# Fast Image Reconstruction Technique for Parallel Phase-Shifting Digital Holography
## 2 Theory  
First, we explain the parallel phase-shifting technique using the polarization image sensor [17]. Figure 1a shows the arrangement of the micro-polarizers attached to the pixels of the image sensor. The directions of the polarizing axes of the micro-polarizers are 0\({}^{\circ}\), 45\({}^{\circ}\), 90\({}^{\circ}\), and 135\({}^{\circ}\). When the object and reference waves having circular polarizations in opposite directions interfere on the image sensor, the phase shifts between the two waves captured by the polarized pixels are \(\delta\) = 0, \(\pi\)/2, \(\pi\), and 3\(\pi\)/2, as shown in Figure 1b. Figure 2 illustrates the calculation procedure of the conventional parallel phase-shifting technique [15; 16]. Pixels having the same phase shifts are extracted to obtain four sparse images. The extracted sparse images are represented by \(I_{o}(x,y;0)\), \(I_{o}(x,y;\pi/2)\), \(I_{o}(x,y;\pi)\), and \(I_{o}(x,y;3\pi/2)\) corresponding to the phase shifts of \(\delta\) = 0, \(\pi\)/2, \(\pi\), and 3\(\pi\)/2, respectively. The interpolated images corresponding to the phase shifts of \(\delta\) = 0, \(\pi\)/2, \(\pi\), and 3\(\pi\)/2 are denoted by \(I(x,y;0)\), \(I(x,y;\pi/2)\), \(I(x,y;\pi)\), and \(I(x,y;3\pi/2)\), respectively. We performed image interpolations and obtained four pixel values at identical positions for the four images. The phase-shifting technique provides a complex-amplitude distribution of the object wave on the image sensor by using the following equation:  
\[u(x,y)=[I(x,y;0)-I(x,y;\pi)]+i[I(x,y;\pi/2)-I(x,y;3\pi/2)]. \tag{1}\]  
Figure 1: Parallel phase-shifting technique using the polarization image sensor: (**a**) polarizing axes of micro-polarizers attached to pixels and (**b**) phase shifts for pixels. The diffraction calculation is applied to \(u(x,y)\) to obtain the reconstructed image. Several kinds of diffraction calculations can be used, such as the Fresnel and Fraunhofer diffraction equations. In this study, we use the angular spectrum representation for the diffraction calculation [29; 30]. The angular spectrum method provides the reconstruction image \(v(x,y)\) at a distance of \(z_{\tau}\) using the following equation:  
\[v(x,y)=\text{IFT}\bigg{[}U\big{(}f_{x},f_{y}\big{)}\exp\bigg{(}\frac{i2\pi}{ \lambda}z_{\tau}\sqrt{1-\big{(}\lambda f_{x}\big{)}^{2}-\big{(}\lambda f_{y} \big{)}^{2}}\bigg{)}\bigg{]}, \tag{2}\]  
where IFT denotes the inverse Fourier transform and \(U\big{(}f_{x},f_{y}\big{)}\) is the Fourier transform of \(u(x,y)\). The Fourier transform is performed to obtain \(U\big{(}f_{x},f_{y}\big{)}\) followed by the multiplication of the phase term. Finally, the inverse Fourier transform gives the reconstructed image. Next, we explain the calculation technique proposed in this study. When the resolution of the image sensors increases and the fast Fourier transform (FFT) algorithm is used for the diffraction calculation, the calculation time for the Fourier transform increases logarithmically with the resolution. The calculation time for the interpolations increases linearly with the resolution; therefore, this time needs to be reduced. In this study, the interpolation is represented by the convolution of an image and an interpolation kernel. The four interpolated images are obtained for the phase shifts \(\delta=0\), \(\pi/2\), \(\pi\), and \(3\pi/2\) using the following equation:  
\[I(x,y;\delta)=I_{0}(x,y;\delta)*f(x,y), \tag{3}\]  
Figure 2: Conventional calculation procedure for parallel phase-shifting digital holography. where \(*\) denotes the convolution operation, and \(f(x,y)\) represents the interpolation kernel. Various kinds of interpolation algorithms are obtained by changing the interpolation kernel.