# Algorithmic Coverage Quantification and Visualization in Range-Free Sensor Networks
## 1 Introduction
The authors in [10] developed a WSN-based distributed detection scheme to detect targets that emit an unknown deterministic signal (non-cooperative) from an unknown location in the presence of multiplicative fading, quantized measurements, and non-ideal, non-identical binary symmetric channels. They assumed that the targets emit standard electromagnetic signals within a specific frequency band. While our research also deals with unknown TRs, our approach differs significantly in that we use a completely different type of range-free SRs (inertia SRs, see [11]). In [12], the authors review several localization techniques that focus on neighbor creation, connectivity, and convergence of SR networks rather than on improving the localization capabilities of TRs within the monitored area. They survey several newly proposed algorithms, including a novel three-dimensional localization algorithm, and compare them to existing techniques in terms of effectiveness across various deployment environments. Towards the end of their conclusions, the authors in [3] highlight a significant oversight in the existing literature: the neglect of obstacles. Our research addresses this gap by considering previously detected TRs located within the AoI as obstacles which obstruct certain areas within the SRs' detection range. In the WSNs under study, a network of SRs is deployed across the designated area, each tasked with the detection and analysis of potential TRs entering the designated AoI. Our study focuses on inertia-type SRs, which use a spectroscopic method, as detailed in [11]. As noted in [13], for such types of SRs, the influence of the environment on signal propagation does not significantly impact the simple geometry of the SR's field of view. Essentially, the SR's field of view can be effectively modeled as a disk centered on the SR, with a constant radius that corresponds to the SR's sensing range. The employed SRs operate on a \"range-free\" principle, meaning they detect TR directions rather than their distance from the SR. The aim is to achieve accurate localization through triangulation. Despite our extensive literature search, we have not encountered studies that focus on this specific type of SRs, which is characterized as long-range and direction-only. Moreover, in our case, the geographic coordinates of the deployed SRs are known and fixed. This allows our triangulation techniques not only to detect but also to localize candidate TRs. In the present study, the concept of triangulation is redefined as the condition whereby a part of the AoI is covered by the sensing ability of three or more SRs. This ensures accurate TR localization. For a new TR to be detected upon entering the AoI, it must fall within the combined detection range of at least three SRs simultaneously. Building upon prior research that highlights the impact of existing TRs and defines SR blindness [7], we consider that the existing TRs degrade network coverage performance. Specifically, we define SRs as \"blind\" within specific angular regions around existing TRs. This highlights the importance of visualizing these coverage gaps, which is crucial for reducing the effects of network detection blindness. Although the system under study does not fall into the category of large WSNs, the significance and impact of SR blindness play a similarly crucial role as in WSNs, where such blind areas are often referred to as \"coverage holes\". The authors in [14] provide a comprehensive review of the literature on hole detection in WSNs, covering various types of holes, their causes, the importance of hole detection, and the different algorithms used for this purpose. Figure 1 presents an overview of the problem being addressed in this study. This \"bird's eye view\" figure illustrates the main components, their interactions, and the overall structure of the system under study. Previous research [7] has explored TR detection using rectangular grids of varying sizes, with fixed SRs positioned at grid nodes.