# Review of 3D Imaging by Coded Aperture Correlation Holography (COACH) +
## 2 System Architectures
### Coded Aperture Digital Holography
#### 2.1.4 FOV Extended I-COACH  
In this section, we describe a technique to improve the FOV of I-COACH. The FOV of a typical imaging system is limited by the ratio between the finite area of the image sensor and the magnification of the optical system. For a given magnification, the limiting factor of the FOV is the finite size of the image sensor. Different techniques have been developed for enhancing the FOV of an imaging system, such as convolution techniques [57], particle encoding [58] and multiplexing of interferograms [59]. Recently, a robust yet simpler technique was developed for extending the FOV of I-COACH beyond the limit dictated by the image sensor area [60]. The optical configuration of the I-COACH with extended FOV is shown in Figure 13. In this technique, the GSA is directed to generate on the sensor plane an intensity pattern with the size three times larger than the area of the image sensor. The pinhole is shifted to pre-calculated lateral locations in the same axial plane, such that the different sections of the intensity pattern are projected on the image sensor and recorded. The recorded parts of the intensity pattern are stitched into a larger intensity pattern in the computer. The same process of recording and stitching is repeated using a second CPM. Unlike the methods in [57; 58; 59], the FOV extension procedure in I-COACH is done only once during the recording of the point holograms, with a relatively longer training, following by a stitching procedure in the computer. Once the synthetic PSH library is created, the recording of the object holograms is as simple as a regular I-COACH system. I-COACH is a linear space-invariant system. Thus, the response on the camera plane for a 2D object is a collection of independent point objects expressed as \(\sum_{j}a_{j}I_{1}\left(\tau_{o}-M_{\mathrm{T}}\overline{r}_{j}\right)\), where \(I_{1}\) is the response for a delta-function in the input, \(M_{\mathrm{T}}\) is the transverse magnification given by \(M_{\mathrm{T}}=z_{h}/z_{\mathrm{s}}\), \(z_{\mathrm{s}}\) is the distance between the object and lens \(L_{2}\) and \(a_{j}\) is a constant. The bipolar PSH is \(H_{PSH}=I_{1}\)-\(I_{2}\), and the bipolar object hologram can be expressed as,  
\[H_{OBJ}=\sum_{j}a_{j}\big{[}I_{1}\big{(}\tau_{o}-M_{\mathrm{T}}\overline{r}_{ j}\big{)}-I_{2}\big{(}\tau_{o}-M_{\mathrm{T}}\overline{r}_{j}\big{)}\big{]}= \sum_{j}a_{j}H_{PSH}\big{(}\tau_{o}-M_{\mathrm{T}}\overline{r}_{j}\big{)} \tag{10}\]  
Figure 11: (**a**) System response for the pinhole, (**b**) bipolar PSH, (**c**) system response for the object, and (**d**) bipolar object hologram. Figure 12: Reconstruction results of SCS-I-COACH. where \(I_{2}\) is the impulse response with the second independent CPM. The image of the object inside the FOV is reconstructed as  
\[ \begin{array}{ll}P(\vec{r}_{R})&=\iint\sum_{j}a_{j}H_{PSH}\big{(}\vec{r}_{o}-M _{T}\vec{r}_{j}\big{)}\vec{H}_{PSH}^{*}(\vec{r}_{o}-\vec{r}_{R})d\vec{r}_{0}\\ &=\mathfrak{D}^{-1}\Bigg{\{}\sum_{j}a_{j}|\mathfrak{D}\{H_{PSH}\}|\exp\big{(}i \varphi-i2\pi M_{T}\vec{r}_{j}\cdot\vec{\rho}\big{)}\exp(-i\varphi)\Bigg{\}}\\ &=\sum_{j}a_{j}\Lambda\big{(}\vec{r}_{R}-M_{T}\vec{r}_{j}\big{)}\approx o\left( \frac{\vec{r}_{R}}{M_{T}}\right)\end{array}  \tag{11}\]  
For a transverse magnification of \(M_{T}\) and sensor area of \(D\times D\), the size of FOV is \(S\times S\), where \(S=D/M_{T}\). To extend the FOV by a factor of 3, the pinhole is shifted to nine locations resulting of shifts of \(H_{PSH}\), and thus different areas of \(H_{PSH}\) are overlapped with the image sensor. The corresponding sections of the intensity patterns are detected separately, each time for the input \(\delta(x_{s}-Sk,y_{s}-Sl)\), where \(k,l=-1\),0,1, and the nine intensities are stitched. Besides, the central object point, the remaining 8 pinholes are outside the FOV of the imaging system. The recorded part of the intensity pattern can be expressed as \(I_{1}(x_{o}-Dk,y_{o}-Dl)\text{Rect}[(x_{o},y_{o})/D]\). The stitched intensity pattern is given by,  
\[