# Single-Shot Compressed Imaging via Random Phase Modulation
## 4 Compressed Imaging Using Random Phase Modulation
### Single-Shot Compressed Imaging with Double Random Phase Encoding (CI-DRPE)  
In 2010, [PERSON] et al. [PERSON] et al. (2010) proposed a single-exposure compressed imaging system by combining the well-known optical scheme of double random phase encoding (DRPE), which could be applied to overcome the resolution limit caused by both diffraction and geometrical limitation. The schematic of compressed imaging using double random phase encoding (CI-DRPE) is depicted in Figure 8. As shown in Figure 8, the image is randomly modulated with the two random phase masks (RPMs) placed in the input plane and the Fourier plane of the 4-f optical system.  
The modulation scheme of CI-DRPE can be expressed as follows  
\[\mathbf{y}=\mathcal{D}_{\downarrow}\mathbf{\mathcal{F}}_{\rm 2D}^{-1}\mathbf{\Sigma}_{\rm RPM 2}\mathbf{\mathcal{F}}_{\rm 2D}(\mathbf{\Sigma}_{\rm RPM1}\odot\mathbf{x}) \tag{24}\]  
where \(\odot\) denotes Hadamard product. \(\mathbf{\Sigma}_{\rm RPM1}=\exp(jP_{1})\) and \(\mathbf{\Sigma}_{\rm RPM2}=\exp(jP_{2})\) denotes two random phase masks, both the size of \(P_{1}\) and \(P_{2}\) are the same as the image \(\mathbf{x}\), \(\mathcal{D}_{\downarrow}\) denotes a random subsampling operator.  
The Equation (21) can be reformulated as a universal measurement operator in matrix-vector form as follows  
\[\mathbf{y}=\mathcal{D}_{\downarrow}F^{\rm H}\Theta_{\rm RPM2}F\Theta_{\rm RPM1}\bm {x} \tag{25}\]  
where \(\mathcal{D}_{\downarrow}\) is the random subsampling operation, \(\mathbf{F}\) denote the discrete Fourier matrix (DFT) with size \(N\times N\) (\(N=n^{2}\)) and \(\mathbf{F}^{\rm H}\) denotes its inverse, \(\mathbf{\Theta}_{\rm RPM1}=\text{diag}(\Sigma_{\rm RPM1})\) and \(\mathbf{\Theta}_{\rm RPM2}=\text{diag}(\Sigma_{\rm RPM2})\), i.e., \(\mathbf{\Theta}_{\rm RPM1}\) take the elements of random phase mask \(\Sigma_{\rm RPM1}\) on the diagonal, so as the \(\mathbf{\Theta}_{\rm RPM1}\). \(\mathbf{x}\) denotes the vector rearranged the input image lexicographically, and \(\mathbf{y}\) denotes the compressed measurements. Note that the operation \(\mathbf{F}\mathbf{\Theta}_{\rm RPM1}\) in Equation (25) denotes random scrambling in the Fourier domain in the matrix form as:  
\[\mathbf{F}\mathbf{\Theta}_{\rm RPM1}=\left(\begin{array}{cccc}1&1&\cdots&1\\ 1&W_{N}&\cdots&W_{N}^{(N-1)}\\ \vdots&\vdots&\ddots&\vdots\\ 1&W_{N}^{(N-1)}&\cdots&W_{N}^{(N-1)(N-1)}\end{array}\right)\left(\begin{array} []{cccc}e^{2\pi p_{1}}&&\\ &e^{j2\pi p_{2}}&&\\ &&\ddots&\\ &&&e^{j2\pi p_{N}}\end{array}\right) \tag{26}\]  
where \(W_{N}=\exp(j2\pi/N)\). The entries \(\{p_{i}\}_{i=1}^{N}\) are drawn independently from a uniform distribution \([0,1)\), which leads to inter-column statistical independence for \(\mathbf{F}\mathbf{\Theta}_{\rm RPM1}\) and equivalence to the random Gaussian measurement operator. Numerical experiments are shown in Figure 9. In Figure 9, the parameters are set as follows: Cameramen image is selected as the test image, the size is \(512\times 512\), the subsampling rate is 0.5, the measurements are complex, the type of subsampling is random location. The sparsifying basis is RWT with a wavelet scale is 3. The performance of CI-DRPE under different subsampling  
Figure 8: Optical setup for compressed imaging using double random phase encoding (DPRE).  
rates is tested; the results are depicted in Table 3. As shown in Table 3, we can find that the quality of the recovered image can be improved as the subsampling rate increases.