Principles of the Complete Voronoi Diagram Localization

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, and 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 Θ(n2) and Θ(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(n4) 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 quasi Analog-to-Digital Conversion (qADC) 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.