## 2 Principle of PSIFTIS and the Information Reconstruction Method
### Principle of Polarization Pattern Decoupling and Information Reconstruction in PSIFTIS  
Because the data structure obtained by this system is relatively complex and the spatial, spectral, and polarization information are coupled together, in order to obtain the three-dimensional information of the target respectively, it is necessary to decouple and reconstruct the map of the polarization interference image array. The process of obtaining the reconstructed polarization spectrum and the polarization fusion image through the processing of the polarization interference image array data is shown in Figure 3. \begin{table}
\begin{tabular}{c c} \hline \hline
**Design Parameters** & **Value** \\ \hline Wavelength & 3.7–4.8 μm \\ Left/right steps & 8 \\ Low-step height & 10 μm \\ High-step height & 80 μm \\ Width of steps & 4 mm \\ Total length of reflector & 64 mm \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Design parameters of rooftop-shaped multi-stage micro-reflector. Figure 2: PSIFTIS experimental platform construction: 1—carbon–silicon rod light source and target, 2—collimation system, 3—micro-lens array, 4—high-order rooftop-shaped multi-stage micro-mirror, 5—low-order rooftop-shaped multi-stage micro-mirror, 6—beam splitter, 7—post-position primary imaging system, 8—polarizer array, 9—post-position secondary imaging system, 10—detector. The process is divided into two processing routes: the first route includes image segmentation, optical path difference matching, and image registration of a polarization interference image array to obtain the polarization interference intensity cube datasets \(I_{0}[x,y,\Delta(m,n)]\), \(I_{45}[x,y,\Delta(m,n)]\), \(I_{90}[x,y,\Delta(m,n)]\), and \(I_{135}[x,y,\Delta(m,n)]\), in polarization directions of \(0^{\circ}\), \(45^{\circ}\), \(90^{\circ}\), and \(135^{\circ}\), where (\(x\),\(y\)) represents the spatial coordinate axis, \(\Delta\) represents the optical path difference axis, \(m\) and \(n\) are the step orders of the high- and low-step micro-mirrors, (\(m\),\(n\)) is the position coordinates of the corresponding phase modulation unit, and \(\Delta(m,n)\) represents the modulated optical path difference generated by the phase modulation unit [18; 19]. The interference intensity corresponding to the polarization direction is:  
\[I_{\theta}[x,y,\Delta(m,n)]=\int_{\
u_{\min}}^{\
u_{\max}}B_{\theta}(x,y,\
u) \exp[j2\pi\
u\Delta(m,n)]dv \tag{1}\]  
Here, \(\
u\) is the wave number, \(\
u_{\min}\) and \(\
u_{\max}\) represent the minimum and maximum wave number respectively, \(B_{\theta}(x,y,\
u)\) is the spectrum of each polarization direction, and the expression of the optical path difference is:  
\[\Delta(m,n)=2(Qn-m)d \tag{2}\]  
Here, \(Q\) is the series of rooftop-shaped stepped micro-mirrors, and \(d\) is the height of the lower step of the rooftop-shaped stepped micro-mirrors. With the array center as the zero point, the optical path difference increases in the horizontal and vertical directions. After addressing baseline correction and apodization, the polarization interference intensity data cube set \(I_{\theta}[x,y,\Delta(m,n)]\) is subjected to two-dimensional discrete Fourier transform to obtain  
Figure 3: Decoupled polarographic reconstruction process of PSIFITS. the three-dimensional spectral data cube set \(B_{\theta}(x,y,\
u)\) of the target in each polarization direction, and the result is as follows:  
\[B_{\theta}(x,y,\
u)=2d{\sum_{m}}\sum_{n}I_{\theta}(x,y,\Delta(m,n))\exp[-i4\pi \
u(Qn-m)d] \tag{3}\]  
Each interferogram unit corresponds to a specific interference order under a specific polarization state, and polarization channel and interference channel are matched. According to the Stokes vector expression, the spectral information corresponding to each of the Stokes vectors \(S_{0}\), \(S_{1}\), \(S_{2}\), and \(S_{3}\) can be obtained as shown in Equation (4).