## 3 Proposed Methods
### The Continuous Object Area Computation
#### 3.2.1 Compute the Rough Area of the Continuous Object  
All sensors are treated as the vertices (or nodes) on the geometrical plane. All sensors are classified into inner and outer nodes, denoted as the _iNode_ or the _oNode_. The sensors enclosed within the coverage area of the continuous object are iNodes, and those uncovered are oNodes. In the real world, an iNode is a sensor that detects the concentration of the event higher than a predefined threshold. Here, the notations \(\Psi_{i}\) and \(\Psi_{o}\) are represented as the set of the iNodes and oNodes, respectively.  
Next, the MDT applies the Delaunay triangulation to figure out the rough contour of the continuous object. The iNodes, which have any one-hop neighbor in \(\Psi_{o}\), are retrieved as inner boundary nodes, denoted as ibNodes. Here we use \(\Psi_{i}{}^{d}\) to represent the set of ibNodes. The one-hop neighbors of iNodes which belong to \(\Psi_{o}\) are outer boundary nodes, denoted as obNodes. We use \(\Psi_{o}{}^{d}\) as the set of obNodes. Note that a neighbor of a node indicates that its physical distance is shorter than the communication range. All nodes are the vertices of the geographical plane. The Delaunay triangulation of vertices in set \(\Psi_{o}{}^{d}\cup\Psi_{i}{}^{d}\) is constructed. The summarized areas of these triangulations in the Delaunay triangulation architecture will be the rough area of the continuous object shown in Figure 1. The reason for including the iNodes in set \(\Psi_{i}{}^{d}\) is to reduce the computation. The iNodes, which are not in \(\Psi_{i}{}^{d}\), must be in the area enclosed by the sensors in \(\Psi_{i}{}^{d}\). Therefore, computing the Delaunay triangulation can exclude them to reduce the computation. Involving the oNodes included in \(\Psi_{o}{}^{d}\) is a way of trying to select the sensors, which are the closest ones to the boundary of the continuous object, for covering all iNodes.  
MCH uses the convex- hull algorithm to enclose the continuous object, shown in Figure 2. After finding out the sets \(\Psi_{i}\) and \(\Psi_{o}\), MCH applies the Graham scan algorithm to obtain the convex hull of the nodes in \(\Psi_{i}\), shown in Figure 2a. Those nodes, which construct the convex hull of the vertices in \(\Psi_{i}\), are marked as ibNodes. Let \(\Psi_{i}{}^{c^{\prime}}\) be the set of these ibNodes. Similarly, the nodes which belong to \(\Psi_{o}\) and are the one-hop neighbors of the vertices in \(\Psi_{i}{}^{c^{\prime}}\) will be the candidate outer boundary nodes, and we denoted \(\Psi_{c}{}^{c}\) as the set of these nodes. The nodes which organize the convex hull of the vertices in \(\Psi_{o}{}^{c}\) will be the rough area of the continuous object, shown in Figure 2b. To simplify the presentation, let \(\Psi_{o}{}^{c}\) be the set of these sensors.  
Figure 1: The continuous object’s rough area is sketched by MDT. This rough area is figured out by the deployed sensors using their original locations instead of their adjusted locations. The sensors enclosed within the coverage area of the continuous object are iNodes, and those uncovered are oNodes. The nodes in both sets, _iNodes_ and _oNodes_, closing to the practice boundary of the continuous object, are picked as the vertices in the plane graph to create the Delaunay triangulation.  
Figure 2: _Cont_.