# Data-Decoupled Scattering Imaging Method Based on Autocorrelation Enhancement
## 2 Materials and Methods  
### Principle of Autocorrelation  
The scattering imaging process is determined within a certain field of view, when the angle of the light source is constant. The scattering-based imaging system mainly consists of a laser light source, target, scattering medium, and detector, as shown in Figure 1.  
The relationship between the target image and corresponding output speckle image captured by the sensor can be described as follows:  
\[E^{out}=K\cdot E^{in}, \tag{1}\]  
where \(E^{out}\) represents the received speckle image, \(K\) represents the point-spread function of the scattering media (i.e., frosted glass), \(E^{in}\) represents the target image of the input vector, and * represents the convolution of the target with the point-spread function. Even though the target cannot be directly identified by the speckle image, the speckle image contains the target information according to the convolution theorem. According to the autocorrelation principle, the autocorrelation operation on the target speckle image is written as follows:  
\[E^{out}=K\cdot E^{in},E^{out}\otimes E^{out}=\left(K*E^{in}\right)\otimes \left(K*E^{in}\right)=\left(E^{in}*E^{in}\right)\otimes\left(K*K\right), \tag{2}\]  
Since the point-diffusion function of an imaging system is a peak function, the above equation can be written as  
\[E^{out}\otimes E^{out}=\left(K*E^{in}\right)\otimes\left(K*E^{in}\right)= \left(E^{in}*E^{in}\right)+C \tag{3}\]  
where \(\otimes\) denotes the autocorrelation operation and \(C\) is the noise term.  
The autocorrelation of the target and its energy spectral density form a Fourier transform pair is given as:  
\[FT\Big{\{}E^{in}\otimes E^{in}\Big{\}}=\left|FT\Big{(}E^{in}\Big{)}\right|^{2} \tag{4}\]  
where \(FT\{\}\) represents the Fourier transform operation, from which the amplitude information of the target in the Fourier transform spectrum is obtained. Owing to the lack of phase information, the target image cannot be recovered by the inverse Fourier transform. Hence, the HIO-ER phase-recovery algorithm is used to recover the phase information of the target [21; 22]. The structure is shown in Figure 2.  
Figure 1: Principle of scatter imaging.  
This method initializes its amplitude information and then obtains the amplitude distribution of the target via iterative transformation in the Fourier domain. At the same time, the phase distribution is the same as that of the target, and the real target can be recovered by inverse Fourier transform. The detailed iterations are written as follows:  
\[\begin{array}{l}G_{k}(u,v)=FT\{g_{k}(x,y)\},\\ G^{\prime}_{k}(u,v)=|FT\{O(u,v)\}|e^{i\arg G_{k}(u,v)},\\ g^{\prime}_{k}(x,y)=IFT\{G^{\prime}_{k}(u,v)\},\end{array} \tag{5}\]  
The initial value of the iteration \(g_{1}(x,y)\) is assigned an arbitrary value, \(g_{k}(x,y)\) is the \(k^{\prime}th\) iteration, \(IFT\{\}\) represents the Fourier-inverse-transform operation, and the \((k+1)^{\prime}th\) iteration is obtained from the result of the \(k^{\prime}th\) iteration, \(g^{\prime}_{k}(x,y)\). In this process, the \"mixed input-output method\" is used for the following physical constraints:  
\[g_{k+1}(x,y)=\left\{\begin{array}{cl}g^{\prime}_{k}(x,y)&(x,y)\
otin\Gamma\\ g_{k}(x,y)-\beta g^{\prime}_{k}(x,y)&(x,y)\in\Gamma\end{array}\right. \tag{6}\]  
where \(\Gamma\) contains all points that do not satisfy the constraint, which is a non-negative real number for the imaging target, and \(\beta\) is the decay factor constant. New estimated objects are generated continuously during the iterative process until all of them satisfy the condition.