# A Review of Ghost Imaging via Sparsity Constraints
## 3 Applications of GISC
### X-ray Fourier-Transform GISC  
Since X-ray diffraction in crystals was discovered in 1914, X-ray crystallography has become a powerful tool in exploring and analyzing the internal structures of complex materials, such as biomolecules and nanomaterials [79; 80; 81]. However, the structure information of many important molecular materials, such as membrane proteins, is still out of reach, because it is difficult to grow macroscopic crystals for these materials. Meanwhile, with the rapid development of nanoscience and biology, there is an urgent need to observe the internal structures of samples in their natural states instead of in crystals. In 1999, the coherent diffraction imaging (CDI) method was proposed to extend X-ray crystallography to allow imaging noncrystalline structures in nanoscale by illuminating the samples with coherent X-rays and recording the diffraction patterns in the far field [82; 83; 84]. Hereafter, Fresnel CDI and ptychography techniques were proposed to circumvent the intrinsic restriction of sample size in classical CDI [85; 86; 87; 88]. Nevertheless, the diffraction pattern within the beam stop is missing in classical CDI, and an additional technology is needed for low-frequency information. Figure 4: Experimental results of a natural scenes at 1 km away. (**a**) Reconstructed spectral images by the GISC spectral camera; (**b**) the original image recorded by the monochrome monitoring camera. Most of all, due to the requirements for high coherence and brightness, currently, only synchrotron radiation and X-ray free electron laser sources can be used to X-ray CDI applications, and coherent imaging using laboratory X-ray sources remain unexplored. In Fourier-transform GI (FGI), a Fourier-transform diffraction pattern of a sample can be obtained at the Fresnel region by calculating the second-order correlation of spatially incoherent light [10]. It has been demonstrated using an X-ray pseudo-thermal source [21], which provides a feasibility to achieve high-resolution microscopy with a tabletop X-ray source. Since lack of effective X-ray beam splitters, in practice the correlated light fields are generated by simply shuttling the sample in and out of the optical path between the source and the detector. In 2015, [PERSON] group proposed a method of phase retrieval from encoded diffraction patterns to achieve better image quality in the classical CDI, where the sample is encoded by a group of random masks in real space. However, because of the destructive effect of X-ray irradiation, it is almost impossible to multiply encode the sample in the classical CDI. Fortunately, different from CDI techniques, in Fourier-transform GISC, the diffractive pattern of a sample in the testing path can be non-locally encoded by inserting modulation components in the reference path [89]. Therefore, multiple encoding of the sample can be easily realized and better image quality may be expected. Figure 5 shows the principle of coded Fourier-transform GISC [89]. A thermal light beam is split into two paths. The sample is in the testing path and a point detector records the intensity signals of this path. In the reference path, a mask is inserted and a panel detector records the intensity distributions. The distances between the beam splitter and the detector in both paths are equal, and the mask's position in the reference arm is the same as the sample's position in the testing arm. The correlation function between the intensity fluctuations of these two arms is proportional to a modulus of a convolution, which can be expressed as [90]  
\[\left\langle\Delta I_{t}(x_{t})\Delta I_{r}(x_{r})\right\rangle\propto\left| \mathcal{F}\left\{s^{*}(x_{r})O(x_{r})\right\}\right|^{2}, \tag{9}\]  
where \(s^{*}(x)\) is the conjugate of the mask's transmission function, \(\mathcal{F}\) represents a Fourier transform. Therefore, the Fourier-transform diffraction pattern of the sample encoded by the mask in real space can be obtained from the intensity correlation between the two paths.