## 2 Circular Fringe Pattern and Their Applications  
The general formula of a 2-D fringe pattern with a constant phase shift \(\varphi_{0}\) is shown as follows  
\[f(x,y)=\frac{1}{2}\{I_{0}+I_{1}\cos[\varphi(x,y)+\varphi_{0}]\}\times g(x,y), \tag{3}\]  
where \(I_{0}\) denotes the background intensity, \(I_{1}\) is the modulation amplitude, \(g(x,y)\) is a 2-D Gaussian distributed function and \(\varphi(x,y)\) denotes the phase function that can be used to generate a fringe pattern whose center is located at the coordinates \((x_{0},y_{0})\). Equations (4) and (5) show the definitions of \(\varphi(x,y)\) and \(g(x,y)\), respectively. \[\varphi(x,y)=a\big{[}(x-x_{0})^{2}+(y-y_{0})^{2}\big{]} \tag{4}\]\[g(x,y)=\frac{1}{\sigma\sqrt{2\pi}}\exp[-\frac{\left(x-x_{0}\right)^{2}+\left(y-y_{0} \right)^{2}}{2\sigma^{2}}] \tag{5}\]  
Here, \(a\) is the maximum width of the fringe pattern and \(\sigma\) is the standard deviation of the 2-D Gaussian function. Figure 1a,b show the two 2-D circular fringe patterns with \(180^{\circ}\) phase difference in between, i.e., \(\varphi_{0}=\pi\). The optimal fractional order \(\alpha_{\text{opt}}\) for a given Gaussian fringe pattern of size \(N\) by \(N\) can be determined as [10]  
\[\alpha_{\text{opt}}=-\frac{2}{\pi}\tan^{-1}\left(\frac{1}{2a}\right), \tag{6}\]  
for continuous-time cases and  
\[\alpha_{\text{opt}}=-\frac{2}{\pi}\tan^{-1}\left(\frac{f_{s}^{2}/N}{2a} \right), \tag{7}\]  
for discrete-time cases, where \(f_{s}\) is the sampling frequency in the 2-D spatial domain. Figure 1a demonstrates the case of \(\varphi_{0}=0\) in Equation (3), showing that the center of the fringe pattern is bright. On the other hand, Figure 1b demonstrates the case of \(\varphi_{0}=\pi\) in Equation (3), the center of the fringe pattern is dark and located at a different position. Figure 1c shows the mixed fringe pattern, which is the mixture of the ones shown in Figure 1a,b. Note that the contrast of the mixed fringe pattern has been decreased. If more circular fringe patterns are mixed, the number of fringes will not be easily recognizable. There are various related works applying circular fringe patterns, especially in applications of the optical metrology. For low-quality fringe pattern enhancement and normalization, [PERSON] et al. proposed an automatic selective reconstruction method to decompose a fringe pattern into a set of empirical modes using enhanced fast empirical mode decomposition [14]. [PERSON] et al. proposed a method to estimate the parameters of optical fringes with a quadratic phase using the FrFT [15]. [PERSON] and [PERSON] evaluated the basic assumptions of the standard methods utilized in fringe pattern analysis and presented the mathematical models to determine the displacement information encoded in fringe patterns [16]. [PERSON] et al. proposed a method that combines the advantages of the FrFT with the least-squared fitting method in analyzing the fringe patterns [17]. Better parameter estimation can be achieved compared with the conventional FrFT method. To retrieve the phase information from a single-shot spatial carrier fringe pattern, [PERSON] and [PERSON] proposed an advanced FT analysis method [18], which can mitigate the problems in the traditional FT analysis method. To measure the whole-field out-of-plane deformation of targets, [PERSON] et al. proposed a new class of circular fringe projection methods that employ circular fringes for three-dimensional shape measurement [19]. Consider the optical elements with full-field or circular pupils, and that the phase presents a quadratic behavior such as astigmatic, elliptical, or circular fringes. [PERSON] presented an FT-based method for phase recovery from a single interferogram with closed fringes [20]. [PERSON] et al. Figure 1: Gaussian-based circular fringe patterns: (**a**) a 2-D fringe pattern; (**b**) the fringe pattern with \(180^{\circ}\) phase difference from (**a**) at a different location; (**c**) the fringe pattern which is the mixture of (**a,b**).