## 2 Plenoptic Imaging with Correlations: From Working Principle to Recent Advances
Interestingly, recent studies have shown that, in the case of chaotic light illumination, the plenoptic properties of the correlation function do not need to rely strictly on ghost imaging: correlations can be measured between any two planes where ordinary images (see Figure 1b) are formed [35]. This discovery has led to the intriguing result that the SNR of CPI improves when ghost imaging of the object is replaced by standard imaging [40]. In particular, excellent noise performances are expected in the case of images of birefringent objects placed between crossed polarizers. This kind of source is particularly relevant in view of applications in fields like biomedical imaging (cornea, zebrafish, invertebrates, biological phantoms such as starch dispersions), security (distance detection, DOF extension), and satellite imaging. * Momentum-position entangled beams, generated by spontaneous parametric down-conversion (SPDC), which have the potential to combine QPI with sub-shot noise imaging [41], thus enabling high-SNR imaging of low-absorbing samples, a challenging issue in both biomedical imaging and security. The design of both quantum plenoptic devices are currently undergoing optimization by implementing a novel protocol that enables further mitigating the resolution vs. DOF compromise with respect to the one shown in Figure 2a: this protocol is based on the observation that, for any given resolution, the DOF can be maximized by correlating the standard images of two arbitrary planes, chosen in the surrounding of the object of interest, instead of imaging the focusing element [36]. Moreover, we are investigating the possibility to merge quantum plenoptic imaging with the measurement protocols developed in the context of differential ghost imaging [42]. In the following section, we will discuss all the technical solutions, both on the hardware and on the software side that we are investigating to speed up the process of generating a correlation plenoptic image. The problematic aspects can be clarified by a description of the practical process of CPI imaging that is made of the following steps:  
1. Collecting \(N_{f}\) synchronized pairs of frames from a sensor \(\mathrm{D}_{a}\), with resolution \(N_{ax}\times N_{ay}\), and \(\mathrm{D}_{b}\), with resolution \(N_{bx}\times N_{by}\). Synchronization of two separate digital sensors entails technical complications that can be overcome by using two disjoint parts of the same sensor [16]. The total acquisition time is \[T=N_{f}(\tau_{\mathrm{exp}}+\tau_{\mathrm{d}}),\] (3) where \(\tau_{\mathrm{exp}}\) is the frame exposure time that must be shorter than the coherence time of impinging light in order to exploit the maximal information on intensity fluctuations, while \(\tau_{\mathrm{d}}\) is the dead time between subsequent frames, usually fixed by the employed sensors. 2. Each acquired frame is transferred to a computer to be processed. This step can occur either  
where \(I_{j}^{(k)}(\mathbf{\rho}_{j})\), with \(j=a,b\), is the intensity measured in frame \(k\) in correspondence of the pixel on the detector D\({}_{j}\) centered on the coordinate \(\mathbf{\rho}_{j}\). In this way, an information initially encoded in \(N_{f}(N_{ax}N_{ay}+N_{bx}N_{by})\) numbers is used to reconstruct a correlation function determined by \(N_{ax}N_{ay}N_{bx}N_{by}\) values. The accuracy of the correlation function estimate (4) increases with the number of frames like \(\sqrt{N_{f}}\)[40]. However, increasing \(N_{f}\) also linearly extends the total acquisition time \(T\). Therefore, the combination of (1) fast and low-noise sensors, (2) methods to extract a good quality signal from smaller number of frames, and (3) tools for efficient data transfer and elaboration, is crucial in order to speed-up the acquisition process and make the CPI technology ready for real-world applications.