## 4 Experimental Results  
The experiments are performed with MATLAB R2013b and a PC with Intel Core 2 Quad Q8200 CPU, DDR3 2G RAM, and Windows 7 OS. For simplicity, both the background intensity \(I_{0}\) and the modulation amplitude \(I_{1}\) in Equation (3) are set as unity. As shown in Figure 2, only two cases of the constant phase shifts, \(\mathbf{\varphi}_{0}=0^{\circ}\) and \(\mathbf{\varphi}_{0}=180^{\circ}\) are considered. Figure 3 shows the two mixed fringe patterns used in our experiments, which comprises six and eight 2-D fringe patterns, respectively. The image is of size 512 \(\times\) 512. According to the block diagram shown in Figure 2, we apply the FrFTs on these pattern images with the fractional orders \(\alpha=[0.01,1]\) with an increment step 0.01 at the \(x\)- and \(y\)-axis directions. The orders with the maximum projections in the \(x\)- and \(y\)-direction of the signal are recorded as \(\mathbf{\alpha}_{x,\mathbf{p}}\) and \(\mathbf{\alpha}_{y,\mathbf{p}}\), respectively. Consider the mixed fringe pattern shown in Figure 3a. Figure 4a-f shows part of the projection results with the various fractional orders. Figure 4a-c shows the projection results in the \(x\)-direction when the order is 1 in the \(y\)-direction, while Figure 4d-f shows the results in the \(y\)-direction when the order is 1 in the \(x\)-direction. By applying the derivative operations on the projection values shown in Figure 5, the fractional orders of the local maxima in the \(x\) and \(y\) directions are \(\alpha_{x}=\{0.14,\,0.16,\,0.18,\,0.20,\,0.22,\,0.25,\,0.37,\,0.44,\,0.51,\,0.53,\, 0.73,\,0.76,\,0.79,\,0.81\}\) and \(\alpha_{y}=\{0.73,\,0.76,\,0.79,\,0.82\}\), respectively. We have assumed that the fractional orders \(\alpha_{x}\) and \(\alpha_{y}\) are identical. Consider that there could be small errors in the determined maximal orders in both the \(x\) and \(y\) directions. We determine the intersections between the two sets of the fractional orders \(\{\alpha_{x-0.01},\,\alpha_{x},\,\alpha_{x+0.01}\}\) and \(\alpha_{y}\). Finally, we obtain  
Figure 2: The systematic block diagram of the proposed method. the results of best orders \(\alpha_{\text{best}}=\{0.73,0.76,0.79,0.82\}\), which are then used to determine the corresponding fringe widths \(a=\{45.39,39.80,34.42,29.21\}\), respectively. To find out how many possible fringe patterns are contained in the mixed one, we determine the fractional Fourier spectra (i.e., intensities of the FrFT) of the signals with the  
Figure 4: The fractional Fourier spectra of the mixed signals in Figure 3a with the various fractional orders: (**a**) \(\alpha\)= (0.73,1); (**b**) \(\alpha\)= (0.76,1); (**c**) \(\alpha\)= (0.79,1); (**d**) \(\alpha\)= (0.82,1); (**e**) \(\alpha\)= (1,0.73); (**f**) \(\alpha\)= (1,0.76); (**g**) \(\alpha\)= (1,0.79); (**h**) \(\alpha\)= (1,0.82). Figure 5: The maximal projection values of the 2-D FrFTs along: (**a**) the \(x\)-direction and (**b**) the \(y\)-direction, while the fractional order at the other direction is fixed as one. The intersection orders of maxima between (**a**,**b**) can be determined. Figure 3: Two artificially generated 2-D mixed multiple fringe patterns composed of (**a**) six and (**b**) eight circular fringe patterns. determined best orders and corresponding fringe widths. By applying a peak detection filter, we find six peaks \(g_{\text{max}}\) and the corresponding positions of fringe patterns, \(g_{\text{bw}}\) in the four spectra. The six separated fringe patters can thus be reconstructed. Figure 6a,c,e,g show the normalized fractional Fourier spectra \(G(u,v)\) for the determined best fractional orders \(\alpha_{\text{best}}=\{0.73,\,0.76,\,0.79,\,0.82\}\) and the filtered results are shown in Figure 6b,d,f,h. There are six peak values observed in the filter results, which correspond to six single fringe patterns in the mixed one. By using Equations (9) and (10), we can determine the maxima in the filter results and then the center positions.