# Single-Shot Compressed Imaging via Random Phase Modulation
## 4 Compressed Imaging Using Random Phase Modulation
### Single-Shot Compressed Imaging with Fourier-Domain Single Random Phase Mask (CI-FSRPM)  
Figure 5 is a schematic diagram of a 2D image in a 4-f optical setup with a random phase modulated in the frequency domain. The first lens transforms the standard Lena image and is then followed by a phase modulation that may be performed by an SLM after another lens transforms back into the time domain. Finally, a low-resolution detector array records the measurements. In Refs. [15; 16], we proposed a novel compressed imaging using single random phase modulation in the Fourier domain. The optical setup of our proposed is depicted in Figure 6. As depicted in Figure 6, the system is based upon the architecture based on a 4-f optical setup. A phase mask that may be manipulated by an SLM placed in the frequency domain plays a crucial role in maintaining the amplitude of the Fourier transform of the original image unchanged; only the phase of the Fourier transforms of the image makes the type uniformly random delay point-wise. Such modulation can reduce the sampling burden in a spatial domain because the entire information is spread out in the spatial domain. Figure 4: Optical setup of random convolution compressed sensing (RCCS). Figure 5: Schematic of random phase modulation in the Fourier domain. For a 2D image, the forward measurement operator of our proposed CI-FSRPM scheme can be expressed as  
\[\mathbf{\Phi}=\mathcal{D}_{\downarrow}\mathbf{\mathcal{F}}_{\text{2D}}^{-1}(\mathbf{\Sigma} \odot\mathbf{\mathcal{F}}_{\text{2D}}(\cdot))=\mathcal{D}_{\downarrow}\mathbf{\mathcal{ M}} \tag{18}\]  
where \(\odot\) denotes Hadamard product. \(\mathbf{\mathcal{F}}_{\text{2D}}\) and \(\mathbf{\mathcal{F}}_{\text{2D}}^{-1}\) denote 2D Fourier transform and 2D inverse Fourier transform operator and matrix \(\mathbf{\Sigma}\) is a \(N\times N\) matrix whose entries are the (renormalized) 2D Fourier transform coefficients of \(h\), i.e., \(\mathbf{\Sigma}=\mathbf{\mathcal{F}}_{\text{2D}}(h)\). Nevertheless, we first generate \(\mathbf{\Sigma}\) at random by taking  
\[\mathbf{\Sigma}=\left[ \begin{array}{cccc}\sigma_{11}&\sigma_{12}&\cdots&\sigma_ {1N}\\ \sigma_{21}&\sigma_{22}&\cdots&\sigma_{2N}\\ \vdots&\vdots&\ddots&\vdots\\ \sigma_{N1}&\sigma_{M2}&\cdots&\sigma_{NN}\end{array} \right] \tag{19}\]  
where the entries \(\left\{\sigma_{ij}\right\},i=1,\cdots,N,j=1,\cdots,N\) are unit magnitude complex numbers with random phases which means pure phase mask. Note that the entries of \(\mathbf{\Sigma}\) must satisfy conjugate symmetric property for simplicity so that real object \(\mathbf{x}\) can still be real after performing the operator \(\mathbf{\Phi}\). Specifically, the entries \(\mathbf{\Sigma}\) are generated as follows:  
For \(N\) even,  
\[\sigma_{ij}=\left\{ \begin{array}{ll}\pm 1\text{ with equal probability}&(i,j)\in\left\{(1,1)\right\}\\ \exp(\theta_{i}),\theta_{i}\in(0,2\pi]\text{ yields uniform distribution}&i=1,2\leq j \leq N/2\\ \sigma_{iN-i+2}^{*}\text{, the conjugate of }\sigma_{iN-i+2}&i=1,N/2+1\leq j\leq N\\ \exp(\theta_{i}),\theta_{i}\in(0,2\pi]\text{ yields uniform distribution}&j=1,2\leq i<N/2\\ \sigma_{N-i+2,N-i+2}^{*}\text{, the conjugate of }\sigma_{N-i+2,N-i+2}&j=1,N/2+1\leq i\leq N \\ \exp(\theta_{i}),\theta_{i}\in(0,2\pi]\text{ yields uniform distribution}&2\leq i<N/2,2\leq j \leq N\\ \exp(\theta_{i}),\theta_{i}\in(0,2\pi]\text{ yields uniform distribution}&N/2+1\leq i \leq N,2\leq j\leq N\\ \end{array} \right. \tag{20}\]  
For \(N\) odd,  
\[\sigma_{ij}=\left\{