## 3 Proposed Method  
According to the definition shown in Equation (1), the fractional orders \(\alpha_{x}\) and \(\alpha_{y}\) in the \(x\) and \(y\) directions, respectively, can be different. For simplicity, we assume that the fractional orders of Gaussian fringe patterns are identical in the proposed method. That is, \(\alpha_{x}=\alpha_{y}=\alpha_{\text{best}}\). Figure 2 shows the systematic block diagram of the proposed method. The detailed procedures are listed as follows:  
1. First, given an image of size \(N\) by \(N\), which consists of mixed multiple 2-D fringe patterns, we apply the 2-D FrFTs to the whole image with the fractional orders of the range within [0.1] with the increment step 0.01 in either the \(x\) or \(y\) direction. When we perform the FrFTs in the \(x\)-direction, the order in the \(y\)-direction is fixed as 1;
2. The orders with the maximum projection in the \(x\)-direction of the signal are calculated and recorded;
3. Then, we perform the FrFTs in the \(y\)-direction and, similarly, the order in the \(x\)-direction is fixed as 1;
4. The orders with the maximum projection in the \(y\)-direction of the signal are calculated and recorded as well;
5. We calculate the orders of maximum projection, \(\alpha_{x}\) and \(\alpha_{y}\), from the local maximum values of fractional Fourier spectra through a differential operation. Then, the set of best orders, \(\alpha_{\text{best}}\), and the number of the paired orders in the set, \(M\), can be determined by finding the intersections of the maximum projection orders \(\alpha_{x,\text{p}}\) and \(\alpha_{y,\text{p}}\). Equations (8) and (9) show the definitions of the maximal projection orders for the \(x\) and \(y\) directions, respectively. \[\alpha_{x,\text{p}}=\text{argmax}_{\alpha_{x}}\left\{\sum_{y=0}^{N-1}F^{( \alpha_{x},1)}[f(x,y)]\right\}\] (8) \[\alpha_{y,\text{p}}=\text{argmax}_{\alpha_{y}}\left\{\sum_{x=0}^{N-1}F^{( 1,\alpha_{y})}[f(x,y)]\right\}\] (9)  
Here, we have assumed that the fractional orders of \(\alpha_{x}\) and \(\alpha_{y}\) should be identical in the FrFT. Therefore, the maximum width of for each fringe pattern, \(a\), can be determined based on the best order \(\alpha_{\text{best}}\) by using Equation (6). By enumerating their absolute values and normalizing the transformed results \(F_{r}(u,v)\), we deal with it by using a filter mask and searching the peak-value position in the filtered results. The filter mask is of size \(3\times 3\) and its weighting coefficients are \([-1\)\(-1\); \(-1\)\(8\)\(-1\); \(-1\)\(-1\)\(-1\)\(-1\)]. The filtered image result is denoted as \(G(u,v)\) and its maximum \(g_{\text{max}}\) is then determined. Through setting up a threshold value \(T\) between 0 and 1, we can get a binary map \(g_{\text{bw}}\) from the filtered transformed result by using the criterion shown in Equation (10)  
\[g_{\text{bw}}=\left\{ \begin{array}{ll}0,\text{ if }G(u,v)<g_{\text{max}}\times T \\ 1,\text{ if}G(u,v)\geq g_{\text{max}}\times T\end{array} \right.. \tag{10}\]  
In this binary map, we consider the neighboring area near the pixel value 1 as the same fringe pattern by searching the 8-connected positions. 7. The original signal is FrFTed with the best order, \(\alpha_{\text{best}}\), to calculate the center position of a specific circular fringe pattern. If the positions \((u_{m},v_{m})\) of the \(k_{f}\) maximal values in the binary map derived from the filtered fractional Fourier spectra \(G(u,v)\) are detected, it means that we can separate \(k_{f}\) fringe patterns from the original mixture signal;
8. The center points \((x_{0},y_{0})\) in each corresponding circular fringe pattern can be determined according to Equation (11) \[u_{m}=x_{0}\cos\frac{\alpha_{\text{best}}\pi}{2},v_{m}=y_{0}\cos\frac{\alpha_{ \text{best}}\pi}{2}.\] (11)
9. After calculating the center points and maximum widths of the \(k_{f}\) fringe patterns, we can reconstruct \(k_{f}\) fringe patterns separately.