## 2 Materials and Methods  
The optical configuration of the proposed DSPI system is illustrated in Figure 1. A coherent laser, with a wavelength of \(\lambda\), is expanded to illuminate the object surface. The object is observed from four directions simultaneously and symmetrically around the z-axis. The four imaging beams are denoted as x\({}_{1}\), x\({}_{2}\), y\({}_{1}\), and y\({}_{2}\), and the corresponding wave vectors are \(\overrightarrow{K_{x}^{1}}\), \(\overrightarrow{K_{x}^{2}}\), \(\overrightarrow{K_{y}^{1}}\), and \(\overrightarrow{K_{y}^{2}}\). \(\overrightarrow{K_{x}^{1}}\) and \(\overrightarrow{K_{x}^{2}}\) are located in the XZ plane, whereas \(\overrightarrow{K_{y}^{1}}\) and \(\overrightarrow{K_{y}^{2}}\) are located in the YZ plane. The angle between each of observation directions and the z-axis is \(\theta\). After being reflected by the single mirror and the four-sided mirror, each image passes through the corresponding aperture stop and polarizer, and is imaged onto the image surface by the same lens. The four aperture stops are spatially offset from each other, and a polarizer is placed in front of each aperture stop. The polarization directions of each beam, passing through four polarizers, are illustrated in Figure 2. After passing through the polarizers, the polarizing directions of imaging beams x\({}_{1}\) and x\({}_{2}\) are the same, and the polarization directions of the imaging beams y\({}_{1}\) and y\({}_{2}\) are perpendicular to those of beams x\({}_{1}\) and x\({}_{2}\). Therefore, imaging beam x\({}_{1}\) interferes with x\({}_{2}\), and imaging beam y\({}_{1}\) interferes with y\({}_{2}\). No cross interference occurs between the two pairs of beams. As shown in Figure 3, \((\xi,0),(-\xi,0),(0,\eta),(0,-\eta)\) are the center coordinates of the four apertures of imaging beams x\({}_{1}\), x\({}_{2}\), y\({}_{1}\), and y\({}_{2}\), respectively. The phases of the waves, which pass through the apertures, have a spherical part plus a speckled part [28]. The four beams that pass through each aperture and reach the image plane have amplitudes given by Equation (1) [29; 30; 31]:  
\[ \begin{array}{l}u_{x1}(x,y)=\big{|}u_{x1}(x,y)\big{|}\exp\big{[}i\phi_{x1}(x,y)+\frac{2\pi i}{\lambda D}(\xi x)\big{]}\\ u_{x2}(x,y)=\big{|}u_{x2}(x,y)\big{|}\exp\big{[}i\phi_{x2}(x,y)+\frac{2\pi i}{ \lambda D}(-\xi x)\big{]}\\ u_{y1}(x,y)=\big{|}u_{y1}(x,y)\big{|}\exp\big{[}i\phi_{y1}(x,y)+\frac{2\pi i}{ \lambda D}(\eta y)\big{]}\\ u_{y2}(x,y)=\big{|}u_{y2}(x,y)\big{|}\exp\big{[}i\phi_{y2}(x,y)+\frac{2\pi i}{ \lambda D}(-\eta y)\big{]}\\ \end{array} \, \tag{1}\]  
Figure 1: Schematic diagram of digital speckle pattern interferometry (DSPI) to simultaneously measure two in-plane displacements. where D is the distance from the image plane to exit pupil plane [32]. \(\frac{2\pi i}{\lambda\Omega}\xi x\) and \(\frac{2\pi i}{\lambda\Omega}\eta y\) are the spatial carrier frequencies introduced by aperture misalignment. The interferogram on the imaging plane can be expressed as follows:  
\[I=I_{x}+I_{y}=(u_{x1}+u_{x2})(u_{x1}+u_{x2})^{*}+\Big{(}u_{y1}+u_{y2}\Big{)} \Big{(}u_{y1}+u_{y2}\Big{)}^{*}. \tag{2}\]  
By applying Fourier transformation to the interferogram, Equation (2) can be transformed into the Equation (3),  
\[ \begin{array}{rl}FT(I)=&U_{x1}\otimes U_{x1}^{*}+U_{x2}\otimes U_{x2}^{*}+U_ {y1}\otimes U_{y1}^{*}+U_{y2}\otimes U_{y2}^{*}\\ &+U_{x1}\otimes U_{x2}^{*}+U_{x1}^{*}\otimes U_{x2}\\ &+U_{y1}\otimes U_{y2}^{*}+U_{y1}^{*}\otimes U_{y2}\end{array} , \tag{3}\]  
where \(\otimes\) is the convolution operation, \(U_{x1}=FT(u_{x1})\), \(U_{x2}=FT(u_{x2})\), and \(U_{y1}=FT\Big{(}u_{y1}\Big{)}\), \(U_{y2}=FT\Big{(}u_{y2}\Big{)}\). The schematic diagram of the spectrum after Fourier transformation is illustrated in Figure 4. The term \(U_{x1}\otimes U_{x1}^{*}+U_{x2}\otimes U_{x2}^{*}+U_{y1}\otimes U_{y1}^{*}+U_ {y2}\otimes U_{y2}^{*}\) contains background information, which is located at the center of the spectrum.