# Overview of Compressed Sensing: Sensing Model, Reconstruction Algorithm, and Its Applications
## 4 Compressed-Sensing Applications  
Compressed sensing has been widely applied in data compression, image encryption, cryptography, complex network reconstruction, channel estimation, analog-to-information conversion, channel coding, radar reconstruction, radar remote sensing, and digital virtual-asset security and management. Figure 4 presents an example of a data-encryption transmission system based on compressed sensing. CS is often used as a data-encryption and -compression method in networks with energy confinement and open links, such as sensor [74] and body-area [75] networks, and the Internet of Things (IoT) [76]. CS has a natural advantage in image encryption due to the sparsity of image data under specific bases or dictionaries. [PERSON] et al. verified that the image-encryption scheme based on CS was effective against noise [77]. They analyzed the security of the model from brute force and structured attacks. In addition, CS is used in the construction of various cryptography schemes. Considering the three main problems of image authentication, i.e., tamper detection, location, and recovery, [PERSON] et al proposed semifragile image authentication based on CS [78]. [PERSON] et al. proposed an image-reconstruction and identity-authentication scheme based on CS in cloud computing. Their scheme outsourced complex reconstruction calculations to the cloud server, and did not reveal the image's private information [79]. [PERSON] et al. made use of compressed sensing to provide a perspective for the solution of parameter-identification problems in coupled map lattices (CMLs) [80]. They used the sparse-recovery method of underdetermined linear systems to solve the CML parameter-identification problem. Generally speaking, widely used CMLs include the diffusive CML (DCML) and global CML (GCML) models. The GCML model is given as  
\[x_{t+1}(i)=(1-\epsilon)f(x_{t}(i))+\frac{\epsilon}{N}\sum_{j=1}^{N}c_{ij}g(x_{ t}(j)), \tag{30}\]  
where \(x_{t}(i)\), state of lattice element \(i\) on discrete time step \(t\); \(\epsilon\), coupling parameter; \(f\) and \(g\), maps in regard to local dynamics and nonlocal system; and \(c_{i}=(c_{i1},c_{i2},\cdots,c_{iN})\), weighted vector of element \(i\). From Equation (30), the following equation is obtained:  
Figure 4: Data-encryption transmission system based on compressed sensing, which can simultaneously realize data encryption and compression. \[\frac{N}{e}[x_{t+1}(i)-(1-e)f(x_{t}(i))]=\sum_{j=1}^{N}c_{ij}g(x_{t}(j)) \tag{31}\]  
Denoting \(y_{i}(t)=(N/\epsilon)[x_{t+1}(i)-(1-\epsilon)f(x_{t}(i))]\), then  
\[y_{t}(i)=\sum_{j=1}^{N}c_{ij}g(x_{t}(j)) =g(x_{t}(1))\ \cdots\ g(x_{t}(N))\left( \begin{array}{c}c_{i1}\\ \vdots\\ c_{iN}\end{array} \right) \tag{32}\]  
If each element \(i\) is sampled \(M\) times, then we know  
\[\left( \begin{array}{c}y_{1}(i)\\ \vdots\\ y_{M}(i)\end{array} \right) =\left( \begin{array}{ccc}g(x_{1}(1))&\cdots&g(x_{1}(N))\\ \vdots&\ddots&\vdots\\ g(x_{M}(1))&\cdots&g(x_{M}(N))\end{array} \right)\left( \begin{array}{c}c_{i1 }\\ \vdots\\ c_{iN}\end{array} \right) \tag{33}\]  
Equation (33) could be written as an underdetermined linear system \(Y=BC\). Thus, GCML identification is equal to the reconstruction problem of compressed sensing. In this way, all weighting parameters can be recovered by utilizing \(M\) samples, which is much smaller than number \(N\) of lattice elements. This method still performs well when various kinds of noise affect the original data. [PERSON] et al. proposed an approach of combining QR decomposition and compressed sensing to recover complex networks with the help of input noise [81], which is shown in Figure 5. The linear network system is defined as  
\[\dot{X}(t)=AX(t)+Bu(t), \tag{34}\]  
where matrix \(A\) with dimensions \(N\times N\) is the structure of the network nodes, and vector \(X(t)\) is the state of \(N\) nodes in a network system at time \(t\). As is shown in Figure 5, they transformed the linear system model into a compressed-featured equation, and the dynamic structure could be reconstructed by CS.