Tight Two-Dimensional Outer-Approximations of Feasible Sets in Wireless Sensor Networks

Abstract

Finding a tight ellipsoid that contains the intersection of a finite number of ellipsoids is of interest in positioning applications for wireless sensor networks (WSNs). To this end, we propose a novel geometrical method in 2-dimensional (2-D) space. Specifically, we first find a tight polygon which contains the desired region and then obtain the tightest ellipse containing the polygon by solving a convex optimization problem. For demonstrating the usefulness of this method, we employ it in a distributed algorithm for elliptical outer-approximation of feasible sets in cooperative WSNs. Through simulations, we show that the proposed method gives a tighter bounding ellipse than conventional methods, while having similar computational cost.

Localization via Visible Light Systems

“…Then, the problem of cooperative localization in VLP systems is equivalent to identifying a point inside the intersection of Lambertian sets as defined in (73) and (75). Assuming that the Lambertian function in (71) is quasi-convex, 14 the (76) 12 We can always subtract a constant value from the obtained RSS measurement to satisfy the assumption of negative measurement noise [136], [137]. For PDFs having finite support, this value can be chosen as…”

A novel approach for ellipsoidal outer-approximation of the intersection region of ellipses in the plane

“…This method is denoted as Expanded in the rest of this paper. We also consider the technique proposed in [19] where a bounding polygon is obtained in a different way than the method proposed in this paper. However, the optimization problem in (17) is finally solved to find the smallest area ellipse covering the polygon.…”

“…In this work, we extend the work of [19] for outer-approximation of the intersection of several ellipses in 2-D space in the following ways, such that a tight bounded and convex polygon can be formed, and the drawbacks mentioned above can be avoided:…”

“…-For every elliptic arc forming the boundary of the intersection of ellipses, we find the two end points, which are the intersection points of two or more ellipses, and generate a desired number of points on that arc. Compared to [19], in this way, we avoid generating unnecessary discrete points, and thus improve the efficiency. -The tangent lines corresponding to every two neighbouring points intersect, leading to a new set of points.…”

Cooperative Localization in Hybrid Infrared/Visible Light Networks: Theoretical Limits and Distributed Algorithms

“…In order to satisfy the negative error assumption, a constant value can always be subtracted from the actual RSS measurement[55]. Decreasing the value of an RSS measurement is equivalent to enlarging the corresponding feasible set.…”