## 2 Plenoptic Imaging with Correlations: From Working Principle to Recent Advances
In general, given an object placed at a distance \(s\) from the focusing element, and characterized by the light intensity distribution \(\mathcal{A}(\mathbf{\rho})\), its images are encoded in the function \(\Gamma(\mathbf{\rho}_{a},\mathbf{\rho}_{b})\), in the geometrical-optics limit, as  
\[\Gamma(\mathbf{\rho}_{a},\mathbf{\rho}_{b})\sim\mathcal{A}^{n}\bigg{(}\frac{s}{s_{0}} \frac{\mathbf{\rho}_{a}}{M}+\bigg{(}1-\frac{s}{s_{0}}\bigg{)}\frac{\mathbf{\rho}_{b}}{M _{L}}\bigg{)}, \tag{2}\]  
where \(M\) and \(M_{L}\) are the magnifications of the images of the reference object plane and of the focusing element, respectively, while the power \(n\) is equal to 1 or 2, according to whether the object lies in only one [16,33,35] or both [36,37] optical paths. Experimental CPI based on pseudo-thermal light is shown in Figure 2c, where both the acquired out-of-focus image and the corresponding refocused image are shown [16]. It was demonstrated in this proof of principle that CPI is characterized by diffraction-limited resolution on the object plane focused on the sensor. Details on the resolution limits are shown in Figure 2a, where one can observe an even more striking effect: thanks to its intrinsic coherent nature, CPI enables an unprecedented combination of resolution and DOF [16]. However, the low-noise sCMOS camera employed in the experiment, working at 50 fps at full resolution, requires several minutes to acquire 30,000 frames used to reconstruct the plenoptic correlation function, and a standard workstation has taken over 10 h for elaborating the acquired data and perform refocusing. The resulting image was also rather noisy, due to the well-known resolution vs. noise compromise of chaotic light ghost imaging that keeps also affecting CPI [35]. We are addressing these issues by employing two kinds of sources:  
* Chaotic light sources, such as pseudothermal light, natural light, LEDs and gas lamps, and even fluorescent samples, operated either in the high-intensity regime or in  
Figure 2: (a) shows the resolution limits, as a function of the longitudinal position, of the image of a double-slit mask with center-to-center distance \(d\) equal to twice the slit width; here, CPI outperforms both conventional imaging and standard PI with \(3\times 3\) directional resolution. The evident asymmetry of the CPI curve is due to the existence of two planes in which the object is focused: one at \(z_{b}=z_{a}\) and one at \(z_{b}=0\) (see [16]). Plots in (**b**) show a result of a simulation: the target is moved from the focused plane (top left) to an out-of-focus plane (top right). Starting from this position, we show the results of PI refocusing with \(3\times 3\) directional resolution (bottom left) and the CPI refocusing (bottom right); (**c**) shows the results of an experiment [16] in which the standard image of a triple slit was completely blurred (top), while the image obtained by CPI (bottom) was made fully visible by exploiting information on light direction. Plots in (**c**) are reproduced with the permission from Ref. [16], copyright American Physical Society, 2017.  
the \"two-photon\" regime, in which an average of two photons per coherence area propagates in the setup. Chaotic light sources are well known to be characterized by EPR-like correlations in both momentum and position variables [38; 39], to be exploited in an optimal way to retrieve an accurate plenoptic correlation function in the shortest possible time. In order to efficiently retrieve spatio-temporal correlations, tight filtering of the source can be necessary to match the filtered source coherence time with the response time of the SPAD arrays that can be as low as 1 ns. Alternatively, pseudorandom sources with a controllable coherence time, made by impinging laser light on a fast-switching digital micromirror device (DMD), can be employed.