Principles of the Complete Voronoi Diagram Localization

Abstract

This paper explores the rationale behind the Complete Voronoi Diagram (CVD) Localization, which is a computational geometry approach to the wireless network localization. Our work consists mainly of three parts. The first part focuses on the analysis of CVD's mathematical properties. We characterize CVD's central tendency as the mirror-image distribution and provide mathematical formula for its probability density function. We also provide a closed formula for the relationship between CVD's vertices, chords and faces, the average chord length, the average edge number of a CVD polygon. And the expressions for the average overall and local positioning error are also provided. Based upon these findings, we show that the convergence speed for a CVD based localization scheme is quadratic, and the optimal time and space complexities are (n 2 ) and n), respectively. The second part proposes a novel approach, called Polling, which utilizes the concept of the Error Region, to further improve the accuracy. Polling, in theory, enables us to make use of the topology information with the quantity up to O(n 4 ) provided by CVD for localization, while a conventional CVD scheme can use only O(1) such information. The third part, through simulations, shows how to use the qADC (quasi Analog-to-Digital Conversion) strategy to handle signal errors. Combined with Polling and qADC, a CVD scheme can provide a simple, robust and powerful solution to the wireless network localization. Some of our findings and methods may also contribute to the field of computational geometry its own.

A node localization algorithm based on Voronoi diagram and support vector machine for wireless sensor networks

“…29 The Voronoi diagram is used to divide the spatial region to construct the order list of the virtual anchor node, and the RSSI method between the beacon nodes is used as the reference to correct the measured distance and the position sequence of the unknown node. Lu et al 30 proposed a new localization algorithm called Basic Complete Voronoi Diagram localization scheme (BCVD). The full text mainly introduces the principle problem behind the complete Voronoi diagram.…”