# A Review of Ghost Imaging via Sparsity Constraints
## 3 Applications of GISC
### Lensless Wiener-Khinchin Telescope  
Image resolution is always an important issue in various fields of scientific researches and engineering applications, many great work and discussions of resolution have been made by researchers. For conventional imaging technologies, which is based on the point-to-point correspondence between the object and the image planes, the resolution can be accurately analyzed with the transmission function of the imaging systems [103; 104]. However, for imaging systems based on the wavefront random phase modulation [105; 106; 107; 108; 64; 108], since the point-to-point correspondence doesn't exist anymore, their resolution ability is difficult to be directly obtained from the transmission function, but can be obtained by analyzing the high-order correlation of light fields. Based on high-order spatial autocorrelation of thermal light, a lensless Wiener-Khinchin telescope was proposed [22]. It can acquire an image of an object in a single-shot measurement simply with a spatial random phase modulator.  
As shown in Figure 8, the lensless Wiener-Khinchin telescope consists of a spatial random phase modulator and a CCD detector, whose positions are fixed. The object is illuminated by a true thermal light source.  
Figure 7: A photograph of the tabletop X-ray GISC microscope facility in SIOM.  
The second-order spatial autocorrelation of the detected light is [22]  
\[G_{I_{t}}^{(2)}\left(r+\Delta r,r\right)\propto\left\{\left[1+g_{h}^{(2)}\left( \frac{\Delta r_{0}}{2\lambda z_{1}}\right)\right]\otimes\mathcal{F}^{-1}\left\{ \left|\mathcal{F}\left\{I_{0}(r_{0})\right\}_{r_{0}\to f_{r_{0}}}\right|^{2} \right\}_{f_{r_{0}}\rightarrow\Delta r_{0}}\right\}_{-\frac{z_{1}}{z_{2}} \Delta r}, \tag{10}\]  
where \(I_{0}(r_{0})\) is the intensity in the object plane, \(r_{0}\) and \(r\) are the coordinates in object plane and the sensitive surface of the CCD detector, respectively. \(g_{h}^{(2)}\left(\frac{\Delta r_{0}}{2\lambda z_{1}}\right)\) is the normalized second-order correlation of the incoherent impulse response distribution. Equation (10) indicates that the Fourier transform of the intensity distribution \(I_{0}(r_{0})\) on the object plane \(\left|\mathcal{F}\left\{I_{0}(r_{0})\right\}_{r_{0}\to f_{r_{0}}}\right|\) can be separated from \(G_{I_{t}}^{(2)}\left(r+\Delta r,r\right)\), and the resolution is determined by \(g_{h}^{(2)}\left(\frac{\Delta r_{0}}{2\lambda z_{1}}\right)\). The object \(I_{0}(r_{0})\) can be reconstructed by utilizing phase retrieval algorithms with sparsity constraints [92; 93; 94; 95; 96; 97; 98; 99; 100; 101]. In some cases, only amplitude information of an object is interested, which can be used as a constraint to greatly improve the reconstruction quality in phase retrieval [109].  
Images of two targets (a double slit and a starfish), illuminated by a xenon lamp, detected by the lensless Wiener-Khinchin telescope are shown in Figure 9. Different parameters are selected. For the double slit, \(z_{1}=0.15\) m, \(z_{2}=12\) mm, and for the starfish, \(z_{1}=1\) m, \(z_{2}=4\) mm.  
Comparing with lensless CS imaging [72; 105; 110] and lensless GI [21; 74; 111], neither a measurement matrix nor a calibration process is required in lensless Wiener-Khinchin telescope, while the FOV of the lensless Wiener-Khinchin telescope is limited by the memory effect [112; 113; 114; 115] of the system. The lensless Wiener-Khinchin telescope may have many applications, such as astronomical observation and X-rays remote sensing. Moreover, considering the scattering medium as the spatial random phase modulator, the lensless Wiener-Khinchin telescope may also open a door for a quantitatively description for imaging through scattering media [116; 117; 118; 119; 120].  
Figure 8: Schematic of a lensless Wiener-Khinchin telescope based on high-order spatial autocorrelation of thermal light [22].