# Single-Shot Compressed Imaging via Random Phase Modulation
## 4 Compressed Imaging Using Random Phase Modulation
### Single-Shot Compressed Imaging with Random Convolution Using Double Random Phase Mask (CI-DRPM)  
In 2009, [PERSON] [4] proposed a compressed imaging system by random convolution, that the convolution with a specific-designed random pulse followed by random sub-sampling was considered as a universal compressed sampling operator, which was equivalent to implementing a random phase modulation in the Fourier-domain of the well-known 4-f system.  
Figure 3: Lena image reconstruction of CI-SSRPM.  
Figure 2: Optical setup of compressed imaging with double-lens phase mask modulation.  
In Ref. [4], [PERSON] proposed a novel strategy for compressed sensing which utilizes random convolution as a universal and efficient random measurement operator. Such measurement operator is constructed by firstly convolved with a specific-designed pulse and successively by random sub-sampling. The random convolution for compressed sensing can be algebraically written as  
\[\mathbf{x}_{\Sigma} =\mathbf{x}\otimes\mathbf{h} \tag{13}\] \[=N^{-1/2}\mathbf{F}^{\mathrm{H}}\Sigma\mathbf{F}\mathbf{x}\] \[=\mathbf{H}\mathbf{x}\]  
where \(\otimes\) denotes convolution operator, \(\mathbf{F}\) denotes the discrete Fourier matrix with entries \((F_{ij})=\left[\exp^{-\beta\tau(i-1)(j-1)/N}\right]_{0\leq i,j\leq N-1}\) and \(\mathbf{F}^{\mathrm{H}}\) denotes the inverse, and \(\Sigma\) is the Fourier transform version of pulse \(\mathbf{h}\) in the following form  
\[\mathbf{\Sigma}=\mathbf{F}\mathbf{h}=\left[\begin{array}{cccc}\sigma_{1}&&&\\ &\sigma_{2}&&\\ &&\ddots&\\ &&&\sigma_{N}\end{array}\right] \tag{14}\]  
where \(\mathbf{\Sigma}\) is a diagonal matrix with the entries \(\sigma_{i}\)\((i=1,2,\cdots,N)\) has a unit amplitude and random phases, which can be generated as follows:  
\[\sigma_{i}=\left\{\begin{array}{ll}\pm 1\text{ with equal probability }&i=1\\ \exp(j\theta_{i}),\theta_{i}\in(0,2\pi]\text{ yields uniform distribution }&2\leq i<N/2\\ \pm 1\text{ with equal probability }&i=N/2\\ \sigma_{N-i+2}^{*},\text{the conjucate of }\sigma_{N-i+2}&N/2+1\leq i\leq N \end{array}\right. \tag{15}\]  
The random demodulation (RD) method is often used for random subsampling. The image \(\mathbf{x}_{\Sigma}\) can be divided into blocks, and successively a measurement vector can be obtained by multiplying the entries in each block by a sequence of random signs and summing. Let \(\Xi\) be a diagonal matrix with the entries independently drawn from \(\pm 1\) Bernoulli distribution with equal probability, and \(\mathbf{P}\in\mathbb{R}^{M\times N}\) denotes an accumulate-and-dump subsampling operation that can be expressed as follows:  
\[\mathbf{P}=\left[\begin{array}{cccccccc}1&1&\cdots&&&&\\ &&1&1&\cdots&&\\ &&&\cdots&&\\ &&&&1&1&\cdots\end{array}\right] \tag{16}\]  
For simplicity, a renormalization factor is introduced into the random demodulation process, and finally, the random measurement vector can be algebraically rewritten as  
\[\mathbf{y}_{\Xi} =(M/N)^{-1/2}\mathbf{P}\mathbf{\Xi}\mathbf{x}_{\Sigma} \tag{17}\] \[=M^{-1/2}P\Xi\mathbf{F}^{H}\Sigma\mathbf{F}\mathbf{x}\] \[=\mathbf{\Phi}\mathbf{x}\] \[=\mathcal{D}_{\downarrow}(\mathbf{x}\otimes\mathbf{h})\]  
where \(\mathbf{\Phi}\) denotes the forward measurement operator (i.e., measurement matrix) in compressed sensing via random convolution followed by random demodulation.  
The subsampling strategy of random convolution compressed sensing (RCCS) has several attractive advantages, such as universality, numerical structure, physically realizable, which can provide an excellent strategy to perform universal CS acquisition, fast reconstruction, and a simply equipped setup to record the compressed measurements in a single exposure. The scheme of RCCS is depicted in Figure 4.  
As illustrated in Figure 4, an SLM coded with the pulse \(h\) is placed in the Fourier domain, and another SLM with random sign entries in the image plane. Numerical results will be presented in the next subsection for comparison.