Medial Axis Construction and Applications in 3D Wireless Sensor Networks Su Xia, Ning Ding, Miao Jin, Hongyi Wu, and Yang Yang Abstract—The medial axis of a shape provides a compact abstraction of its global topology and a proximity of its geometry. The construction of medial axis in two-dimensional (2D) sensor networks has been discussed in the literature, in support of several applications including routing and navigation. In this work, we first reveal the challenges of constructing medial axis in a three-dimensional (3D) sensor network. With more compli- cated geometric features and complex topology shapes, previous methods proposed for 2D settings cannot be extended easily to 3D networks. Then we propose a distributed algorithm with linear time complexity and communication cost to build a well- structured medial axis of a 3D sensor network without knowing its global shape or global position information. Furthermore we apply the computed medial axis for safe navigation and dis- tributed information storage and retrieval in 3D sensor networks. Simulations are carried out to demonstrate the efficiency of the proposed medial axis-based applications in various 3D sensor networks. I. I NTRODUCTION The medial axis of a shape is the set of all points that have more than one closest point on the boundary of the shape. For a given shape, its medial axis provides a compact abstraction of its global topology and a proximity of its geometry. The medial axis of a two-dimensional (2D) sensor network has been discussed and applied for applications including routing and navigation [1], [2]. In [1], an approximated medial axis of a 2D sensor network is constructed and represented compactly by a graph with size proportional to the number of the geometric features of the network field. A greedy routing scheme with guaranteed packet delivery and load balance is then proposed with local decisions guided by the computed medial axis. In [2], the medial axis of a distributed 2D sensor network is dynamically maintained to provide guidance for users to escape from dangerous areas. While most earlier studies assume sensor networks on a plane, there has been increasing interest in deploying wireless sensors in three-dimensional (3D) space for applications such as underwater reconnaissance, environmental monitoring and exploration [3]–[15]. With significantly higher complexity in geometric features and topology shapes, a 3D sensor network needs more urgently the guidance provided by a medial axis for numerous applications including but not limited to 3D rout- ing, 3D navigation, and data storage and retrieval. Although the construction of medial axis has been discussed for 2D sensor networks, it is anti-intuitively nontrivial to extend it This work is partially supported by National Science Foundation under Award Number CNS-1018306. The authors are with the Center for Advanced Computer Studies, University of Louisiana at Lafayette, Lafayette, LA. Fig. 1. The true medial axis points are mistakenly considered as noises when simply extending a 2D medial axis algorithm to 3D. (a) A 2D sensor network. (b) The cross-section of a 3D network. to 3D settings. Next we show the fundamental challenges for constructing medial axis in 3D sensor networks. A. Challenges in 3D Networks An algorithm for constructing medial axis in 2D sensor networks has been proposed in [1]. In a nutshell, each boundary node floods a control packet over the network. Upon receiving such flooding packets, an internal node can discover its shortest distance (in hops) to the boundary and the corresponding closest boundary node. According to the definition of medial axis, an internal node is identified as part of the medial axis if it has two or more closest boundary nodes. However, noises exist in discrete sensor networks, due to the lack of accurate distance information. For example, Node B in Fig. 1(a) can be misinterpreted as a medial axis node because it has equal hops to two closest boundary nodes (i.e., B 1 and B 2 ). To filter out such noise, the algorithm considers the hop distance between B 1 and B 2 along the boundary (i.e., the distance of the shortest path between B 1 and B 2 along the boundary, which is marked as green color). Node B is deemed a non-medial axis node if the distance is less than a threshold. On the other hand, a true medial axis node (e.g., Node A) will not be filtered out, because it has at least two closest boundary nodes (e.g., A 1 and A 2 ) that are well separated along the boundary (i.e., the shortest path along the boundary is long, which is marked as red color). The filtering process is critical for the success of the medial axis algorithm. At the first glance, it appears straightforward to extend this approach to 3D networks. However, it is surprisingly hard for such an approach to distinguish true medial axis nodes and noises based on hop distance only in 3D. Fig. 1(b) illustrates a cross-section of a 3D sensor network. When the same filtering strategy discussed above is applied here, Node C will be considered as noise, because the shortest path between Node 978-1-4673-5946-7/13/$31.00 ©2013 IEEE 2013 Proceedings IEEE INFOCOM 305