# Experimental Investigation of Anisotropic Diffusion Applied in Ghost Imaging Reconstruction
## 2 Experimental Setup and Principle  
The experimental schematic is shown in Figure 1; the pseudo-thermal light source is generated by a beam of the laser (with wavelength \(\lambda=650\) nm) irradiating a rotating ground glass. The beam is then divided into the reference beam path and the object beam path by a 50:50 beam splitter. In the object beam path, the total light intensity of scattered light or transmitted light after a beam of light irradiated an object is collected by CCD1, and the nth measurement value is denoted as Bn. The transfer function of the measured object is \(T(x,y)\). At the same time, the optical intensity distribution of the other beam is recorded by the CCD2 in the reference beam path, and the nth measurement value is denoted as \(I_{n}(x,y)\). PGI uses these sampled data to reconstruct objects and the reconstructed results are recorded as \(\mathbf{T}_{IPGI}^{(0)}\), the initial image of IPGI. \(\mathbf{T}_{IPGI}^{(0)}\) firstly is smoothed by anisotropic diffusion (denoted as AD_ denoising in short as shown in Figure 1), then the denoised image is used to construct the noise interference item of IPGI. Finally, the noise interference term is substituted into the equation of IPGIfor iterative computation. Theoretically, the reconstructed result of the current iteration is clearer than the previous iteration, so the reconstructed image of the current iteration can be used to construct the noise interference term of next iteration after denoised by AD_ denoising, which is much closer to the actual noise term. Anisotropic diffusion, combining the image filtering process with the image edge detection process, shows better performance than that of the traditional denoising methods. Suppose the initial image before denoising is \(u\) (\(x\), \(y\)); the time variable t is introduced to represent the changing process of the image and the change of the image can be expressed as a partial differential equation:  
\[\frac{\partial u}{\partial t}=F[u(x,y,t)], \tag{1}\]  
where \(F\) represents an operator corresponding to a particular algorithm. The solution to the equation \(u\) (\(x\), \(y\), \(t\)) denotes the image at time \(t\). The anisotropic diffusion equation is expressed as:  
\[\left\{ \begin{array}{c}\frac{\partial u(x,y,t)}{\partial t}=div[g([\
abla u] )\
abla u]\\ u(x,y,0)=u(x,y)\end{array} \right., \tag{2}\]  
where div and \(\
abla\) are respectively denoted as the divergence operator and the gradient operator. \(|\
abla u|\) is the length of the gradient \(\
abla u\), and it plays an important role in determining if the current region is a smooth or edge area. If the current region is a smooth one, the value of \(|\
abla u|\) is small and the diffusion ability will be intensified. On the contrary, the opposite happens when the current region is an edge region. The term \(g(|\
abla u|)\) is the diffusion coefficient, which meets the following conditions:  
1. \(g(|\
abla u|)\) is a monotonically decreasing function and \(g(|\
abla u|)\in(0,1)\). 2. \(g(0)=1\) denotes that the diffusion is enhanced in the smooth area and \(\lim\limits_{|\
abla u|\rightarrow\infty}g(|\
abla u|)=0\) shows that the diffusion is stopped in the edge area. According to the nature of \(g(|\
abla u|)\), its specific expression is presented as follows:  
\[g(|\
abla u|) =\exp(-(\frac{|\
abla u|}{m})), \tag{3}\]  
Figure 1: The schematic diagram of the ghost imaging and anisotropic diffusion denoising system. L1 is the distance between the rotating ground glass and the measured object, L2 is the distance between the rotating ground glass and the CCD2 (L1 = L2), and f is the focal length of the Lens. where \(m\) is an artificial parameter, and its value plays an important role in image denoising. When \(|\
abla u|\leq m\), the diffusion ability is strengthened, and the image has a good effect of denoising; otherwise, the diffusion coefficient is suppressed but the edge information will be well preserved.