# A Review of Ghost Imaging via Sparsity Constraints
## 3 Applications of GISC
### X-ray Fourier-Transform GISC
The mask in the reference path can \"non-locally\" encode the information of the sample in the test path. For pseudo-thermal light, it is easy to modulate the speckle fields in a predetermined manner, so we can alter the masks in the reference path to get different encoded patterns. Obviously, when \(s(x)=1\), Equation (9) degenerates to the conventional Fourier diffraction pattern of the sample. Based on recombination and reutilization of the correlated intensity distributions of light fields, a spatial multiplexing reconstruction method  
Figure 5: Principle of X-ray Fourier-transform GISC with coded diffraction patterns [89]. has been proposed to improve the sampling efficiency and image quality of Fourier-transform GISC. It can greatly reduce the scale of the sensing matrix in the sensing equation, and provide the feasibility of reconstruction with just a few measurements [91]. Figure 6 shows the simulation results of an encoded X-ray Fourier-transform GISC, with a wavelength of 1 nm, distances \(d_{1}=10\) cm and \(d_{2}=30\) cm, respectively. The sample in the testing path is a double-slits with a separation \(d_{slit}=200\)\(\mu m\) as shown in Figure 6a, and two masks were adopted in the simulation. The first one is the same double-slit with a \(90^{\circ}\) rotation as shown in Figure 6b. Figure 6c is the corresponding non-locally encoded Fourier-transform GISC diffraction pattern. Clearly, it is a coherent diffraction pattern of the encoded sample. Using the second one, a concentric-circle phase mask with a period of 75 \(\mu\)m (Figure 6d), an encoded FGI diffraction patterns as shown in Figure 6e was obtained, which is almost the same as the encoded diffraction pattern in Figure 6f obtained from the conventional coherent diffraction imaging simulation. Then the image can be reconstructed by utilizing phase retrieval algorithms with sparsity constraints [92; 93; 94; 95; 96; 97; 98; 99; 100; 101]. Figure 7 is a photograph of our tabletop X-ray GISC microscope facility. The whole system is placed in a vacuum chamber and controlled remotely. A micro-focus X-ray tube with center energy at 1.25 keV is adopted as the light source. X rays emitted from the tube is filtered to get monochromatic X-rays, then passes through a pinhole array and illuminates a designed diffuser to produce high contrast X-ray speckles. A controllable pseudo-thermal X-ray source is obtained by shifting the diffuser transversely. A translation device can switch the sample and the modulation masks in and out of the beam automatically. A CCD camera with a pixel size of 15 \(\upmu\)m is arranged in the beam to detect the X-ray intensity fluctuations. Figure 6: Simulation results of X-ray Fourier-transform GISC. (**a**) is the sample, (**b**,**d**) are the masks, (**c**,**e**) are the non-locally encoded X-ray Fourier-transform GISC diffraction patterns reconstructed from the intensity correlations with (**b**,**d**) inserted in the reference path, respectively. (**f**) is the corresponding coded diffraction pattern obtained from the coherent diffraction imaging [89]. By exploiting X-ray Fourier-transform GISC, it is of great hope to realize tabletop high-resolution 3D imaging of complex materials, such as biomolecules and nanomaterials. As a unique lensless imaging scheme, Fourier-transform GISC may also be introduced to neutron optics [102], its novel capability to image crystalline and noncrystalline samples, especially the micro magnetic structures, will bring important applications to various scientific frontiers.