# 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)
\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+1)/2\\ \sigma_{i,N-i+2}^{*}\text{, the conjugate of }\sigma_{iN-i+2}&i=1,(N+1)/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+1)/2\\ \exp(\theta_{i}),\theta_{i}\in(0,2\pi]\text{ yields uniform distribution}&j=1,(N+1)/2+1\leq i \leq N\\ \sigma_{N-i+2,N-j+2}^{*}\text{, the conjugate of }\sigma_{N-i+2,N-j+2}&2\leq i<(N+1)/2,2\leq j \leq N\\ \end{array} \right. \tag{21}\]  
And the adjoint operator can be constructed as  
\[\mathbf{\Phi}^{\text{H}}=\mathbf{\mathcal{F}}_{\text{2D}}^{-1}\Big{(}\mathbf{\Sigma}^{ \text{H}}\odot\mathbf{\mathcal{M}}_{\text{2D}}\big{(}\mathcal{D}_{\uparrow}( \cdot)\big{)}\Big{)} \tag{22}\]  
where \(\mathbf{\Sigma}^{\text{H}}\) is the conjugate of \(\mathbf{\Sigma}\), \(\mathcal{D}_{\uparrow}\) is the integral upsampling. Note that one advantage of the construction is that the modulation scheme \(\mathbf{\mathcal{M}}\) is orthogonal, i.e.,  
\[\mathbf{\mathcal{M}}^{\text{H}}\mathbf{\mathcal{M}}=\mathbf{\mathcal{F}}_{\text{2D}}^{-1}( \mathbf{\Sigma}^{\text{H}}\odot\mathbf{\mathcal{F}}_{\text{2D}}(\mathbf{\mathcal{F}}_{ \text{2D}}^{-1}(\mathbf{\Sigma}\odot\mathbf{\mathcal{F}}_{\text{2D}}(\cdot))))=N\mathbb{I} \tag{23}\]  
Figure 6: Compressed imaging with Fourier-domain single random phase mask (CI-FSRPM). where \(\mathbb{I}\) is the matrix with all entries equal to one. In this section, numerical experiments are presented to verify the effectiveness of CI-DRPM and CI-FSRPM. The test environment is set with a 32-bit Windows7 operating system, the Intel (R) Core (TM) i5-2320 and 4-core processor, a basic frequency 3.00 GHz, an effective memory of 3.0 GB, and the test software Matlab2010a. Figure 7 shows the potential applications of random phase encoding in frequency domain simulation based on 4-f optical architecture. We create 1536 \(\times\) 1536 original images using Lena and Phantom and take them using Figure 6 to achieve a random phase encoding and integral subsampling. Figure 7a,b are standard images of Lena and Phantom, and Figure 7c,f are compressed measurements of Lena and Phantom images, respectively. The measurement size is 384 \(\times\) 384; that is to say, the sampling rate is four in the horizontal and vertical directions. Figure 7d,e are the reconstruction results of Lena with the measurements shown in Figure 7c using iterative soft thresholding (IST [17]) algorithm and gradient projection sparse reconstruction (GPSR [18]) algorithm. The SNR is 29.84 dB and 29.68 dB. Figure 7g,h are the reconstruction results from measurements shown in Figure 7f using the IST and GPSR algorithm. The SNR is 39.87 dB and 39.62 dB. The sparse operators **V** have chosen the Daubechies 10 wavelet of Rice University (rice wavelet toolbox, RWT) [19]. The wavelet scale is the same as 3. In another numerical experiment, we compare our proposed method with Random Demodulation (RD) method proposed by [PERSON] [16] and the RecPC method proposed by [PERSON] [20]. The test images are Lena, Cameraman, and Phantom image; the image size is chosen into seven levels of 256 \(\times\) 256, 512 \(\times\) 512, 768 \(\times\) 768, 1024 \(\times\) 1024, 1280 \(\times\) 1280, 1536 \(\times\) 1536, and 1792 \(\times\) 1792, the total downsampling rate is 1/8 (the reason for this rate selection is due to the hybrid measurements of RD method, which mean two low-resolution measurements, one is the measurement with random demodulation, another is a direct low-resolution version of the original image, both sampling rates are 1/16, with the amount to 1/8. For a fair comparison, while RecPC method uses random location downsampling, that is, in all positions, the uniform random decimation measurement value of the position. In this experiment, the sample rate for horizontal and vertical are 1/2 and 1/4, respectively.