## 2 Materials and Methods  
### Fourier and Statistical Optics Theory  
The spatial filter geometry is shown in Figure 1. This simple geometry depicts a random field passing through a two-lens system. The source plane is one focal length \(f_{1}\) in front of \(\mathrm{L}_{1}\). The field passes through \(\mathrm{L}_{1}\) and then a complex amplitude filter \(T\) placed at the rear focus of \(\mathrm{L}_{1}\). The location of \(T\) also corresponds to the front focal plane of \(\mathrm{L}_{2}\). The field then transits \(\mathrm{L}_{2}\) before finally being observed in \(\mathrm{L}_{2}\)'s rear focal plane. Assuming that \(\mathrm{L}_{1}\) is large enough in diameter to collect all the light emitted from the source, the field in \(\mathrm{L}_{1}\)'s focal plane, right before \(T\), is  
\[ \begin{split} U\left(\mathbf{\rho},f_{1}^{-}\right)&= \frac{\exp\left(\mathrm{i}kf_{1}\right)}{\mathrm{i}\lambda f_{1}}\iint_{-\infty }^{\infty}U^{\mathrm{src}}\left(\mathbf{\rho}^{\prime}\right)\exp\left(-\mathrm{i }\frac{k}{f_{1}}\mathbf{\rho}\cdot\mathbf{\rho}^{\prime}\right)\mathrm{d}^{2}\mathbf{\rho} ^{\prime}\\ &=\frac{\exp\left(\mathrm{i}kf_{1}\right)}{\mathrm{i}\lambda f_{1 }}\tilde{U}^{\mathrm{src}}\left(\frac{\mathbf{\rho}}{\lambda f_{1}}\right)\end{split} , \tag{5}\]  
where \(\mathbf{\rho}\) is the observation vector, \(\mathbf{\rho}^{\prime}\) is the source vector, \(k=2\pi/\lambda\), and \(\lambda\) is the wavelength [22]. The random source field \(U^{\mathrm{src}}\) is assumed to be a sample function drawn from a wide-sense stationary random process (the dependence of \(U^{\mathrm{src}}\) on radian frequency \(\omega\) is assumed and suppressed), and \(\tilde{U}^{\mathrm{src}}\) is the spatial Fourier transform of \(U^{\mathrm{src}}\):  
\[\tilde{U}^{\mathrm{src}}\left(\mathbf{f}\right)=\iint_{-\infty}^{\infty}U^{ \mathrm{src}}\left(\mathbf{\rho}\right)\exp\left(-\mathrm{i}2\pi\mathbf{f}\cdot\mathbf{ \rho}\right)\mathrm{d}^{2}\rho, \tag{6}\]  
where \(\mathbf{f}=\hat{\mathbf{x}}f_{x}+\hat{\mathbf{y}}f_{y}\) is the spatial frequency vector. Figure 1: Spatial filter geometry. The field immediately to the right of the complex amplitude filter is  
\[U\left(\mathbf{\rho},f_{1}^{+}\right)=T\left(\mathbf{\rho}\right)U\left(\mathbf{\rho},f_{1}^{ -}\right)=\frac{\exp\left(\mathrm{j}kf_{1}\right)}{\mathrm{j}\lambda f_{1}}T \left(\mathbf{\rho}\right)\tilde{U}^{\mathrm{src}}\left(\frac{\mathbf{\rho}}{\lambda f _{1}}\right). \tag{7}\]  
Lastly, assuming that \(\mathrm{L}_{2}\) is large enough to capture all the light that passes through \(T\), the field at the output of the spatial filter \(U^{\mathrm{out}}\) is [22]  
\[ \begin{split} U\left(\mathbf{\rho},f_{2}\right)&=U^{ \mathrm{out}}\left(\mathbf{\rho}\right)\\ &=\frac{\exp\left(\mathrm{j}kf_{2}\right)}{\mathrm{j}\lambda f_{2 }}\iint_{-\infty}^{\infty}U\left(\mathbf{\rho}^{\prime},f_{1}^{+}\right)\exp \left(-\mathrm{j}\frac{k}{f_{2}}\mathbf{\rho}\cdot\mathbf{\rho}^{\prime}\right)\mathrm{ d}^{2}\rho^{\prime}\\ &=-\frac{\exp\left[\mathrm{j}k\left(f_{1}+f_{2}\right)\right]}{ \lambda^{2}f_{1}f_{2}}\iint_{-\infty}^{\infty}T\left(\mathbf{\rho}^{\prime} \right)\tilde{U}^{\mathrm{src}}\left(\frac{\mathbf{\rho}^{\prime}}{\lambda f_{1}} \right)\exp\left(-\mathrm{j}\frac{k}{f_{2}}\mathbf{\rho}\cdot\mathbf{\rho}^{\prime} \right)\mathrm{d}^{2}\rho^{\prime}\end{split} . \tag{8}\]  
The output field given in Equation (8) is for a single realization of the random source field \(U^{\mathrm{src}}\). Taking the autocorrelation of \(U^{\mathrm{out}}\) produces  
\[