## 3 Exposure Level Estimation and Adaptive Selection of Suitable \(\beta\) Value
### Adaptive Selection of Suitable \(\beta\) Value  
In the reconstruction algorithm, our constraint is that the pixels must be non-negative, for pixels that violate the constraint, we use Equation (3) [18] to replace them. Among them, \(\gamma\) is the set of points which violates the object-domain constraints, \(\beta\) is a constant, which prevents the algorithm from falling into the local optimum points. \[g_{k+1}(x)=\left\{ \begin{array}{ll}g^{\prime}_{k}(x),&x\
otin\gamma\\ g_{k}(x)-\beta g^{\prime}_{k}(x),&x\in\gamma\end{array} \right. \tag{3}\]  
So, we change the points which violate the object-domain constraints, which use the second line of Equation (3). Suppose the \(kth\) input \(g_{k}(x)\) is a \(3\times 3\) matrix A. The output \(g^{\prime}_{k}(x)\) is another \(3\times 3\) matrix B, hence, the part of violates the object-domain constraints (the pixel value must be non-negative) can be expressed as:  
\[g_{k+1}(x)=\left[ \begin{array}{ccc}a_{11}&a_{12}&a_{13}\\ a_{21}&a_{22}&a_{23}\\ a_{31}&a_{32}&a_{33}\end{array} \right]-\beta\left[ \begin{array}{ccc}b_{11} &b_{12}&b_{13}\\ b_{21}&b_{22}&b_{23}\\ b_{31}&b_{32}&b_{33}\end{array} \right]=\left[ \begin{array}{ccc}a_{11}+\beta \big{|}b_{11}&a_{12}+\beta\big{|}b_{12}&a_{13}+\beta\big{|}b_{13}\big{|}\\ a_{21}+\beta\big{|}b_{21}&a_{22}+\beta\big{|}b_{22}&a_{31}+\beta\big{|}b_{32} \big{|}\\ a_{31}+\beta\big{|}b_{31}&a_{32}+\beta\big{|}b_{32}&a_{33}+\beta\big{|}b_{33} \big{|}\end{array} \right] \tag{4}\]  
Among them, '\(||\)' is an absolute value symbol, each element of the matrix in Equation (4) is an independent positive proportion function. As a positive proportional coefficient, \(\beta\) determines the change of the whole iteration process. As \(\beta\) becomes larger, the multiple change of pixel value makes too much difference between the \((k+1)th\) and \(kth\) inputs, which will lead to the instability of the whole system and fails to reconstruct the object image finally. Therefore, the \(\beta\) value should be selected carefully, a single \(\beta\) value cannot make all coherent power-spectrum patterns reconstruct clear objects, and there is a corresponding \(\beta\) interval for input \(g_{k}(x)\). We validate the above conclusions by simulation. According to Equation (4), using the 'cameraman' image with a pixel size of \(256\times 256\) as the object to be reconstructed. The value of \(\beta\) traverses between 0.1-2 intervals and 0.01 intervals, and 100 iterations. According to the result of reconstruction, the object with \(\beta\) value in the range of 0.1-1.3 is very clear (the reconstruction effect can refer to Figure 3c. The recovery images appear blurred when \(\beta\) is bigger than 1.4, as shown in Figure 3e. When \(\beta\) is selected up to 2, no matter how many iterations are performed, the object images cannot be reconstructed successfully. The simulation results also show that \(\beta\) reconstruct a clear object not only in a the range of 0.5-1.0 proposed by [PERSON]. For the image of the 'cameraman', \(\beta\) is in the range of 0.1-1.3. According to the above analysis, it is summarized as follows: in the single-shot coherent power-spectrum scattering imaging experiment, the exposure time of the camera determines the exposure degree of the collected power-spectrum. The exposure degree is also a reflection of the overall pixel value. The larger the exposure degree, the set of violations of the constraints needs to be adjusted by setting a larger \(\beta\), making adjusted input is closer to the real object. So we choose to adjust the feedback coefficient \(\beta\) to solve the problem of acquiring single coherent power-spectrum scattering imaging with different exposure. We use the above conclusion to reconstruct the experimental power-spectrum (with the increase in the exposure level, the \(\beta\) value also needs to increase, accordingly).