aggregation schemes provide inaccurate results, compression efficiency is dramatically reduced, data storage skews storage load among nodes and incurs significantly greater routing overhead. To mitigate the impact of irregularity, we outline a spectrum of solutions. For data aggregation and compression, we propose the use of spatial interpolation of data (first suggested by in [3] and temporal signal segmentation followed by alignment. To reduce the cost of data-centric storage and routing, we propose the use of virtualization, and boundary detection.
summary: The field reconstruction at the sink node with spatial and temporal correlation among sensor measures is studied, for instance, in @cite_20 @cite_23 @cite_17 @cite_7 @cite_4 . Other interesting studies can be found in @cite_34 @cite_19 , which address the perturbations of regular sampling in shift-invariant spaces @cite_34 and the reconstruction of irregularly sampled images in presence of measurement noise @cite_19 .