# Correlation Plenoptic Imaging: An Overview
## 2 Correlation Plenoptic Imaging with Chaotic Light--First Scheme
### Point-Spread Function and Plenoptic Properties
This indicates the crucial role played in CPI by the high-resolution detector D\({}_{b}\), as opposed to the bucket detector of ghost imaging. In fact, magnified and displaced images obtained by the correlation measurements can be reshaped and realigned (i.e., _refocused_) by employing the following scaling property,  
\[\Gamma_{(z_{a},z_{b})}\left[\frac{z_{a}}{z_{b}}\mathbf{\rho}_{a}-\frac{\mathbf{\rho}_{ b}}{M}\left(1-\frac{z_{a}}{z_{b}}\right),\mathbf{\rho}_{b}\right]\sim\mathcal{S} \Big{(}-\frac{\mathbf{\rho}_{b}}{M}\Big{)}^{2}\big{|}A(\mathbf{\rho}_{a})\big{|}^{2}\quad, \tag{11}\]  
as shown in Figure 3b. The independence of the image \(|A|^{2}\) of \(\mathbf{\rho}_{b}\) guarantees that integration over \(\mathbf{\rho}_{b}\) does not compromise the image quality. In fact, as shown in Figure 3d, the final image  
\[\Sigma^{\text{ref}}_{(z_{a},z_{b})}(\mathbf{\rho}_{a})=\!\int\!d\mathbf{\rho}_{b} \Gamma_{(z_{a},z_{b})}\left[\frac{z_{a}}{z_{b}}\mathbf{\rho}_{a}-\frac{\mathbf{\rho}_{ b}}{M}\left(1-\frac{z_{a}}{z_{b}}\right),\mathbf{\rho}_{b}\right]\sim\Sigma^{\text{ foc}}(\mathbf{\rho}_{a}) \tag{12}\]  
now approximates, up to an intensity rescaling, the focused incoherent image in Equation (4). The refocusing algoritm of Equation (11) is formally identical to the one employed in standard plenoptic imaging [3]. In fact, in analogy with standard plenoptic imaging, the key for refocusing is the information on the direction of light propagating between the object and the focusing element. In our CPI scheme, by measuring correlations, we can reconstruct the path of light rays from the source to the object, as clarified in Figure 2b. Hence, once again, we find that the coherent images associated with any fixed value of \(\mathbf{\rho}_{b}\), as described by Equations (10) and (11), have a wider depth of field with respect to the incoherent images obtained by a mere integration over \(\mathbf{\rho}_{b}\). However, such images are formed by a very small fraction of light propagating through the object, and are thus affected by a low signal-to-noise ratio. The SNR highly improves when summing all the rescaled images of Equation (11) over D\({}_{b}\), to get the final refocused image of Equation (12). Finally, both the result in Equation (10) and the plot in Figure 3a indicate that, when the image is out of focus, each pixel on the sensor D\({}_{b}\) represents a different point of view from which the image of the object is projected onto the sensor D\({}_{a}\). Hence, imaging the light source on the sensor D\({}_{b}\), enables acquiring multi-perspective images of the scene of interest. The change of viewpoint is a common feature of both PI and CPI, with the only difference that, in CPI, it is obtained with a single lens (\(L_{b}\)) rather than by the microlens array of standard PI; therefore, potentially, it is significantly larger in CPI than in standard PI. Despite not using it in the present work, it is worth emphasizing the key role played by the wide change of perspective and its achievable resolution, for implementing 3D imaging. Based on Equations (11) and (12), to get a refocused image, one should know with sufficient precision both the source-to-sensor distance \(z_{a}\) and the source-to-object distance \(z_{b}\). In reality, while \(z_{a}\) is determined by experimental settings, \(z_{b}\) is generally unknown. In this case, the refocusing algorithm must be applied for different values of \(z_{b}\), until the optimal sharpness of the image is found. This is shown in Figure 4, where we present a simulation of different images of an out of-focus object [67]: (a) the blurred ghost image; (b) the refocused CPI image; (c) and (d) two refocusing attempts with the correct value of \(z_{a}\) and an incorrect value of \(z_{b}\). The last two images are evidently less sharp than the one in (b). This also indicates that the refocusing capability of CPI can also be exploited for measuring distances.