# On Ghost Imaging Studies for Information Optical Imaging
## 3 Ghost Imaging Studies for Information Optical Imaging
### Resolution Analysis in the High-Dimensional Light-Field Domain  
For the GI system, since it enables to map information of higher-dimensional light-field domains into lower dimensions and to achieve imaging of the high-dimensional image information via the second-order correlation, the description of its resolution should be reconsidered from a different perspective. Intuitively, it is the resolution in the high-dimensional light field domain rather than the 2-D spatial domain that would be limited by the optical diffraction effect. In this case, some analyses on the resolution of GI from the viewpoint of information theory have been conducted. For example, taking the GI camera  
Figure 3: Schematic diagram of the GI spectral camera system [75]. The object illuminated by natural lights is firstly imaged by a front imaging module, then diffracted by a spatial random phase modulator (SRPM), and finally recorded by a detector as a pattern of intensity distribution \(I_{t}(\mathbf{r}_{t})\). By the spatial–dimensional second-order correlation between this intensity distribution and the pre-measured impulse response \(I_{t}(\mathbf{r}_{t};\mathbf{r},k)\) corresponding to a monochromatic point source \(\delta(\mathbf{r},k)\) where \(k=1/\lambda\), the object’s spatial–spectral information can be achieved.  
as the research object, [PERSON] et al. [63; 81] proposed a criterion to measure its discernibility on high-dimensional light-field image information by combining with the compressed sensing (CS) theory [82; 83]. Specifically, the relationship between the normalized second-order correlation in the high-dimensional light-field domain \(g^{(2)}\left(\left|\boldsymbol{\tau}_{i}-\boldsymbol{\tau}_{j}\right|\right)\) (\(\boldsymbol{\tau}\) denotes a high-dimensional coordinate that could contain spatial, spectral and polarization dimensions) and the mutual coherence \(\mu\) of system's transmission matrix is first constructed as  
\[\mu=\underset{i\
eq j}{\text{max}}\Big{|}g^{(2)}\left(\left|\boldsymbol{\tau}_ {i}-\boldsymbol{\tau}_{j}\right|\right)\Big{|}, \tag{14}\]  
and the condition for resolving sparse point-like targets is given as the minimal distance of two points that establishes  
\[g^{(2)}\big{(}\left|\boldsymbol{\tau}_{i}-\boldsymbol{\tau}_{j}\right|\big{)} <\frac{1}{2K-1} \tag{15}\]  
by combining with the exact recovery condition [84], where \(K\) is the sparsity level of those point-like targets. Through this description, it can be roughly seen that multiple dimensions of light fields do interact with each other when considering the resolving ability. Since the resolution is fundamentally limited by the system's transmission function which transfers the high-dimensional image information, it is possible to greatly increase the spatial resolving ability beyond the Rayleigh criterion by utilizing discrepancies of the imaging object in other dimensions [81]. Similar to the typical deconvolution operation [85; 86] for traditional imaging that exploits the prior information of the system's transmission function, it is possible to achieve spatial super-resolution imaging in GI. By performing regularized preconditioning operations [87; 88; 89] on both the detection signal as well as the transmission matrix and solving the preconditioned problem via optimization algorithms, it is experimentally demonstrated that a high-quality imaging result with resolution beyond the diffraction limit can be obtained, and further theoretical analysis from the perspective of Fourier frequency domain also verifies its capability [90].