## 3 Numerical Verification  
In this section, numerical simulations are used to validate the effectiveness of MIPO-CSGI. The image resolution of each target is set to 64 \(\times\) 64. The simulations are carried out on Matlab R2018a on the Windows 11 operating system. The hardware system is a laptop with a 3.20 GHz central processor unit (AMD Ryzen 7 5800H) and 16.0 GB of random access memory. Additionally, we use the peak signal-to-noise ratio (\(PSNR\)) as an objective evaluation, as defined by the following definition [29; 33]:  
\[PSNR=10\ log_{10}\left[\frac{255^{2}}{\frac{1}{N_{x}\cdot N_{y}}\sum_{x,y}( \hat{T}(x,y)-T(x,y))^{2}}\right], \tag{9}\]  
where \(T(x,y)\) and \(\hat{T}(x,y)\) are the original and recovered image intensity values, respectively. \(N_{x}\) and \(N_{y}\) represent the object's horizontal and vertical dimensions. In general, the higher the \(PSNR\) value, the higher the quality of the reconstructed image. Meanwhile, we characterize the compressive sampling ratio by the parameter \(\beta=M/P\), which is defined as a ratio of the number of measurements \(M\) to the total pixels \(P\) of the multiple images. We first perform numerical simulations by imaging the periodic moving object to verify the feasibility of MIPO-CSGI. The MIPO-CSGI reconstruction results are compared to those of CSGI using a fast bucket detector (F-CSGI) and CSGI using a slow bucket detector (S-CSGI). In F-CSGI, the sampling rate of the fast bucket detector is high or equal to the frequency of the state switching of the object, and each bucket detector value corresponds to one speckle pattern acting on each state of the object. The sampling rate of the slow bucket detectors in S-CSGI and MIPO-CSGI is slower, so they can only record the total energy of each group of the speckle patterns acting on the object at each cycle time \(T\). Assume the original objects are two two-grayscale objects such as a jumping \"square\", a moving \"\(T\)\", and an eight-grayscale moving \"spacecraft\". Taking the jumping \"square\" as an example, in order to obtain four frames of the object, we use the computer to simulate DMD to produce four 64 \(\times\) 64-resolution random speckle patterns in each cycle. The sampling rate of the bucket detector used in F-CSGI is four times that in MIPO-CSGI and S-CSGI, which also means that to get the same number of the bucket detector signals, the number of speckle patterns and the optical measurements time in MIPO-CSGI and S-CSGI is four times that of F-CSGI. Then, one by one, each speckle pattern is used to illuminate each state of the periodic moving object. When \(\beta\) is set to 60.01%, each scheme will obtain 9,832 differential bucket detector signals. We will obtain a 9832 \(\times\) 16,384-resolution random matrix in MIPO-CSGI. As the reconstruction results are shown in Figure 2, four frames of the \"square\" reconstructed by MIPO-CSGI and F-CSGI both have a high \(PSNR\). Because the fast bucket detector is fast at making enough measurements of each state of the moving object, F-CSGI can reconstruct the images of the object with high quality. Although the slow bucket detector cannot capture each state of the moving object, MIPO-CSGI programmed a random matrix, constructed from four speckle patterns used in each cycle time \(T\). MIPO-CSGI makes the total energy value detected by the slow bucket detector in one cycle time \(T\) equivalent to one measurement of the effect of a large speckle pattern composed of a row of the constructed matrix on a large static image composed of each state of a moving object. With a certain amount of measurements, MIPO-CSGI can also reconstruct a large image with each state of the object of high quality. Although the \(PSNR\) of F-CSGI is almost the same as the MIPO-CSGI, the sampling rate of the bucket detector in F-CSGI is four times that in MIPO-CSGI. For S-CSGI, each detector value of the slow detector of each cycle \(T\) is equivalent to one state of the object detected by the bucket detector, and noise composed of other state detections is added.