# Single-Shot Compressed Imaging via Random Phase Modulation
## 4 Compressed Imaging Using Random Phase Modulation
### Single-Shot Compressed Imaging with Spatial-Domain Single Random Phase Mask (CI-SSRPM)  
In this section, single-shot compressed imaging with spatial-domain single random phase mask (CI-SSRPM) is presented. The optical setup is depicted in Figure 2. As illustrated in Figure 2, the random phase mask is attached to the middle between the two lenses \(L_{1}\) and \(L_{2}\). The object plane is at the position \(z_{1}\) in front of the lens \(L_{1}\), and the right side of the lens is a random phase mask with the phase distribution \(\varphi(r_{\varphi})\). The right side of the phase mask is closely attached to the lens \(L_{2}\), and the image plane is \(z_{2}\) from the right side of the lens. The optical field emitted by the object is focused by the lens \(L_{1}\) and then randomly modulated by a random phase mask; the scattered field is then focused by the lens \(L_{2}\). Finally, the complex field of the object reaches the CCD detection array behind the lens. The entire measurement process is achieved in a single exposure. The advantage of this system is that all the required measurements can be obtained in a single exposure. The forward projection operator (i.e., measurement matrix) between the object point \(r_{o}\) and the image point \(r_{i}\) can be expressed as  
\[ \begin{array}{rl}u(r_{i},r_{o})&=K_{0}\int\exp\left[j\frac{2\pi t}{\lambda z _{1}}(r_{o}-r_{\varphi})^{2}\right]\times\exp\left[-j\frac{\pi}{\lambda f_{L_ {1}}}r_{\varphi}^{2}\right]\exp\left[j\varphi(r_{\varphi})\right]\times\\ &\exp\left[-j\frac{\pi}{\lambda f_{L_{2}}}r_{\varphi}^{2}\right]\exp\left[j\frac {2\pi}{\lambda z_{2}}(r_{i}-r_{\varphi})^{2}\right]d\varphi\end{array}  \tag{8}\]  
Figure 1: Illustration of the importance of Fourier phase for image recovery. where \(\lambda\) denotes the wavelength of the incident light, \(\varphi(\mathbf{r}_{\varphi})\) denotes the random phase mask, \(f_{L_{1}}\) and \(f_{L_{2}}\) denotes the focal length of Lens \(L_{1}\) and \(L_{2}\), and \(C\) denotes a multiplicative constant. Equation (8) defines the relationship between the object point and image point as integrated overall random phase mask pointing \(\varphi(\mathbf{r}_{\varphi})\). Such relation is spatial random and shift-variant due to the randomness of the phase mask. The relation in Equation (9) can be discretized into the following form  
\[ \begin{array}{ll}\Phi_{m,n}&=K_{0}\int_{-L/2}^{L/2}\exp\left[j\frac{\pi}{ \lambda z_{1}}\left(n\Delta_{o}-\xi\right)^{2}\right]\bullet\exp(-j\frac{\pi}{ \lambda f_{L_{1}}}\xi^{2})\exp[j\phi(\xi)]\\ &\bullet\exp\left[-j\frac{\pi}{\lambda f_{L_{2}}}\xi^{2}\right]\exp\left[j\frac {\pi}{\lambda z_{2}}\left(m\Delta_{i}-\xi\right)^{2}\right]d\xi\end{array}  \tag{9}\]  
where \(-N/2\leq n\leq N/2-1,-M/2\leq m\leq M/2-1\). \(\Delta_{o}\) and \(\Delta_{i}\) are the pixel size of the object and the CCD. As the element of the Gaussian matrix has too much freedom, which cannot be physically achieved or cost too much, some deterministic matrices [13] with fewer independent elements have important practical significance and are easy to implement to satisfy the specific distribution of optical phase mask (such as Toeplitz or Circulant distribution), which can be expressed as follows Equations (10)-(12):  
\[\varphi_{\text{Gauss}}=\left( \begin{array}{cccc}a_{11}&a_{12}&\cdots&a_{1N} \\ a_{21}&a_{22}&\cdots&a_{2N}\\ \vdots&\vdots&\ddots&\vdots\\ a_{M1}&a_{M2}&\cdots&a_{MN}\end{array} \right) \tag{10}\]  
\[\varphi_{\text{Toeplitz}}=\left( \begin{array}{cccc}a_{N}&a_{N-1}&\cdots&a_{ 1}\\ a_{N+1}&a_{N}&\cdots&a_{2}\\ \vdots&\vdots&\ddots&\vdots\\ a_{N+M-1}&a_{N+M-2}&\cdots&a_{M}\end{array} \right) \tag{11}\]  
\[\varphi_{\text{Circulant}}=\left( \begin{array}{cccc}a_{N}&a_{N-1}&\cdots&a_ {1}\\ a_{1}&a_{N}&\cdots&a_{2}\\ \vdots&\vdots&\ddots&\vdots\\ a_{N-1}&a_{N-2}&\cdots&a_{N}\end{array} \right) \tag{12}\]  
The numerical experiments of CI-SSRPM are depicted in Figure 3. Lena image is chosen as the test image with the size 64 \(\times\) 64. Daubechies 5 wavelet transform is chosen as the sparsifying basis.