# Correlation Plenoptic Imaging: An Overview
## 4 A Different Architecture of CPI with Chaotic Light
### Plenoptic Properties of the Correlation Function  
The typical refocusing capability of PI characterized also the the CPI protocol due to the spatial and angular information encoded in the correlation function of Equation (27). To explicitly see this point, let us start by considering the simple case of a focused image, as obtained when the distance between the object and the source \(z_{b}=z_{bF}\) satisfies the two-photon thin lens equation [61; 75]  
\[\frac{1}{z_{a}+z_{bF}}+\frac{1}{z_{a}^{\prime}}=\frac{1}{f}. \tag{29}\]  
In this situation, the correlation function of Equation (27) gives the (incoherent) ghost image of the object, upon integration over the detection plane of \(\text{D}_{b}\)[61; 75]:  
\[\Sigma^{\text{foc}}(\mathbf{\rho}_{a})=\int d\mathbf{\rho}_{b}\Gamma_{(z_{a},z_{bF})}( \mathbf{\rho}_{a},\mathbf{\rho}_{b})\propto\int d\mathbf{\rho}_{o}|A(\mathbf{\rho}_{o})|^{2} \left|h_{\text{tr}}\left[\frac{\Omega}{cz_{bF}}\left(\mathbf{\rho}_{o}+\frac\mathbf{{\rho}_{a}}{m}\right)\right]\right|^{2}, \tag{30}\]  
where \(m=z_{a}^{\prime}/(z_{a}+z_{bF})\) is the image magnification. This result holds if \(h_{\text{tr}}\) is more peaked around the origin than the Fourier transform of the imaging lens \(L_{a}\); otherwise, it would be affected by the finite size of the lens. Such image is formally equivalent to the incoherent image obtained in an ordinary imaging system, were the PSF \(h_{\text{tr}}\) is given by the Fourier transform of the imaging lens pupil. However, as in the case of chaotic light, the correlation function of Equation (27) contains much more information than the ghost image: the main physical differences are once again related with the coherent nature of the images it encodes. Coherence is the immediate consequence of measuring coincidences between the spatial sensor \(\text{D}_{a}\) and any pixel of the angular sensor \(\text{D}_{b}\). This can be better understood in terms of the Klyshko picture [75] reported in Figure 13: light illuminating the object and contributing to the coincidence detection between any two pairs of pixels \(\mathbf{\rho}_{a}\) and \(\mathbf{\rho}_{b}\) has a well defined propagation direction; hence, it is responsible for the formation of a coherent image. Now, to explicitly demonstrate both the refocusing and three-dimensional imaging capabilities of this scheme, and to better highlight the plenoptic properties of the correlation function of Equation (27), we shall consider the more general out-of-focus situation (\(z_{b}\
eq z_{bF}\)). The stationary points of the phase defined in Equation (28) enable to determine the geometrical correspondence between points on the object and the source with points on sensors \(\text{D}_{a}\) and \(\text{D}_{b}\), respectively. In particular, the stationarity of \(\varphi\)with respect to \(\rho_{s}\) determines the object point that gives the predominant contribution to the integral of Equation (27), which is  
\[\boldsymbol{\rho}_{o}=-\frac{z_{b}}{z_{bF}}\frac{\boldsymbol{\rho}_{a}}{m}-\frac {\boldsymbol{\rho}_{b}}{M}\left(1-\frac{z_{b}}{z_{bF}}\right). \tag{31}\]  
When the focusing condition of Equation (29) is satisfied, this object point becomes independent of the specific sensor pixel \(\boldsymbol{\rho}_{b}\); hence, the focused ghost image is not sensitive anymore to the change of perspective enabled by the high resolution of the angular sensor \(\mathrm{D}_{b}\).