## 2 Inconsiderateness in Previous Single-Shot Coherent Power-Spectrum Imaging Methods  
By applying [PERSON]'s single-shot power-spectrum imaging method for imaging a target hidden in an opaque scattering medium, with a fixed \(\beta\) value, we found that the reconstruction performance of various power-spectrum patterns captured with different exposure levels is different which usually occurs in practice, as shown in Figure 1. At the same time, we also show the power spectrum collected without a scattering layer in Figure 1(g1,g2) for comparison.  
The images shown in the upper part of Figure 1(a1-f1), are the captured power-spectrum patterns of the character '5' on a United States Air Force (USAF) 1951 resolution test chart, with the different exposure times of the camera. In the experiments, we made the illuminating laser power to be the  
Figure 1: The reconstruction results from various power-spectrum patterns with fixed \(\beta\) = 0.9. (**a1–f1**) are power-spectrum patterns captured with exposure times of 3000 \(\upmu\)s, 4000 \(\upmu\)s, 6000 \(\upmu\)s, 10,000 \(\upmu\)s, 12,000 \(\upmu\)s, and 20,000 \(\upmu\)s, respectively; (**a2–f2**) are the results reconstructed from (**a1–f1**) by 100 iterations, (**g1,g2**) represent the power spectrum with and without scattering media, respectively. (**b1–f1**) has the same scale bar as (**a1**); (**b2–f2**) has the same scale bar as (**a2**); (**g1**) has the same scale bar as (**g2**).  
same. In the reconstructions, we used the fixed \(\beta\) = 0.9, and the results are shown in the lower part of Figure 1, i.e., Figure 1(a2-f2), which are considered the best output for various patterns.  
In order to check the effect of the value of the parameter \(\beta\) on the reconstruction output, we went through the feedback coefficient \(\beta\) in the range of 0.1-1.5 with intervals of 0.1 for various input patterns, to get the best reconstruction outputs, as shown in Table 1, Y denotes reconfigurability, N denotes non-reconfigurability. Y or N is determined by the following criteria with the normalized root-mean-square error (NRMS) [16] expressed in Equation (1), the threshold was set to NRMS = 0.03. We record the NRMS of each reconstructed image with the power spectrum under the same exposure condition at different \(\beta\) values as \(N[\beta_{1},\beta_{2},\cdots\beta_{n}]\), then calculate \(\Delta_{i}=N[\beta_{i+2}]-N[\beta_{i}]\), \(i\in[1,n-2]\), and find out the max value of \(\Delta_{i}\). After this, we use the average of \(N[\beta_{i+2}]\) and \(N[\beta_{i}]\) corresponding to the maximal \(\Delta_{i}\) as the threshold of the successful reconstruction.  
\[NRMS=\begin{bmatrix}\sum\limits_{x\in\mathbb{C}}\left|S_{k}^{\prime}(x)\right| ^{2}\\ \sum\limits_{x}\left|S_{k}^{\prime}(x)\right|^{2}\end{bmatrix}^{1/2} \tag{1}\]  
The power-spectrum pattern captured at different exposure levels could reconstruct objects with suitable feedback coefficient \(\beta\) values. The suitable \(\beta\) values for different exposure level patterns are distinguished from each other. It should be pointed out that, when the input power-spectrum pattern size is 480 \(\times\) 480, each reconstruction in Figure 1(a1-f1) took 13 min with a normal laptop (ASUS N75S, Intel (R) Core i7-2670 QMCPU, 16G RAM) by going through the \(\beta\) value range of 0.1-1.5. Most of the time was spent was going through the wide \(\beta\) value range to find out the suitable value, which is much longer for larger size input power-spectrum pattern, hence, it hinders the real-time imaging applications. However, if the suitable \(\beta\) value can be pre-estimated properly, the reconstruction time is much shorter, which is only about 8 s for each successful reconstruction in Table 1. Therefore, pre-estimating the suitable \(\beta\) value for object image reconstruction is significant for the single-shot coherent power-spectrum scattering imaging in practical applications.