# On Ghost Imaging Studies for Information Optical Imaging
###### Abstract
In the famous Hanbury Brown-Twiss (HB-T) intensity interference experiment [39], light from the star is split into two beams and recorded by two separated detectors, and the desired Fourier-domain information of the distant star's intensity distribution \(I_{\rm s}(\mathbf{r}_{\rm s})\) is reproduced by the intensity correlation of the two beams (essentially the second-order auto-correlation of the star's light fields)  
\[ \begin{split}&\langle E^{*}(\mathbf{r}_{1},t)E^{*}(\mathbf{r}_{2},t+\tau)E( \mathbf{r}_{1},t)E(\mathbf{r}_{2},t+\tau)\rangle\\ =&\langle E^{*}(\mathbf{r}_{1},t)E(\mathbf{r}_{1},t)\rangle \langle E^{*}(\mathbf{r}_{2},t+\tau)E(\mathbf{r}_{2},t+\tau)\rangle,\\ &+\left|G^{(1)}(\mathbf{r}_{1},\mathbf{r}_{2};\tau)\right|^{2}\end{split}  \tag{6}\]  
where \(\left|G^{(1)}(\mathbf{r}_{1},\mathbf{r}_{2};\tau)\right|^{2}\propto\left|\mathcal{F} \{I_{\rm s}(\mathbf{r}_{\rm s})\}\right|^{2}\). On the other hand, GI [40; 41; 42] (shown as Figure 1) is also a representative high-order correlation imaging technique that has been widely considered and applied. In GI, illumination light with spatiotemporal fluctuations is split into two beams, one of which passes through the object (called the object beam) and then is recorded by a detector as \(I_{\rm t}(\mathbf{r}_{\rm t})\), while the other propagates without passing through the unknown object (called the reference beam) and is detected by another detector as \(I_{\rm r}(\mathbf{r}_{\rm r})\). For the incoherent thermal source, the point-to-point image information reproduction can be achieved by the second-order intensity correlation between the two detection signals (essentially the second-order cross-correlation) [43]  
\[\Delta G^{(2)}(\mathbf{r}_{\rm r},\mathbf{r}_{\rm t})=\langle\Delta I_{\rm r}(\mathbf{r}_ {\rm r})\Delta I_{\rm t}(\mathbf{r}_{\rm t})\rangle\propto\left|\int d^{2}\mathbf{r}h_ {\rm r}^{*}(\mathbf{r}_{\rm r};\mathbf{r})h_{\rm t}(\mathbf{r}_{\rm t};\mathbf{r})\right|^{2}, \tag{7}\]  
where \(h_{\rm r}(\mathbf{r}_{\rm r};\mathbf{r}),h_{\rm t}(\mathbf{r}_{\rm t},\mathbf{r})\) are CTFs of light fields in the reference and object paths, respectively (It is worth mentioning that, there exists another imaging scheme similar to GI that encodes the image information with light-field fluctuations, while retrieving the desired information via the first-order cross-correlation. This scheme is mostly used in microwave imaging and commonly referred to as \"correlated imaging\" [44] or \"radar coincidence imaging\" [45; 46]). Several studies have shown that, for GI with thermal light fields, by designing proper CTFs, various types of point-to-point signal reproduction between the desired object space and the image space can be realized, resulting in different GI systems, such as spatial-domain GI [47] (related to but different from single-pixel imaging [48; 49]), which constructs mapping between the image space and the object space via first-order light-field correlation), diffraction GI [50], phase contrast GI [51; 52] and holographic GI [53]. In practical optical imaging systems, commonly used optical detectors are mostly based on [PERSON]'s principle of photoelectric detection [54; 55], which indicates that the mean number of radiated electrons is proportional to the integrated light intensity during the exposure time. Since the characteristic time of thermal light field fluctuations is usually much smaller than the integration time of the detector, the output signal of optical detectors is actually proportional to the average light intensity (the first-order auto-correlation). Hence, the first-order light-field correlation can be directly detected to immediately realize traditional imaging, while the high-order correlation of light fields cannot be directly measured so that the desired image information in imaging techniques based on high-order correlation is hard to directly record.