## 3 Definitions and Basic Concepts  
### Area Coverage  
The WVSN considered in this article is composed of a set of visual sensor nodes \(VS\ =\ \{vs_{1},vs_{2},...,vs_{n}\}\), which are deployed over a two-dimensional area \(\mathbb{A}\). Each camera may assume different values for their parameters, however, without loss of generality, we consider in this work that all visual sensors are identical and so they are configured with the same values [21]. Each sensor is expected to be equipped with a camera, having a viewing angle \(\theta_{vs}\) and an orientation \(\alpha_{vs}\), as shown in Figure 1, where the camera is represented by the small red circle. Each camera also presents a sensing radius \(R_{s}\) as the approximation of the distance between the nearest and farthest point that can be sharply sensed. The field of view of any visual sensor is defined as the area of an isosceles triangle composed of three vertices, \(A\), \(B\) and \(C\) (see Figure 1), being \((A_{x},A_{y})\) the Cartesian coordinates of the sensor node. The coordinates of vertices \(B\) and \(C\) can be computed according to Equation (1) and the FoV of any visual sensor \(vs\) is the area of the triangle \(\triangle ABC\), which can be computed using trigonometry, as expressed in Equation (2) [20]. \[ \begin{array}{rcl}B_{x}&=&A_{x}+R_{s}\cos\left(\alpha_{vs}\right)\\ B_{y}&=&A_{y}+R_{s}\sin\left(\alpha_{vs}\right)\\ C_{x}&=&A_{x}+R_{s}\cos\left(\left(\alpha_{vs}+\theta_{vs}\right)\text{ mod }2\pi\right)\\ C_{y}&=&A_{y}+R_{s}\sin\left(\left(\alpha_{vs}+\theta_{vs}\right)\text{ mod }2\pi\right) \\ \end{array}  \tag{1}\]  
\[FoV_{vs}\ =\ \frac{R_{s}^{2}\cdot\sin\left(\theta_{vs}\right)}{2}. \tag{2}\]  
The monitoring field \(\mathbb{A}\) may be composed of one or more _Monitoring Areas_ (_MA_), each one described as a rectangle defined by its origins (\(x1\), \(y1\)), a width \(w\), a height \(h\) and a rotation angle \(\beta\), as shown in Figure 2. The coordinates of the other vertices of a _MA_ (vertices 2, 3 and 4) can be computed as presented in Equation (3). A _MA_ is the area of interest of a visual application and only visual information from this sub-region is relevant to the considered WVSN. \[ \begin{array}{rclrclrcl}x2&=&x1+w.\cos(\beta),&&y2&=&y1+w.\sin(\beta)\\ x3&=&x1+h.\sin(\beta),&&y3&=&y1-h.\cos(\beta)\\ x4&=&x1+h.\sin(\beta)+w.\cos(\beta),&&y4&=&y1-h.\cos(\beta)+w.\sin(\beta).\\ \end{array}  \tag{3}\]  
A _MA_ can be divided into smaller regions, called _Monitoring Blocks_ (_MB_), each one defined as a rectangle represented by its origins (\(x1s\), \(y1s\)), a width \(ws\), a height \(hs\), a centroid (\(xc\), \(yc\)) and the same rotation angle \(\beta\) of the _MA_, as shown in Figure 2. The coordinates of the other vertices of a _MB_  
Figure 1: Field of View of a visual sensor. (vertices \(2s\), \(3s\) and \(4s\)) can be calculated similarly to Equation (3). Thus, a \(MA\) is composed of a grid of \(MB\) with size \(M\times N\). In this case, the \"area coverage\" problem can be indirectly approached as several \"target coverage\" problems, where each point (\(xc\), \(yc\)) is a target with infinitesimal size. This is an important abstraction aimed at higher efficiency, while keeping the computational cost low, as it was discussed in Reference [22]. In that paper, the monitoring blocks approach for area calculation has been proven to be a good approximation in terms of accuracy, with low computational cost. A single monitoring block \(mb\) is considered to be covered by a visual sensor node \(vs\) if its center \((xc,yc)\) is within the polygon area of \(FoV_{vs}\). In this configuration, the area of a \(mb\) (\(ws\times hs\)) is accounted to the total coverage area [22]. A covered \(MB\) is represented by notation \(mb\in FoV_{vs}\). This definition can be extended for a set of visual sensors \(VS\) according to Equation (4). The total area covered by \(VS\) is computed according to Equation (5), where \(mb_{j,l}\) is a \(MB\) at the position \((j,l)\) of the grid. \[