# 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  
Recently, a self-interference incoherent digital holography technique using a coded aperture was developed in the optical regime [46] and is shown in Figure 3. Unlike other imaging techniques with coded apertures [49], in COACH, the aperture is a phase-only mask. Thus, the aperture does not absorb the incoming light and consequently, more light power participates in the hologram recording. In addition, the detected pattern is a digital hologram containing the 3D image of the observed scene. COACH can be considered as a generalization of FINCH [23; 26] in the sense that instead of the quadratic phase mask of FINCH, a pseudorandom coded phase mask (CPM) modulates the beam. In COACH, the light emitted from each object point splits into two beams in which one of the beams is modulated by the CPM while the other beam is not. The two beams with the common origin are mutually coherent and thus are interfered. Three phase-shifted holograms are recorded for  
Figure 1: Recording and reconstruction of a hologram in a general self-interference digital holography system. Figure 2: Coded aperture imaging system. phase values \(\theta_{1,2,3}=0\), \(2\pi/3\) and \(4\pi/3\) and are superposed into a complex hologram. As in FINCH, this procedure is done in order to remove the twin image and bias terms during the reconstruction. In COACH, unlike FINCH, a one-time training procedure is required. The training is done by recording a library of point spread holograms (PSHs) of a point object positioned at various axial locations. For a 3D object placed within the axial boundaries of the PSH library, three phase-shifted holograms are once again recorded, using the same CPM with the above three phase values, and superposed into a complex object hologram. Any axial plane of the object space is reconstructed by a cross-correlation between the object hologram and the corresponding PSH from the library. The holograms recorded by COACH cannot be classified as Fourier or Fresnel holograms because neither Fourier transform nor Fresnel back-propagation can reconstruct the image. Instead, this hologram is generated from the interference between the plane and chaotic waves. Therefore, COACH can be considered as a generalized correlation self-interference holography technique in which FINCH is only a special case when the CPM is the special mask of the diffractive spherical lens. Another dissimilarity is that in FINCH, the object image can be reconstructed by a Fresnel back-propagation from the hologram. Therefore, the imaging characteristics of COACH are also different from those of FINCH as analyzed in the following. While COACH, being a generalized technique, is operational with any random phase mask, it was noticed that for an arbitrary CPM, the reconstruction via a cross-correlation generated a disturbing background noise. In order to reduce the background noise and at the same time to retain the randomness aspect of the CPM, the CPM is synthesized using a modified Gerchberg-Saxton algorithm (GSA) [50] to render a pure phase function for the CPM and a uniform magnitude over some desirable area in the spectrum domain, as shown in Figure 4. The constraint on the CPM plane is required since the CPM is displayed on a pure-phase SLM. On the other hand, the constraint on the spectrum domain (sensor plane) reduces the background noise on the reconstructed image. This is because the noise originated from the bias level of the spectral intensity. This bias level is considerably reduced by the superposition of two, or more, spectral intensities, if this level is approximately the same in all two, or more, spectral intensities. The constraint on the spectrum domain satisfies this last condition of approximately equalizing the bias level. Based on Figure 3, the mathematical formulation is as follows. For a point object located at the front focal plane of lens \(L_{2}\), a collimated beam passes through the polarizer \(P_{1}\) oriented \(45^{\circ}\) with respect to the active axis of the SLM.