# Overview of Compressed Sensing: Sensing Model, Reconstruction Algorithm, and Its Applications
## 2 Sensing Methods
### Deep-Learning Compressed Sensing  
The combination of deep learning and compressed sensing has attracted much attention. [PERSON] et al. presented a deep-learning approach for block CS. They deployed a fully connected network to be performed on the block-based linear-perception and nonlinear-reconstruction section [19]. They employed a deep neural network that performed BCS by independently processing each block as per Equation (9). They proposed a fully connected network with four layers:  
1. input layer with \(B^{2}\) nodes (\(B\) is block size);
2. compressed-sensing layer, \(B^{2}R\) nodes, \(R\ll 1\) (its weights form the sensing matrix);
3. \(K\geq 1\) reconstruction layers, \(B^{2}T\) nodes, each followed by a rectified linear unit (ReLU) [20] activation unit where \(T>1\) is the redundancy factor; and
4. output layer, \(B^{2}\) nodes.  
Compared with popular BCS methods such as block-compressed sensing smooth Landweber with dual-tree discrete wavelet transform (BCSSPL-DDWT) [21], multiscale block-compressed sensing smooth Landweber (MS-BCS-SPL) [22], and multihypothesis block-compressed sensing smooth Landweber (MH-BCS-SPL) [23], this method performed well with regard to recovery quality and calculation time.  
Learning from convolutional networks, a deep-learning-based sparse-measurement matrix was presented to reduce the sampling calculation complexity and improve CS reconstruction quality [24]. The method had two subnetworks, the sample and reconstruction subnetworks. They assumed that block size \(N_{B}\) in block CS was \(B\times B\), and measurement size for every block was \(N_{b}=\lfloor\frac{M}{N}N_{B}\rfloor\). The \(k\)-th line of sensing matrix \(\Phi\) was denoted as  
\[\Phi(k)=\{a_{k,1},a_{k,2},\cdots,a_{k,N_{B}}\}. \tag{16}\]  
The sparse degree is \(\Gamma\left(\Phi\right)=\frac{\
u}{N_{b}N_{B}}=\alpha(0\leq\alpha<1)\), where \(\
u\) is the number of nonzero elements in \(\Phi\), and \(N_{b}N_{B}\) is the whole elements in \(\Phi\). To generate the target sample matrix, a sparsity constraint was added as follows:  
\[S(a_{k,i})=\left\{\begin{array}{l}0_{r}\mid a_{k,i}\mid\leq\mu\\ a_{k,i}\mid a_{k,i}\mid>\mu,\end{array}\right. \tag{17}\]  
where \(k=1,2,\cdots,N_{b},i=1,1,\cdots,N_{B}\), and \(\mu\) is the \((1-\alpha)N_{b}N_{B}\)-th smallest element in \(\mid\Phi\mid\). \(\Phi(k)\) is the \(k\)-th kernel of the convolutional layer in the sample subnetwork, and \(a_{k,i}\) is the \(i\)-th value of this kernel. The normalization constraint for \(k\)-th kernel was formulated as  
\[\Gamma(s_{k,j})=\frac{s_{k,j}}{\sqrt{\sum_{i=1}^{N_{B}}{s_{k,j}}^{2}}},j=1,2, \cdots,N_{B}, \tag{18}\]  
where \(s_{k,j}=S(a_{k,j})\) and its derivative performed as  
\[\Gamma(s_{k,j})^{\prime}=\frac{\sqrt{\omega}-\frac{s_{k,j}^{2}}{\sqrt{\omega} }}{\sqrt{\omega}},\omega=\sum_{i=1}^{N_{B}}{s_{k,i}}^{2} \tag{19}\]On the basis of Equation (18), the normalized sampling matrix was obtained.  
[PERSON] et al. presented a deep-learning method for quantizing CS called BW-NQ-DNN [25]. The BW-NQ-DNN framework consists of three parts: a nonuniform quantizer, a binary sensing matrix, and a noniterative recovery solver. These three parts have joint optimization through BW-NQ-DNN training. BW-NQ-DNN not only saves a lot of storage and energy, but it also surpasses the most advanced CS-based approaches. When the compression ratio is very high, this method still performs well in recovery performance and classification accuracy.