# Review of 3D Imaging by Coded Aperture Correlation Holography (COACH) +
## 3 System Applications
### Imaging through Scatterers  
Imaging through scatterers is often considered a challenging task [65; 66; 67; 68] and it is necessary to find new techniques for imaging through scatterers. In this section, the COACH principles are applied for imaging through scatterers on one hand and on the other hand, we show how scatterers can be used for 3D imaging. Recently, several techniques of imaging through scatterers using incoherent light have been proposed [65; 66; 67; 68]. Based on I-COACH, we have recently proposed an interferenceless incoherent digital holography technique for imaging objects through a thin scattering medium [69]. The optical configuration of the proposed technique is shown in Figure 27.  
An incoherent source illuminates a point object and the light emitted from the point object is scattered by the scattering sheet. The scattered light is collected by a lens \(L_{2}\) with a focal length \(f\)\(=\)\((1/z_{s}+1/z_{h})^{-1}\) and is focused on the image sensor. In the absence of the scattering sheet, the optical  
Figure 25: (a) Direct imaging result, (b) reconstruction result of FINCH and (c–i) C-FINCH reconstruction results for different values of the scattering degree \(\sigma\). Figure 26: Reconstruction results of C-FINCH for different values of the scattering degree \(\sigma\). configuration becomes a single lens imaging system. The recorded intensity pattern for the point object is termed as the point spread hologram \(I_{PSH}\) given as,  
\[ \begin{array}{ll}I_{PSH}(\overline{r}_{0})&=C\Big{|}\
u\Big{[}\frac{1}{\Delta _{h}^{2}}\Big{]}\Im\Big{[}L\Big{(}\frac{\overline{r}_{s}}{z_{s}}\Big{)}\exp(i \Phi_{r})\Big{]}\Big{|}^{2}\\ &=C\Big{|}\
u\Big{[}\frac{1}{\Delta_{h}^{2}}\Big{]}\Im\Big{[}\exp(i\Phi_{r}) \Big{]}\Big{|}^{2}*\delta\Big{(}\overline{r}_{0}-\frac{z_{h}}{z_{s}}\overline{ r}_{s}\Big{)}\end{array}  \tag{16}\]  
where \(C\) is a constant, \(\overline{r}_{0}=(x_{0},y_{0})\) is the transverse location vector on the sensor plane, and \(\Phi_{r}\) is the chaotic phase profile of the scatterer. A library of PSHs is recorded at various axial locations. An object is placed at the same axial location and the corresponding recorded intensity pattern is given by  
\[I_{OBJ}(\overline{r}_{0})=\sum_{j}a_{j}I_{PSH}\Big{(}\overline{r}_{0}-\frac{ z_{h}}{z_{s}}\overline{r}_{j}\Big{)} \tag{17}\]  
where each \(a_{j}\) is a positive real constant. The Fourier transform of \(I_{OBJ}\) is given by  
\[I^{\prime}_{OBJ}=\Im\{O*I_{PSH}\}=O^{\prime}\cdot\big{|}I^{\prime}_{PSH}\big{|} \exp\big{(}\text{iarg}\big{\{}I^{\prime}_{PSH}\big{\}}\big{)} \tag{18}\]  
where, \(I^{\prime}_{OBJ},\ O^{\prime}\) and \(I^{\prime}_{PSH}\) are 2D Fourier transforms of \(I_{OBJ},O\) and \(I_{PSH}\), respectively. The object image \(I_{IMG}\) is reconstructed by cross-correlating \(I_{OBJ}\) with the filtered version of \(I_{PSH}\), as follows  
\[ \begin{array}{ll}I_{IMG}(\overline{r}_{R})&=\Im^{-1}\big{\{}I^{\prime}_{OBJ }\,|I^{\prime}_{PSH}|^{\gamma}\exp(-i\text{arg}\{I^{\prime}_{PSH}\})\big{\}}\\ &=\sum_{j}a_{j}\Lambda\Big{(}\overline{r}_{R}-\frac{z_{h}}{z_{s}}\overline{r}_ {j}\Big{)}\approx o\Big{(}\frac{\overline{r}_{s}}{M_{T}}\Big{)}\end{array}  \tag{19}\]  
where \(\overline{r}_{R}=(x_{R},y_{R})\) is the transverse location vector on the reconstruction plane and \(\gamma\ (-1\leq\ \gamma\ \leq 1)\) is chosen to maximize the SNR. The SNR can be improved by using two or three intensity recordings as shown in the case of I-COACH techniques [54; 64]. The experiment was carried out using a pinhole with a diameter of \(\approx\)100 \(\upmu\)m and USAF objects (element 6 of group 2 and numeric digit 6 of group 2). The reconstructing filter was given as \(|I^{\prime}_{PSH}|^{\gamma}\exp(-j\text{arg}\{I^{\prime}_{PSH}\})\) where the optimal \(\gamma\) for the tested object was \(\gamma=-0.3\). The intensity patterns for the pinhole and object, the image of the filter magnitude for \(\gamma=-0.3\) and the reconstruction of a single camera shot are shown in Figure 28a-d, respectively.