## 3 Problem Formulation
### Fundamental Concepts for Energy Consumption  
We consider a WSN composed of \(P\) hop-by-hop wireless paths. Each path \(p\), \(p\) = 1, \(\ldots\), \(P\), comprises \(H_{p}\) intermediate nodes, where data packets flow from the source node (\(h\) = 0) to the sink of the network (\(h\) = \(H_{p}\) + 1). A number of \(D_{p}\) data packets are transmitted in each path \(p\). The size of the transmitted packets may vary according to the link layer technology and the application requirements, but small data packets are expected (reducing the error probability [19]) with the same size. Considering that most wireless sensor motes communicates through IEEE 802.15.4 wireless link-layer technology, the maximum MAC frame size is assumed to be 127 bytes including all packet overhead. If we exclude the MAC header and the overhead for network, transport and/or application layer protocols, as well as the image fragmentation header required for image decoding at the receiver size, the effective data size in each packet may be lower than 90 bytes in this link-layer technology [20]. Considering that even a very small 64 \(\times\) 64 uncompressed grayscale image sizes 4,096 bytes (excluding the image header of the entire image), it is natural to expect transmission of packets with maximum size to achieve a minimal packet overhead. Thus, in our model, every data packet has the same size in bits, \(s\), corresponding to the entire packet (data payload + headers), since all transmitted bits should be considered when computing the expected energy consumption. This initial idea is further extended to account the expected reduced size of the last packet after fragmentation, as described later in this section. For simplicity, the communication scenario is assumed to be contention-free, considering packet transmission using protocols as TDMA or the CFP (Contention-Free Period) in IEEE 802.15.4. Such assumption is intended to reduce the complexity of our mathematic models, but still keeping it highly realistic, following a trend often found in the literature [15,16,21]. Although not explicitly defined, the routing protocol may be cluster-based, allowing the creation of hop-by-hop paths from the source to the sink node. Moreover, node failure or harming were not considered. A large set of image-based WSN monitoring applications or even traditional scalar monitoring applications where the collected information is complemented by visual data attend such requirements. Considering that it is highly probable that all intermediate nodes in a path \(p\) are homogeneous (excepting the source and the sink), we can expect the same hardware characteristics and energy consumption pattern. Based on [15], \(e_{t}\) is defined as the energy consumed for the transmission of one bit by each relaying node, and \(e_{e}\) as the energy consumption for processing of a single bit. The energy consumption for packet transmissions (\(Et_{ph}\)) and receptions (\(Er_{ph}\)) in an intermediate node \(h\) of the path \(p\) is a function of the energy consumption characteristics of the employed hardware and the total amount of bits to be transmitted/received, as expressed in Equation (1). The variable \(d_{ph}\) is the maximum transmission range (in meters) of the node \(h\), and it is a direct function of the sensor's hardware. \(T_{ph}\) is the total amount of bits to be transmitted from hop \(h\) to the hop \((h+1)\) in path \(p\), and \(T_{ph}=D_{p}\times s\) when there are no packet retransmission. \[ \begin{array}{l}Et_{ph}{=}T_{ph}\Big{(}e_{e}+e_{t}d_{ph}^{2}\Big{)}\\ E_{ph}{=}T_{p(h-1)}e_{e}\end{array}  \tag{1}\]  
As most energy consumption is expected in transmission and reception procedures instead of processing and storage [21, 22, 23], we can roughly state the total energy consumption in path \(p\) as expressed in Equation (2).