## 5 Model Formulation and Description  
### Model Formulation  
Assuming that the field of interest is a two-dimensional square region with area \(Z=A\times B/Z\in R^{2}\), and the network to be homogenous, N sensor nodes are deployed in the target region \(Z\), and the location of sensor \(i\) is denoted \(x_{i}\). We consider that every colored pixel represents a point with interest, a point has no interest when its color is white. A set of points with interest represents a Region of interest as demonstrated in Figure 5. We need to deploy an adequate number of sensor nodes \(X=(x_{1},\ x_{2},\ldots x_{N})\) that will fulfill at least 1-coverage and 1-connectivity. Based on the triangular deployment grid, sensor nodes are deployed line by line while the first is placed randomly on the map. Each node that collects an interest different than zero is selected and saved. Using the gradient method, all sensor nodes on the RoI boundary are detected and set to seek more interest. Finally, the joint interest collected by the given set of sensor nodes \(X\) is calculated. With every iteration, the minimal distance between sensor nodes, which is the distance between each node and its first-degree neighbors', is updated, and the adequate number of nodes is recalculated. If the recalculated joint interest is equal or greater than the previous one, the positions are updated. In order to minimize the processing time, the collection of interest for each sensor node is done using a clustering technique so the search will focus on a cluster which is a square, instead of  
Figure 5: Description of the cluster in the supervised area. the whole supervised area. Each cluster's center is a sensor node position. Let (Xi,Yi) be the Cartesian coordinates of a point Pi, and (Xs,Ys) the sensor node coordinates. The cluster is defined as follows \(\mathrm{C_{k}}=\big{\{}\ (\mathrm{Xi},\mathrm{Yi})\in\mathbb{R}^{2}/\left|\mathrm{Xs -Xi}\right|\ \leq\ \mathrm{R_{s}}\&\left|\mathrm{Ys-Yi}\right|\ \leq\ \mathrm{R_{s}}\big{\}}\). The new objective function described in Equation (9) is developed based on Equation (2) developed in [13]. { If \(f(\mathrm{X^{*}})\ \geq\ f(\mathrm{X})\), with \(f(\mathrm{X^{*}})\) the new output of \(f(\mathrm{X})\)} the function's output is saved to C, otherwise it's equal to the optimum solution stored in C. We updated the interest function given by Equation (8), and we added new sub functions given by Equations (5)-(7). The new sub functions are defined, as follows:  
\[\mathrm{f_{G}(X)}=\
abla\ \mathrm{f_{ref}(x,y).\overset{\rightarrow}{u}} \tag{5}\]  
\[\mathrm{f_{TX}(X)}=\ \mathrm{x_{i}}+\left(\mathrm{k.step.\frac{x_{i}-x_{j}}{d_{ max}}}\ \right) \tag{6}\]  
\[\mathrm{f_{N}(X)}=\ \frac{\mathrm{A}}{d_{min}}.\frac{\mathrm{B}}{d_{min}} \tag{7}\]  
where, the gradient function \(\
abla\ \mathrm{f_{ref}}=\langle\partial\ \mathrm{f_{ref}}\partial\mathrm{x}, \partial\ \mathrm{f_{ref}}\partial\mathrm{y}\rangle\) and (\(\mathrm{f_{ref}}\)) is the class function, the class of covered point is [1], while uncovered point has \(\{-1\}\). \(\overset{\rightarrow}{u}\) is a unit vector. \(d_{max}\) is the distance between unconnected nodes \(x_{i}\)and \(x_{j}\). \(d_{min}\) is the distance between close nodes. \(k\) is the number of nodes needed to achieve connection, step is the distance between new nodes: \(\mathrm{step}=\mathrm{a}\times\mathrm{R_{T}}\). \[\mathrm{f_{S}(X)}=\sum\
olimits_{i=1}^{N}\left[\alpha_{i}.\mathrm{area}(S_{i })-\ \mathrm{area}\Big{(}\bigcap\limits_{j=i}^{N}S_{j}\Big{)}\right] \tag{8}\]  
\[\mathrm{f(X)}=\ \left\{ \begin{array}{c}\mathrm{f_{min}(X)+\ f_{S}(X)+f_{G}(X)+f_{ TX}(X)-f_{N}(X)}\\ \\ \mathrm{C,\ otherwise}\end{array} \right. \tag{9}\]  
where: \(X\) is a set of \(N\) nodes. \(C\) is a constant ensuring that only survivable solutions will be accepted. \(\mathrm{f_{min}(X)}\) denotes the minimum distance between sensor nodes given by Equation (1) where \(d=d_{min}\) and increases by \(\
u\) in every iteration.