A Geometric Modelling Approach to Determining the Best Sensing Coverage for 3-Dimensional Acoustic Target Tracking in Wireless Sensor Networks

Existing 3-dimensional acoustic target tracking methods that use wired/wireless networked sensor nodes to track targets based on four sensing coverage do not always compute the feasible spatio-temporal information of target objects. To investigate this discrepancy in a formal setting, we propose a geometric model of the target tracking problem alongside its equivalent geometric dual model that is easier to solve. We then study and prove some properties of dual model by exploiting its relationship with algebra. Based on these properties, we propose a four coverage axis line method based on four sensing coverage and prove that four sensing coverage always yields two dual correct answers; usually one of them is infeasible. By showing that the feasible answer can be only sometimes identified by using a simple time test method such as the one proposed by ourselves, we prove that four sensing coverage fails to always yield the feasible spatio-temporal information of a target object. We further prove that five sensing coverage always gives the feasible position of a target object under certain conditions that are discussed in this paper. We propose three extensions to four coverage axis line method, namely, five coverage extent point method, five coverage extended axis lines method, and five coverage redundant axis lines method. Computation and time complexities of all four proposed methods are equal in the worst cases as well as on average being equal to Θ(1) each. Proposed methods and proved facts about capabilities of sensing coverage degree in this paper can be used in all other methods of acoustic target tracking like Bayesian filtering methods.