## 2 Methodology
### Carrier Mask Modulation and Window Selection Phase Map Retrieval  
The initial light intensity for the original data captured at the camera side can be expressed as:  
\[I_{0}(x,y)=\frac{1}{2}\bigg{\{}I_{1}(x,y)+I_{2}(x,y)+2\sqrt{I_{1}(x,y)I_{2}(x,y )}\cos\big{[}\phi(x,y)+\varphi_{j}(x,y)\big{]}\bigg{\}} \tag{1}\]  
where \(I_{1}\) and \(I_{2}\) are the intensities provided by the two split beams from the beam splitter in the Michelson interferometer, \(\phi\) is the optical random phase difference between the two beams, \(\varphi_{j}\) represents the four phase values 0, \(\frac{\pi}{2}\), \(\pi\), and \(\frac{3\pi}{2}\) shifted by the system's setup and the micro-polarisation image sensor. The above equation cannot be solved for a phase map using a conventional four-step phase-shift calculation, as the calculation will need  
Figure 1: Pixelated spatial phase shift shearography system setup for dynamic WTB inspection. Figure 2: Illustration of the polarisation directions for the micro-polarisation image sensor applied and the corresponding phase shift. to be carried out in the complex domain with a carrier mask modulated on all the pixels. The carrier modulation on each subordinating pixel is \(e^{-i\varphi_{j}}\). The modulated intensity with the carrier mask in a single pixel could be expressed as in the following equation, which shows the phase shift angles in each of the four small pixels:  
\[I_{m}=\left\{ \begin{array}{ll}(I_{1}+I_{2})+\sqrt{I_{1}I_{2}}\cos(\phi),&(x,y )=(2m+1,\;2n+1)\\ -i[(I_{1}+I_{2})+\sqrt{I_{1}I_{2}}\cos(\phi+\frac{\pi}{2})],&(x,y)=(2m+2,\;2n+1 )\\ [(I_{1}+I_{2})+\sqrt{I_{1}I_{2}}\cos(\phi+\pi)],&(x,y)=(2m+2,\;2n+2)\\ i[(I_{1}+I_{2})+\sqrt{I_{1}I_{2}}\cos(\phi+\frac{3\pi}{2})],&(x,y)=(2m+1,\;2n+2 )\end{array} \right. \tag{2}\]  
where \(m=0,\;1,\;\dots,\;1023\) and \(n=0,\;1,\;\dots,\;1223\) according to the image sensor's coordinate arrangement. Equation (2) can also be expressed in exponential form, for showing the different frequency areas in the frequency domain, as:  
\[I_{m}=I_{0}\cdot e^{-i\varphi_{j}}=\left[\frac{1}{2}(I_{1}+I_{2})+\sqrt{I_{1}I _{2}}\frac{e^{i(\phi+\varphi_{j})}+e^{-i(\phi+\varphi_{j})}}{2}\right]e^{-i \varphi_{j}}=\frac{1}{2}\Big{[}(I_{1}+I_{2})e^{-i\varphi_{j}}+\sqrt{I_{1}I_{2} }e^{i\phi}+\sqrt{I_{1}I_{2}}e^{-i\phi-2i\varphi_{j}}\Big{]} \tag{3}\]  
The second term in the above equation is the lower frequency that could be selected by changing the window size in the frequency domain, while other terms are at the higher frequency that could be separated at the same time, thus the selected term (denoted by \(\mathcal{R}\)) by the window in the frequency domain can be expressed as:  
\[\mathcal{R}=\sqrt{I_{1}I_{2}}e^{i\phi} \tag{4}\]  
To select the lower frequency, \(\mathcal{R}\), the necessary step of 2D Fourier transform (2D-FT), and a window of selecting the designated frequency area in the 2D frequency domain need to be generated. The 2D-FT of the modulated intensity distribution can be expressed as:  
\[F(u,\;v)=\iint_{-\infty}^{+\infty}I_{m}(x,y)e^{-2\pi i(ux+vy)}dxdy \tag{5}\]  
where \(u\) and \(v\) are complex indices in the 2D frequency domain equivalent to \(x\) and \(y\) in the 2D spatial domain. The window for selecting the appropriate lower frequency area can be expressed as \(g(u,v)\). The window function could be used as a Gaussian centre or an ordinary rectangular window, the length and width of which could be changed according to the practical circumstances. In the case here, the rectangular window is used for simplicity of lower frequency selection. This function allows the lower frequency to pass while blocking the higher frequency below the cutoff rectangular edge, and can be expressed as:  
\[g(u,v)=\left\{ \begin{array}{cc}1,\;a\leq A,b\;\leq B\\ 0,\;otherwise\end{array} \right. \tag{6}\]  
where \(a\) and \(b\) represent the window size, i.e., length and width of the filtering window, and \(A\) and \(B\) are the cutoff frequencies along \(u\) axis and \(v\) axis to be filtered in this process.