## 3 Results  
In order to test the three algorithms (FSD, FSD-HSO and FSD-PSO), synthetic interferograms were implemented, which contained Gaussian functions in strategic positions of the interferogram. The normalized interferogram \(I_{n}(x,y)\) now reads  
\[I_{n}(x,y)=\cos[\rho_{1}\varphi(x,y)], \tag{9}\]  
where \(\rho_{1}\) controls the general amplitude of the entire phase, and \(\varphi(x,y)\) is the sum of several Gaussians \(g_{xx}\),  
\[\varphi(x,y)=A\!:\!g_{mw}+B\!:\!g_{ne}+C\!:\!g_{sw}+D\!:\!g_{se}+E\!:\!g_{c}. \tag{10}\]  
The Gaussians \(g_{xx}\) programmed in each synthetic interferogram were placed at the northeast (\(g_{ne}\)), northwest (\(g_{mw}\)), southeast (\(g_{se}\)), southwest (\(g_{sw}\)) and central (\(g_{c}\)) positions. Parameters \(A\) to \(D\) are used to activate each of the Gaussians individually. Internally, the Gaussians can be written relative to their position \(xx\) or \(yy=uw\), \(ne\), \(sw\), \(se\) or \(c\), respectively; that is  
\[g_{xx}=\exp\left(-\rho_{2}\Big{[}F_{xx}(x-N_{xx})^{2}+F_{yy}\big{(}y-N_{yy} \big{)}^{2}\Big{]}\right), \tag{11}\]  
where the parameters \(F_{xx}\) and \(F_{yy}\) individually controls the width of the Gaussian at \(x\) and \(y\) positions, respectively, and the parameter \(\rho_{2}\) determines the width of each Gaussian gen erally in \((x,y)\) positions. To control the positions of each of the Gaussians, the parameters \(N_{xx}\) and \(N_{yy}\) are used. The synthetic wrapped phase is calculated as  
\[\varphi_{w}(x,y)=\arctan\biggl{[}\frac{\sin(\rho_{1}\varphi(x,y))}{\cos(\rho_{1 }\varphi(x,y))}\biggr{]}. \tag{12}\]  
In Figure 2, the results with different types of synthetic interferograms are shown: in rows (a-b), five positive Gaussian examples are presented; in row (c), an example with four positive Gaussians and a negative one is depicted; and in the final row (d), two Gaussian interferograms are seen. Figure 3 shows the 3D phases representations of the interferograms shown in Figure 2, in order to observe the differences between the original phase \(\varphi(x,y)\) with the unwrapped phases calculated \(\phi_{u}(x,y)\) with the FSD and FSD-HSO algorithms, as well as the proposed algorithm FSD-PSO. The fixed parameters in all synthetic interferograms were: \(N=256\), \(\rho_{1}=-30\), \(\rho_{2}=1/1800\), as well as the size of the interferograms was \(N\times N\). To calculate the original phase \(\varphi(x,y)\), Gaussians presented in Figures 2 and 3 rows (a)-(c), the following parameters were used: \(a(x,y)=0\), \(b(x,y)=1\), \(A=1\), \(B=1\), \(C=1\), \(D=1\), \(F_{xx}=1\) and \(F_{yy}=1\) are fixed in \(g_{ne}\), \(g_{nn}\), \(g_{se}\) and \(g_{sw}\), but \(g_{c}\) acts as adjustable Gaussian. In these cases, the \(g_{c}\) parameters change as follows: in row (a) \(E=0.5\), \(F_{xx}=2\) and \(F_{yy}=1\); in row (b) \(E=0.5\), \(F_{xx}=3\) and \(F_{yy}=3\); and in row (c) \(E=-1\), \(F_{xx}=5\) and \(F_{yy}=5\). In row (d), only two Gaussian functions are shown, with \(A=0.8\), \(B=0\), \(C=0\), \(D=0.8\) and \(E=0\), as well as their respective thickness parameters are \(F_{xx}=1\) and \(F_{yy}=1\). A set of interferograms with a single Gaussian function with varying width and amplitude were calculated in order to test the response of the considered methods with respect to the fringe frequency. In Figure 4 it is shown the two cases of the gaussian functions, \(F_{xx}=F_{yy}=1\) and \(10\). A 3D view of the recovered phases is shown in Figure 5, where in row (a) all the methods had success, in row (b) details of the behavior of the different methods in high-frequency interferograms are shown, where it can be observed the Gaussian peaks distortion. In addition to \(F_{xx}\) and \(F_{yy}\), the remaining parameters used to obtain the interferograms and phases of Figures 4 and 5 were: \(A=0\), \(B=0\), \(C=0\), \(D=0\) and \(E=1\). The interpretation of the results is given in the next section. Figure 2: Demodulation results using FSD, FSD-HSO and FSD-PSO.