## 3 SWIFTNET: Protocol for Fast-Reactive Monitoring
### Compressed Sensing  
The basic theory of CS emerged in the works of [18,19]. CS is a simple and yet efficient data acquisition paradigm. Precisely, CS exploits the fact that many natural signals are sparse in the sense that they can be compressed as long as they can be represented in their proper basis \(\Psi\). CS is governed by two principles; sparsity and incoherence.  
Sparsity relates to the fact that the information rate of a continuous signal may be smaller than its bandwidth. On the other hand incoherency, as explained in [41], extends to the duality between time and interval, and therefore it expresses the idea that objects having a sparse representation in \(\Psi\) must be spread out in the domain in which they are acquired. With this knowledge, it is possible to design an algorithm or sensing protocol that captures useful events that are contained in a sparse signal, and then compress it into a smaller amount of data before transmission. It is on the basis of these premises that we argue to use CS in combination with other strategies as a data acquisition scheme for a surveillance application. Let us consider a discrete-time signal \(x\), viewed as a column \(N\times 1\) vector in a \(\mathbb{R}^{n}\) domain. The signal can be represented in terms of a basis of \(N\times 1\) vectors \(\Psi_{i=1}^{N}\). For simplicity sparsity offers  
Figure 1: SWIFTNET in a WSN with star topology.  
that many natural signals can have a concise representation when expressed in a convenient basis. Mathematically, the vector \(x\in\mathbb{R}^{n}\) can be expanded in an orthonormal basis such as Discrete Cosine Transform (DCT) or wavelet given as:  
\[x(t)=\sum_{i=1}^{N}y_{i}\Psi_{i}(t) \tag{4}\]  
where \(y\) is a coefficient of \(x\) in the domain \(y_{i}\) =\([x,\Psi_{i}]\). Similarly, \(\Psi\) can be expressed as an \(N\times N\) matrix with \([\Psi_{1}\Psi_{2}...\Psi_{N}]\) as columns. The implication is that when a signal has a sparse expansion, one can discard the small coefficients without much perceptual loss [18]. Generally, any basis vector \(\Psi\) can be formed without assuming prior knowledge of the signal, apart from the size, since the size determines the size of the measurement matrix \(\Phi\).  
The theory of CS demonstrates that a signal can be compressed using a proper measurement matrix \(\Phi\) = \([\phi_{1},\phi_{2},...,\phi_{N}]\) of size \(M\times N\), with \(M<N\). Having said this, we are interested in _under-sampled_ situations, where the number of measurements \(M\) available are much smaller than the dimension \(N\) of a signal \(y\). Assuming \(y\) is the compressed form of \(f\), then the matrix form is given as:  
\[y=\Phi f \tag{5}\]  
where \(y\) is an \(M\times 1\) column vector. The recovery process of the original signal \(f\) can be achieved through optimization methods. According to [41], suppose a given pair of orthobases of \(\mathbb{R}^{n}\) given as \((\Phi,\Psi)\), where \(\Phi\) is used for the sensing phenomenon and \(\Psi\) is used to represent \(x\), then the coherence between the sensing basis \(\Phi\) and the representation basis \(\Psi\) can be expressed as:  
\[\mu(\Phi,\Psi)=\sqrt{N}\cdot\text{max}_{1\leq k,j\leq N}\left|\left\langle \varphi_{k},\psi_{j}\right\rangle\right\rangle \tag{6}\]  
If \(\Phi\) and \(\Psi\) contain correlated elements, we say the coherence is large. It follows that the smaller the coherency, the better the reconstruction would be. From linear algebra, the upper and lower limit is bounded as \(\mu(\Phi,\Psi)\in[1,\sqrt{n}]\)[41]. This relationship form the basis for which we are able to recover sparse signals.