## 2 Fourier-Transform Method
Thereafter, an inverse Fourier transform is applied to the filtrated image. The value of each pixel of the image obtained by this processing is represented by a complex number, which corresponds to the second term on the right-hand side in Equation (3). By taking the principal value of the logarithm of the complex number, we can obtain the following wrapped phase including the background phase, shown in Figure 1d. \[\mathcal{W}\left\{\delta\mathbf{k}\cdot\mathbf{r}+\phi(\mathbf{r})\right\}=\mathrm{Log} \left\{\exp\left[+\mathrm{j}\left(\delta\mathbf{k}\cdot\mathbf{r}+\phi(\mathbf{r}) \right)\right]\right\}, \tag{5}\]  
where \(\mathcal{W}\left\{\right\}\) denotes the phase wrapping operator which returns the principal value in range \([-\pi,+\pi)\). By subtracting the phase rotation caused by the background fringes, \(\delta\mathbf{k}\cdot\mathbf{r}\), from the wrapped phase, the wrapped phase-shift, \(\phi(\mathbf{r})\), is obtained as shown in Figure 1f. \[\mathcal{W}\left\{\phi(\mathbf{r})\right\}=\mathcal{W}\left\{\mathcal{W}\left\{ \delta\mathbf{k}\cdot\mathbf{r}+\phi(\mathbf{r})\right\}-\delta\mathbf{k}\cdot\mathbf{r}\right\}. \tag{6}\]  
This sequence of the procedures is called the Fourier-transform method. The result of the Fourier-transform method is the wrapped phase and includes the arbitrariness of an integral multiple of \(2\pi\). To resolve the phase discontinuity, a phase unwrapping process should be applied. We employed a phase unwrapping method using localized rotational compensator [14] which is robust to the noisy wrapped phase [22]. As introduced in Section 1, there is a limitation regarding the interval of the background fringe. Since the interval in \(\mathbf{r}\)-space is proportional to the reciprocal of \(|\delta\mathbf{k}|\) in \(\mathbf{k}\)-space, a longer interval corresponds to a shorter \(|\delta\mathbf{k}|\). If a width of \(I_{\phi}(\mathbf{k})\) is larger than the \(|\delta\mathbf{k}|\), since \(I_{\phi}(\mathbf{k}-\delta\mathbf{k})\) spreads to the origin, \(I_{\phi}(\mathbf{k}-\delta\mathbf{k})\) is distorted in the filtering process of the DC peak or the conjugated peak. The carrier peak isolation method [14] can reduce the distortion; however, it cannot perfectly solve the filtering problem. The condition that \(I_{\phi}(\mathbf{k}-\delta\mathbf{k})\) does not spread to the origin equivalent to that \(i(\mathbf{r})\) is monotonic function [21]. Otherwise, \(\phi(\mathbf{r})\) has extrema; that is, the interferogram has closed-fringes, \(I_{\phi}(\mathbf{k}-\delta\mathbf{k})\) spreads to the origin. Figure 2a shows examples of an interferogram including closed-fringe, and its Fourier transform is shown in Figure 2b. We can avoid this problem [21] using a coordinate-transform from the Cartesian coordinate to a polar coordinate where the origin is the extremum. However, it can only be applied in the case of a single extremum. Figure 1: An example of processing of Fourier-transform method: (**a**) interferogram, \(i(\mathbf{r})\). (**b**) \(|I(\mathbf{k})|\): absolute value of Fourier transform of (**a**) where the origin in \(\mathbf{k}\)-space is located at the center of the image and the brightness of the image is shown with logarithm scale. The distribution \(I(\mathbf{k})\) is superposed results of three peaks; one is a DC-peak at the origin, the other two are the spreading distribution around the carrier peak and its complex conjugate. (**c**) \(|I_{\phi}(\mathbf{k}-\delta\mathbf{k})|\): filtered Fourier spectrum around the carrier peak, \(\delta\mathbf{k}\). (**d**) \(\mathcal{W}\left\{\phi(r)+\delta\mathbf{k}\cdot\mathbf{r}\right\}\): wrapped phase with carrier-shift obtained by evaluation of an angle of complex value at each point on inverse Fourier transformed image of (**d**). (**e**) \(\mathcal{W}\left\{\phi(r)\right\}\): wrapped phase-shift after cancellation of carrier frequency. (**f**) \(\phi(\mathbf{r})\): unwrapped phase-shift. Figure 2: Interferogram including closed-fringe and overlapping of Fourier transformed peak.