# Review of 3D Imaging by Coded Aperture Correlation Holography (COACH) +
## 2 System Architectures
### Coded Aperture Digital Holography
#### 2.1.4 FOV Extended I-COACH
\begin{array}{ll}\vec{I}_{1}(x_{o},y_{o})=\sum_{k=-1}^{1}\sum_{l=-1}^{1}I_{1 }(x_{o}-Dk,y_{o}-Dl)Rect\left[\frac{(x_{o},y_{o})}{D}\right]*\delta(x_{o}-Dk,y _{o}-Dl)\end{array}  \tag{12}\]  
The process is repeated using the second CPM and the final bipolar PSH is \(\overline{H}_{PSH}=\vec{I}_{1}-\vec{I}_{2}\). An object located outside the FOV of the system, around the point (\(kS\),\(lS\)), can be expressed as \(\sum_{j}a_{j}\delta(x_{s}-x_{j}+kS,y_{s}-y_{j}+IS)\), for \(k,l=-1\),0,1 [but (\(k,l\
eq\) (0,0))]. The bipolar out-of-FOV object hologogram, obtained by subtraction of the two intensity patterns, is given by  
\[ \begin{array}{ll}H_{OBJ}(x_{o},y_{o})=\sum_{j}\overline{H}_{PSH}\big{(}x_{o} -M_{T}x_{j}-Dk,y_{o}-M_{T}y_{j}-Dl)Rect\left[\frac{(x_{o},y_{o})}{D}\right] \end{array}  \tag{13}\]  
The object is reconstructed by a cross-correlation between the phase-only filtered synthetic \(\overline{H}_{PSH}^{\prime}\) and the \(H_{OBJ}\) as  
\[ \begin{array}{ll}P(\vec{r}_{R})&=\iint\sum_{j}\overline{H}_{PSH}\big{(}x_{o} -M_{T}x_{j}-Dk,y_{o}-M_{T}y_{j}-Dl)Rect\left[\frac{(x_{o},y_{o})}{D}\right]\\ &\times\overline{H}_{PSH}^{*}(x_{o}-x_{R},y_{o}-y_{R})d\vec{r}_{0}=\sum_{j}a_ {j}\Lambda\big{(}\vec{r}_{R}-M_{T}\vec{r}_{j}-D\overline{v}\big{)}\approx o \left(\frac{\vec{r}_{R}}{M_{T}}-D\overline{v}\right)\end{array}  \tag{14}\]  
where \(\overline{v}=(k,l)\). The reconstruction is achieved because there is a high correlation between the synthetic \(\overline{H}_{PSH}\) and \(H_{OBJ}\) at \((x_{R},y_{R})=(Dk,Dl)\). The experiment was carried out using a pinhole with a diameter of 80 \(\upmu\)m. The pinhole was shifted to 8 positions outside the FOV, and the intensity patterns were recorded followed by the stitching procedure. The experiment was repeated for an object made up of three transparent digits '6', '0' and '1' from two optical channels. In channel-1, the object '0' was mounted on the optical axis to be within the FOV of the imaging system, whereas in channel-2, the objects '1' and '6' are mounted outside the FOV of the imaging system. The images of the bipolar stitched PSH and zero-padded object holograms are shown in Figure 14a,b, respectively. Even though the optical configuration used for the FOV extension technique involves a diffractive lens to satisfy the GSA condition between the SLM and sensor plane, there is additional background noise due to the zero padding of the object hologram. To reduce the background noise, the averaging technique was implemented with 20 samples. The reconstruction results using only the central part of the PSH and synthetic PSH for the entire objects are shown in Figure 15a,b, respectively. The FOV extension technique is demonstrated in I-COACH system with an extension factor of 3. However, with a longer training process, the FOV can be extended theoretically without limit. While the FOV extension technique is demonstrated on a coded aperture holography system, the technique can be easily adapted to other imaging systems by creating a synthetic point spread function and by stitching the individual responses corresponding to different lateral locations of the pinhole.