We present a new approach to tracking of radiation sources moving on smooth trajectories which can be approximated with piece-wise linear joins or piece-wise linear parabolas. We employ the use of cheap binary proximity sensors which only indicate when a radiation source enters and leaves its sensing range. We present two separate cases. The first is considering that the trajectory can be approximated with piece-wise linear joins. We develop a novel scalable approach in terms of the number of sensors required. Robustness analysis is done with respect to uncertainties in the timing recordings by the radiation sensors. We show that in the noise free case, a minimum of three sensors will suffice to recover one piece of the linear join with probability one, even in the absence of knowledge of the speed and statistics of the radiation source. Second, we tackle a more realistic approximation of trajectories of radiation sources -- piece-wise parabolic joins -- and show that no more than six sensors are required in the noise free case to track one piece of the parabola with probability one. Next we present an upper bound on the achievable error variance in the estimation of the constant speed and the angle of elevation of linear trajectories. Finally, a comprehensive set of simulations are presented to illustrate the robustness of our approach in the presence of uncertainties.
@cite_4: 
@cite_6: We consider the use of binary proximity sensors for tracking targets. Such sensors provide only 1-bit information regarding a target's presence or absence in their vicinity, albeit with less than 100 reliability. A novel tracking method employing such binary sensors is proposed and its performance in different deployment scenarios evaluated. For a given target, the method utilizes the sensor outputs to estimate individual positions in the path of the target in the near past and finds the line which best fits the path points. This line is then used to estimate the target's current position. A performance study has been conducted through comprehensive simulation using parameters collected from a prototype deployment consisting of wireless micro-sensors with binary acoustic detectors.
@cite_11: We explore fundamental performance limits of tracking a target in a two-dimensional field of binary proximity sensors, and design algorithms that attain those limits. In particular, using geometric and probabilistic analysis of an idealized model, we prove that the achievable spatial resolution Δ in localizing a target's trajectory is of the order of 1overρ R, where R is the sensing radius and ρ is the sensor density per unit area. Using an Occam's razor approach, we then design a geometric algorithm for computing an economical (in descriptive complexity) piecewise linear path that approximates the trajectory within this fundamental limit of accuracy. We employ analogies between binary sensing and sampling theory to contend that only a "lowpass" approximation of the trajectory is attainable, and explore the implications of this obervation for estimating the target's velocity.We show through simulation the effectiveness of the geometric algorithm in tracking both the trajectory and the velocity of the target for idealized models. For non-ideal sensors exhibiting sensing errors, the geometric algorithm can yield poor performance. We show that non-idealities can be handled well using a particle filter based approach, and that geometric post-processing of the output of the Particle Filter algorithm yields an economical path description as in the idealized setting. Finally, we report on our lab-scale experiments using motes with acoustic sensors to validate our theoretical and simulation results.
Other existing studies do not consider piece-wise joins, but rely on high density of sensors to guarantee that every motion is within the detection range of multiple sensors @cite_11 , @cite_6 , @cite_4 . These studies also assume implicitly that the source signal strength at unit distance and no shielding is known.