# On Ghost Imaging Studies for Information Optical Imaging
###### Abstract
Hence, in current GI studies, light fields with controllable macro fluctuations, such as pseudo-thermal light fields [56], are used to imitate the true thermal light-field fluctuations, so that the imitated fluctuations can be measured and used to calculate the correlation as  
\[\frac{1}{m}\sum_{i=1}^{m}\Delta I_{\mathbf{r}}^{(i)}(\mathbf{r}_{\mathrm{r}}) \Delta_{\mathbf{t}}^{(i)}(\mathbf{r}_{\mathrm{t}}). \tag{8}\]  
Mathematically speaking, the calculated correlation under finite number of samplings is actually a kind of data mean rather than the ensemble average, and there will thus be significant deviation between them due to the finite number of practical measurements. Therefore, the retrieved image information will deviate from the true one and become degraded. To address this issue, prior information is considered during the image retrieval, and an optimization-type problem is applied [57]. Intuitively, in the case of a bright classical light source, by considering the underlying light-field propagation in the object path, the problem can be roughly expressed as  
\[\min_{O}f_{\mathrm{t}}\Big{[}I_{\mathrm{t}}-\left|T(E_{\mathrm{s}},O)\right|^ {2}\Big{]}+\lambda_{\mathrm{t}}C_{\mathrm{t}}[O]. \tag{9}\]  
Here, \(f_{\mathrm{t}}(\cdot)\) means the noise effect on propagation and detection, \(C_{\mathrm{t}}[O]\) denotes the prior information expressed as the constraint on the object's image information \(O\), and \(T(E_{\mathrm{s}},O)\) is used to implicitly denote the light propagation in the object path. In GI, the ensemble  
Figure 1: Diagram of ghost imaging. Light with spatiotemporal fluctuations is split into two beams (shown as the blue beams), one of which passes through the object (called the object path) and then is recorded by a detector as \(I_{\mathrm{t}}(\mathbf{r}_{\mathrm{t}})\) while the other propagates without passing through the unknown object (called the reference path) and is detected by another detector as \(I_{\mathrm{r}}(\mathbf{r}_{\mathrm{r}})\). The point-to-point image information reproduction can be achieved by the second-order intensity correlation between the two detection signals. The orange arrows with dashed line shows the imagined path such that lights illuminating the object are emitted from the detection plane on the reference path. correlation between two paths will ensure that these two detectors \"see\" the same \"effective light source\" \(E_{\rm s}\), thus the unknown \(E_{\rm s}\) in (9) can be obtained in principle by solving the following phase-retrieval-type [58; 59; 60] problem of the reference path:  
\[\min_{E_{\rm s}}f_{\rm r}\Big{[}I_{\rm r}-\left|R(E_{\rm s})\right|^{2}\Big{]}+ \lambda_{\rm r}C_{\rm r}[E_{\rm r}] \tag{10}\]  
such that it is connected with \(E_{\rm r}\) as \(E_{\rm s}=g(E_{\rm r})\), where \(R(\cdot)\) denotes the effective light propagation on the reference path, \(f_{\rm r}(\cdot)\) means the noise effect, and \(C_{\rm r}[E_{\rm r}]\) is the constraint imposed on reference light fields \(E_{\rm r}\). Substituting that into Equation (9), the optimization-type problem can be written as [61]  
\[\min_{O}f_{\rm t}\Big{[}I_{\rm t}-\left|TR\Big{(}\sqrt{I_{\rm r}}e^{\phi_{\rm r }},O\Big{)}\right|^{2}\Big{]}+\lambda_{\rm t}C_{\rm t}[O], \tag{11}\]  
where \(TR(\cdot)\) denotes the implicit relationship between the reference path and the object path, and \(\{\phi_{\rm r}\}\) denotes unknown phases of light fields at the reference detection plane. Here, since only the \"intensity\" correlation between \(I_{\rm t}\) and \(I_{\rm r}\) are necessary to retrieve the image information of the unknown object in GI, it is intuitively believed that, under the constraint of setup parameters in a GI system, Equation (11) should be insensitive to the accuracy of those light phases \(\{\phi_{\rm r}\}\) (this has been empirically and theoretically verified in spatial-domain GI and diffraction GI [61; 62; 63], but has not been strictly proved for all kinds of GI systems).