# Fast Terahertz Coded-Aperture Imaging Based on Convolutional Neural Network
## 2 Method  
### Signal Model and Learning-Based Approach  
For the convenience, we take TCAI model based on single input single output technology as an example. The schematic diagram is shown in Figure 1 and it mainly contains a controlling and processing terminal, transmitter module, transmitter, coded aperture, receiver module, receiver. The transmitting module includes mixer and frequency multiplier, and the receiving module includes low noise amplifier and mixer. The THz wave transmitted from the transmitter, and then it propagates to the coded aperture. The coded aperture, controlled by the controlling and processing terminal, randomly or pseudo-randomly modulates the amplitude or phase of incident THz wave to produce a spatiotemporal-independent radiation field in the imaging plane, which is divided into M grid-cells. After being reflected from the imaging area, the pseudo-random signal is collected by receiver and then sent to the controlling and processing terminal for reconstruction the target. Suppose the transmitter emits a THz linear frequency modulation signal, the reference signal of the \(m\)-th grid cell at time \(t_{n}\) can be deduced as  
\[S\left(t_{n},m\right)=\sum_{q=1}^{Q}A\exp\left[j2\pi\left(f_{c}\left(t_{n}-\frac {r}{c}\right)+\frac{K}{2}\left(t_{n}-\frac{r}{c}\right)^{2}\right)\right]\exp(j \varphi_{t_{n},q}), \tag{1}\]  
where \(A\) is the amplitude, \(f_{c}\) is the center frequency, \(K\) is the chirp rate, \(c\) is the speed of light, \(r\) represents the signal propagation distance, \(Q\) stands for the number of transmitting element of the coded aperture, \(\varphi_{t_{n},q}\) is the random phase modulation factor for the \(q\)-th coding aperture element at time \(t_{n}\). Then, the echo signal at time \(t_{n}\) is expressed as  
\[S_{b}\left(t_{n}\right)=\sum_{m=1}^{M}\beta_{m}S(t_{n},m), \tag{2}\]  
where \(\beta_{m}\) stands for the scattering coefficient corresponding to the \(m\)-th grid cell. Based on the time discretion of Equation (2), the matrix-imaging formula can be written as  
\[\left[ \begin{array}{c}S_{b}(t_{1})\\ S_{b}(t_{2})\\ \vdots\\ S_{b}(t_{N})\end{array} \right]=\left[ \begin{array}{cccc}S(t_{1},1)&S(t_{1}, 2)&\cdots&S(t_{1},M)\\ S(t_{2},1)&S(t_{2},2)&\cdots&S(t_{2},M)\\ \vdots&\vdots&\cdots&\vdots\\ S(t_{N},1)&S(t_{N},2)&\cdots&S(t_{N},M)\end{array} \right].\left[ \begin{array}[] {c}\beta_{1}\\ \beta_{2}\\ \vdots\\ \beta_{M}\end{array} \right]+\left[ \begin{array}{c}\omega_{1}\\ \omega_{2}\\ \vdots\\ \omega_{N}\end{array} \right], \tag{3}\]  
where \(\mathbf{\omega}=\left\{\omega_{n}\right\}_{n=1}^{n=N}\) is the additive measurement noise, \(\mathbf{S}_{b}=\left\{S_{b}(t_{n})\right\}_{n=1}^{n=N}\) is echo signal vector. The reference signal matrix is  
\[\mathbf{S}=\left[ \begin{array}{cccc}S(t_{1},1)&S(t_{1},2)&\cdots&S(t_{1},M)\\ S(t_{2},1)&S(t_{2},2)&\cdots&S(t_{2},M)\\ \vdots&\vdots&\cdots&\vdots\\ S(t_{N},1)&S(t_{N},2)&\cdots&S(t_{N},M)\end{array} \right]. \tag{4}\]  
The row vector and column vector of \(\mathbf{S}\) are the time-domain samples at \(\left\{t_{n}\right\}_{n=1}^{n=N}\) and the spatial-domain samples at \(\left\{m\right\}_{n=1}^{m=M}\), respectively. Figure 1: Schematic of Terahertz coded-aperture imaging (TCAI) system. For the previous TCAI [23; 32], the compressed sensing [18] recovery algorithms are the standard method to solve \(\beta\) and the imaging problem is treated as an optimization problem after obtaining the evaluation of \(S\)  
\[\overset{\sim}{\beta}=\arg\min\underset{\beta}{\psi}\left(\beta\right)s.t.\ \|S_{b}-S\beta\|_{2}^{2}<\sigma^{2}, \tag{5}\]  
where \(\|S_{b}-S\beta\|_{2}^{2}\) is the fitting error, \(\sigma^{2}\) is usually the variance of \(\omega\) and \(\psi\) represents the prior information on \(\beta\). Typically, \(\psi(\beta)=\|\beta\|_{1}\), where \(\|\cdot\|_{1}\) is \(L_{1}\) Norm that is used to constrain the imaging domain. For the solving (5), a large number of iterations are usually required which essentially limit the imaging speed.