## 5 Cooperative Scheduling for Object Detection in Cluster-Based WMSN.
### Cluster-based \(T_{p}\) and \(T_{interval}\) Computation
Figure 10a,b shows the evolution of T\({}_{\rm p}\) and T\({}_{\rm interval}\) as a function of \(\gamma\) for several C\({}_{\rm size}\). Both parameters are normalized by T. We first have to notice that for a overlapping scale factor \(\gamma\) = 1, T\({}_{\rm p}\)\(=\) T, while for \(\gamma\) \(<\) 1, T \(\leq\) T\({}_{\rm p}\)\(\leq\) T/(1 \(-\gamma\)). Then the duty-cycle frequency at which a specific node is awakened is decreased by a factor that at least is (1 \(-\)\(\gamma\)) times the frequency of the non-collaborative scheme. On the other hand some sensor of the cluster will be on duty every T\({}_{\rm interval}\) seconds. Note that T\({}_{\rm interval}\) will be lower than T and will be smaller as C\({}_{\rm size}\) increases. This means that the area is monitored more frequently although every specific sensor monitors with less frequency. The reason is justified in how clusters are formed. Any sensor of the cluster overlaps with the first-member by at least an area of \(\gamma\)\(\cdot\)A\({}_{\rm FoV}\). Thus, when a sensor enters in duty, he will monitor an area equal to \(\gamma\)\(\cdot\)A\({}_{\rm FoV}\) overlapped with the  
first-member and an area equal to \((1-\gamma)\cdot\mathrm{A_{FoV}}\) that in the worst case does not overlap with any other member of the cluster. Sensing the whole cluster area with \(\mathrm{T_{interval}}\) equal to \(\mathrm{T}\) would result in that an area equivalent to \((1-\gamma)\cdot\mathrm{A_{FoV}}\) would be monitored every \(\mathrm{C_{size}\,T}\), a value that can be very high. However, using Equation (12), monitoring of the area equivalent to \((1-\gamma)\cdot\mathrm{A_{FoV}}\) is guaranteed by a monitoring interval that is not superior to \(\mathrm{T/(1-\gamma)}\), that is much lower than \(\mathrm{C_{size}\,T}\). Sleep/wake up protocols has extensively been studied in the area of wireless sensor networks, mainly for the radio subsystem, [23]. Our clustering algorithm works on the sensing subsystem. It is important to notice that executing object detection does not imply sending packets to the sink. Thus, the sleep/wake up algorithm can be decoupled with the radio subsystem. Sleep/wake up can be based on periodic duty-cycle synchronized by the first-member: every \(\mathrm{T_{p}}\) period, the sensing subsystem wakes up and performs object detection. However, clock drifts can cause cluster de-synchronization. To handle resynchronization, the system makes use of the beaconing scheme for cluster maintenance: nodes receive periodical beacons from the first-member and vice versa in order to detect new members or to detect members that have died. Beaconing duty-cycling belongs to the radio subsystem and it is independent of the sensing subsystem. That means that waking up the sensor to send a beacon is independent of waking up the sensor to take a picture and perform object detection. Thus, the cluster-head may resynchronize cluster members without need of waking up the sensing subsystem. In case the cluster-head dies, cluster members are reassigned to another clusters or form new cluster/s as explained in Section 3.2. Figure 10: (a) Sensor node duty-cycle period \(\mathrm{T_{p}}\)/T, (b) Cluster duty-cycle period \(\mathrm{T_{interval}}\)/T.