Continuous Sensor Placement

Abstract

Abstract-Existing solutions to the sensor placement problem are based on sensor selection, in which the best subset of available sampling locations is chosen such that a desired estimation accuracy is achieved. However, the achievable estimation accuracy of sensor placement via sensor selection is limited to the initial set of sampling locations, which are typically obtained by gridding the continuous sampling domain. To circumvent this issue, we propose a framework of continuous sensor placement. A continuous variable is augmented to the grid-based model, which allows for off-the-grid sensor placement. The proposed offline design problem can be solved using readily available convex optimization solvers.Index Terms-Convex optimization, joint sparsity, sensor placement, sensor selection, sparse sensing, sparsity. I. PROBLEM STATEMENT SENSOR networks are widely used in a variety of applications like environmental monitoring, security and safety, to list a few. Sensors are devices capable of sensing a certain physical phenomenon, processing data, and communicating information to a central unit. Sensors are geographically deployed, and the data acquired by such distributed sensors are often used to solve statistical inference problems like field (e.g., heat, sound) estimation, target localization, and so on.The number of sensors available are often limited due to various factors like availability of physical or data storage space, economical constraints, or due to energy-efficiency reasons. Such a restriction on the number of sensors naturally limits the estimation accuracy. Moreover, selecting different sampling locations for the sensors generally leads to different values of the mean-squared-error. In this article, we are interested in finding the best placement of the sensors such that a desired estimation accuracy is ensured and the number of sensors are as low as possible. In other words, the focus will be on designing a sparse sensing technique to capture only informative data, thereby reducing the costs associated with sensing, data storage, and communication overheads.Let denote the observation signal with a continuousdomain argument, where without loss of generality (w.l.o.g.) denotes the sampling domain. We will restrict ourselves to the one-dimensional spatial domain, but the ideas presented can be applied directly to higher dimensions and even to temporal or spatio-temporal domains 1 . Assume that represents the measured physical field over a continuous one-dimensional space , and it satisfies the linear model (1) where collects the parameters to be estimated, is the known linear model representing the mapping between the parameters and the measurements, and is the noise. For example, in field estimation, might contain the source location and its field intensity. Furthermore, we assume for and . In other words, is completely described by its variation in the interval where we can place the sensors.The fundamental question of interest is-where to place the sensors such that the estimation error is as lo...

Sensor Placement by Maximal Projection on Minimum Eigenspace for Linear Inverse Problems

“…Joshi and Boyd [1] formulated the sensor placement problem as an elegant nonconvex optimization problem, and approximated it as a convex optimization problem by the relaxation of the nonconvex Boolean constraints that represent the sensor placements, to a convex box set. This convex relaxation was then used in many works [2], [11], [16]- [19]. The sensing locations can be easily determined based on the solution of the convex optimization problem.…”

“…Besides the sensor placement for linear inverse problems, many other excellent sensor placement works have focused on the continuous system [11], nonlinear model [16], energy saving [4], [18], state estimation for dynamic system [18], [19], [21]- [23], and Gaussian process interpolation [24]- [27].…”

“…Apart from the enumeration method, the optimal solution can also be obtained by branch-and-bound methods [9], [10], which unfortunately do take a very long time, even for a moderate scale problem [1]. Consequently, in recent years the sensor placement for the linear inverse problem has attracted increasing attention to find a suboptimal solution via computationally more efficient methods [1]- [8], [11]- [15].…”

Sensor Scheduling With Time, Energy, and Communication Constraints

“…Earlier work used information theoretic approaches like mutual information maximization [15], [16] and cross entropy optimization [17] or other search heuristics like genetic algorithms [18], tabu search [19] and branch-and-bound methods [20] to solve the sensor placement problems. Several recent works have also considered nonlinear sensor networks [13], tracking applications [21], [22], distributed sensing scenarios [23], [24], correlated noise models [25], estimation of continuous variables [26]. Additional scenarios where further limitations are added to the network sensing problem include: energy budget constraints [27] and ways to maximize the lifetime per unit cost in wireless sensor networks [28], 2 regularization terms that discourage the selection of the same sensors over a period of time [12], [29] and scheduling [30], [31] over the network.…”

Joint Optimization of Receiver Placement and Illuminator Selection for a Multiband Passive Radar Network

“…Convex optimization is attractive for both computational complexity and convergence. Nevertheless, a crucial step for the convex optimization technique is to transform the non-convex model into convex model [ 32 , 33 , 34 , 35 ]. Therefore, the convex relaxation method is the focus of the study in solving the SCP.…”