# Review of 3D Imaging by Coded Aperture Correlation Holography (COACH) +
## 2 System Architectures
### Coded Aperture Digital Holography
#### 2.1.1 Coded Aperture Correlation Holography Architectures
 \tag{2}\]  
By the phase-shifting procedure, a complex object hologram \(H_{OBJ}\) is synthesized given by,  
\[H_{OBJ}(u,v)=\sum_{j}A_{j}^{*}B_{j}G\left(u-u_{j},v-v_{j}\right) \tag{3}\]  
A particular plane of the object can be reconstructed by the cross-correlation between the \(H_{PSH}\) = \(G(u,v)\) and \(H_{OBJ}\) as  
\[\begin{split} P(u\prime,v\prime)&=\iint\left\{ \sum_{j}A_{j}^{*}B_{j}G\left(u-u_{j},v-v_{j}\right)\right\}G^{*}(u-u\prime,v- v\prime)dudv\\ &\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad \quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad \quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad \quadis shown in Figure 5l. The experiment is repeated using two channels and United States Air Force (USAF) object with elements 2 and 3 of group 2 placed at the same location as the NBS object from the lens \(L_{2}\). A complex hologram is synthesized from three camera shots. The reconstruction result is shown in Figure 5m. The USAF object is shifted by a relative distance of 2 cm from the NBS object. The reconstruction results using \(H_{PSH}\)'s recorded at the location of the NBS and the new location of the USAF are shown in Figure 5n,o, respectively.  
From the reconstruction results, it can be seen that even though the CPMs were synthesized using GSA, there is a substantial amount of background noise. In order to reduce the background noise, additional techniques are necessary. In relation to the topic of optical pat  
Figure 4: CPM synthesis using the modified Gerchberg-Saxton algorithm.  
Figure 5: **(a–c)** Intensity patterns for the pinhole for phase shifts \(\theta\) = 0\({}^{\circ}\), 120\({}^{\circ}\) and 240\({}^{\circ}\), **(d–f)** intensity patterns for the object for phase shifts \(\theta\) = 0\({}^{\circ}\), 120\({}^{\circ}\) and 240\({}^{\circ}\), **(g–h)** amplitudes of \(H_{PSH}\) and \(H_{OBI}\), respectively, **(i)** image of the CPM, **(j–k)** phase of \(H_{PSH}\) and \(H_{OBI}\), respectively, **(l)** reconstructed image, **(m)** reconstruction results of two objects when both are on the same plane. Reconstruction results when the two objects are separated by 2 cm and reconstructed using **(n)**\(H_{PSH}\) of NBS plane and **(o)**\(H_{PSH}\) of USAF plane.  
demonstrated that using a phase-only, instead of a matched filter, the correlation peaks become sharper with reduced side lobes [51]. Therefore, as a first step against the noise, a modified PSH with phase-only Fourier transform, given by, replaced the matched filter based PSH, where \(\mathfrak{G}\) and \(\mathfrak{G}^{-1}\) are a Fourier and inverse Fourier transform, respectively, and \(\overline{\rho}\) is the location vector in the reconstruction plane. The phase-only filter, when used in the correlation with the object hologram, reconstructed sharper images with lower background noise. The reconstruction results [52] of the NBS object 10 _lp/mm_ using the matched filter and the phase-only filter are shown in Figure 6a,b, respectively. However, the phase-only filter is less suitable for reconstruction of objects with greyscale transmittance [52], due to the lower immunity of the phase-only filter against noise.  
\[\widetilde{H}_{PSH}(\overline{\rho})=\mathfrak{G}^{-1}\{\exp[i\cdot\arg( \mathfrak{G}\{H_{PSH}(\overline{\rho})\})]\} \tag{5}\]  
Averaging is another technique, which involves the recording of multiple holograms followed by averaging over the multiple reconstructions [53]. The SNR of the COACH method was further improved by recording several \(H_{OBJ}\) and \(H_{PSH}\) under independent CPMs and averaging over the several complex reconstructions. The averaging technique, in the case of COACH, is based on the assumption that any two CPMs synthesized by GSA from different initial random profiles are independent, i.e., their cross-correlation is negligible compared to their auto-correlation.