## 3 Proposed Methods
### The Continuous Object Area Computation
#### 3.2.2 Refine the Enclosed Area with Mobile Sensors  
* Determine the moving direction for the sensors  
In MDT, the constructed triangulations created by the nodes in sets \(\Psi_{i}\) and \(\Psi_{o}\) must settle on the boundary of the continuous objects. Adjusting the positions of these nodes can refine the bounded area of the continuous object. Let \(\Delta\)QBC be one of the triangulations constructed by MDT, shown in Figure 3a. The vertex Q is selected to adjust its position because the detecting status of Q is different from the other two vertices, A and B. The moving direction of vertex Q, denoted as D, is computed as the resultant vector of \(\overrightarrow{QB}\) and \(\overrightarrow{QC}\). That is, \(\overrightarrow{D}=\overrightarrow{\left(\overrightarrow{QB}+\overrightarrow {QC}\right)}\). In MCH, all the sensors in sets \(\Psi_{i}^{{}^{\prime}}\) and \(\Psi_{o}^{{}^{\prime}}\) are settled at both sides of the continuous object's boundary. MCH only selects them to adjust the positions instead of all nodes in \(\Psi_{i}^{{}^{\prime}}\) and \(\Psi_{o}^{{}^{\prime}}\). This choice can efficiently reduce the number of moving sensors. Considering the moving direction of a sensor \(Q\) in \(\Psi_{o}^{{}^{\prime}}\), \(Q\) firstly collects the position of its one-hop neighbor \(N_{j}\) in \(\Psi_{i}^{{}^{\prime}}\) to compute vector \(\overrightarrow{QN_{j}}\). Let the number of one-hop neighbors be \(k\). Then, the unit vector of the resultant vector, \(\sum_{j=1}^{k}\overrightarrow{QN_{j}}\), will be its moving direction. That is, \(\overrightarrow{D}=\frac{\sum_{j=1}^{k}\overrightarrow{QN_{j}}}{\left| \overrightarrow{x_{j}}-\overrightarrow{QN_{j}}\right|}\), shown in Figure 3b. If sensor \(Q\) is in \(\Psi_{i}^{{}^{\prime}}\) instead of \(\Psi_{o}^{{}^{\prime}}\), the mechanism to determine the moving direction of \(Q\) is similar. \(Q\) uses its one-hop neighbors in \(\Psi_{i}^{{}^{\prime}}\) instead of \(\Psi_{o}^{{}^{\prime}}\). This involves more one-hop neighbors of \(Q\) with distinct detected states, which can make the position adjust to the boundary line more efficiently. Suppose that \(Q\) only selects two sensors in set \(\Psi_{i}^{{}^{\prime}}\) to compute its next target position instead of all one-hop sensors. Without any further information, \(Q\) may choose the sensors \(B\) and \(E\). This decision makes \(Q\) move to location \(Q^{\prime}\), which is farther away from the boundary than \(Q^{\prime}\). * Determine the moving step size of the mobile sensors and location freeze mechanism  
Figure 2: The continuous object’s rough area is sketched by MCH. **(a)** The sensors included in the set of iNode are involved in finding their convex hull. The sensors used to construct this convex hull are the inner boundary nodes. **(b)** The sensors included in the set of oNodes are involved in finding their convex hull. The sensors used to construct this convex hull are the outer boundary nodes. The practice boundary of the continuous object is enclosed by the two convex hull polygons built with the sensors in iNode and oNode. Adopting the appropriate moving step size is very important. Using a small moving step size is helpful to improve the precision of the figured area, but it increases the convergence time of the moving operation. Using a large moving step size can shorten the convergence time of the sensors' moving task, but the sensors will easily cross the boundary of the continuous object. Thus, the maximum value of the moving step size for any sensor \(Q\) must be well controlled. Let \(\gamma\) be the moving step size. The maximum value of the moving step size of mobile sensor Q, \(\gamma_{max}\), is set to \(\gamma_{max}=\frac{1}{2}\times\left|\overbrace{QO_{a}+\overbrace{QO_{b}}}\right|\), where \(Q_{a}\) and \(Q_{b}\) are two of the nodes that determine the moving direction of \(Q\) and are closest to \(Q\). Taking half the size of the resultant vector can reduce the probability that the mobile sensor \(Q\) crosses the boundary of the continuous object in one movement.