## 2 Theory and Methods  
The MIPO-CSGI schematic diagram is illustrated in Figure 1, which has a part to achieve differential detector signals, a part to generate a random matrix, and a part to reconstruct the images. In the achieving differential detector signals part, the light field is modulated by a digital micro-mirror device (DMD) to produce pairs of random speckle patterns. Each speckle pattern \(I_{i}^{k}(x,y)\) and its inverse pattern \(\tilde{I}_{i}^{k}(x,y)\), where, \(x=1,2,\cdots,N_{x}\), \(y=1,2,\cdots,N_{y}\), \(i\) is the \(i\)th number of the groups, \(k\) represents the \(k\)th number of the speckle patterns in each group, and the speckle patterns are either white \(1\) or black \(-1\) for each coordinate (x,y). According to the differential acquisition method [21, 31], we set \(I_{i}^{k}(x,y)=-\tilde{I}_{i}^{k}(x,y)\). In traditional CGI, the sampling rate of the bucket detector is high or equal to the frequency of the state switching of the object. When a slow bucket detector is used to obtain \(N\) frames of the periodic moving/state-changed object, we need to produce \(N\) random speckle patterns \(I_{i}^{k}(x,y),k=1,2,\cdots,N\) as one group according to the cycle time \(T\) of the object and use them to illuminate the object \(\{T(x,y,k)\}_{k=1}^{N}\), which is then measured by the slow bucket detector and recorded as \(B_{i}\):  
\[B_{i}=\eta\iiint I_{i}^{k}(x,y)T(x,y,k)dxdydk+n, \tag{1}\]  
where \(\eta\) is the bucket detector's responsivity, and \(n\) is the noise generated by environmental illuminations. Similarly, corresponding to \(\widetilde{I}_{i}^{k}(x,y),k=1,2,\cdots,N\), we can obtain the detection result \(\widetilde{B}_{i}\) expressed as:  
\[\widetilde{B}_{i}=\eta\iiint\widetilde{I}_{i}^{k}(x,y)T(x,y,k)dxdydk+n. \tag{2}\]  
Thus, the differential detector signal \(\triangle B_{i}\) between the two corresponding detection results \(B_{i}\) and \(\widetilde{B}_{i}\) can be expressed as:  
\[ \begin{split}\triangle B_{i}&=B_{i}-\widetilde{B} _{i}\\ &=\eta\iiint(I_{i}^{k}(x,y)-\widetilde{I}_{i}^{k}(x,y))T(x,y,k) dxdydk\\ &=\eta\iiint\triangle I_{i}^{k}(x,y)T(x,y,k)dxdydk,\end{split}  \tag{3}\]  
where \(\triangle I_{i}^{k}(x,y)=I_{i}^{k}(x,y)-\widetilde{I}_{i}^{k}(x,y)\). It can be seen that the environmental noise can be efficiently removed. With \(M\) measurements, we will obtain \(M\) groups of random speckle patterns \(\{I_{i}^{k}(x,y)\}_{i=1}^{M}\)\((k=1,2,\cdots,N)\) and their corresponding bucket detector signals \(\{B_{i}\}_{i=1}^{M}\). Meanwhile, \(M\) groups' inverse patterns \(\{\widetilde{I}_{i}^{k}(x,y)\}_{i=1}^{M}(k=1,2,\cdots,N)\) also correspond to \(M\) bucket  
Figure 1: A schematic diagram of the MIPO-CSGI. detector signals \(\{\widetilde{B}_{i}\}_{i=1}^{M}\) and will be obtained by other \(M\) measurements. Therefore, we will obtain \(M\) groups of differential speckle patterns \(\{\triangle I_{i}^{k}(x,y)\}_{i=1}^{M}(k=1,2,\cdots,N)\) and their corresponding differential bucket detector signals \(\{\triangle B_{i}\}_{i=1}^{M}\). In the generating random matrix part, each speckle pattern \(\triangle I_{i}^{k}(x,y)\) can be reshaped as a row vector \(\mathbf{I}_{i}^{k}\) with size \(1\times K\), where \(K=N_{x}\times N_{y}\):  
\[ \begin{split}\mathbf{I}_{i}^{k}=\big{[}&\triangle I_ {i}^{k}(1,1)\quad\cdots\quad\triangle I_{i}^{k}(1,N_{y})\quad\triangle I_{i}^{k }(2,1)\quad\cdots\\ &\triangle I_{i}^{k}(2,N_{y})\quad\cdots\quad\triangle I_{i}^{k}( N_{x},1)\quad\cdots\triangle I_{i}^{k}(N_{x},N_{y})\big{]}.\end{split}  \tag{4}\]  
Each group with \(N\) speckle patterns \(\triangle I_{i}^{k}(x,y)\) (\(k=1,2,\cdots,N\)) can form a row vector \(\mathbf{A}_{i}\), with size \(1\times P\), where \(P=N\times K\):  
\[