# Fast Image Reconstruction Technique for Parallel Phase-Shifting Digital Holography
## 1 Introduction  
Digital holography can measure the two-dimensional amplitude and phase distributions of objects without performing spatial scanning. Thus, three-dimensional (3D) structures of objects can be obtained by diffraction analysis, and label-free imaging can be achieved by using phase information. Traditional digital holography uses an interferometer and coherent light [1]; however, the recently developed incoherent digital holography uses a self-interferometer in which incoherent light can be used [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12]. The use of incoherent light helps in a wide range of applications, such as fluorescence microscopy [2, 3, 4, 5, 6, 7, 8], the 3D measurements of objects under natural light or LED illumination [9, 10, 11], and the measurement of radiometric temperature [12].  
The phase-shifting technique has been used for digital holography to obtain the complex amplitude of a wavefront diffracted from objects by eliminating the zero-order diffraction and conjugate image components [13, 14]. Most incoherent digital holography methods use the phase-shifting technique. The phase-shifting technique requires the capture of three or four holograms with different phase shifts. To avoid multiple image exposures, the parallel phase-shifting technique [15, 16] was proposed, which obtained the complex-amplitude distribution of an object wave by a single exposure. In this technique, multiple holograms with different phase shifts were recorded by space multiplexing. The parallel phase-shifting technique can be effectively implemented using polarization image sensors [17]. The recently commercialized polarization image sensors have a high resolution (>1000 \(\times\) 1000) and operate at a video rate or a high frame rate (\(\geq\)60 Hz) [18]. The use of the parallel phase-shifting technique enables real-time digital holography [19, 20, 21]. The single-shot recording enables the measurement of short-time phenomena [22, 23, 24] and rapidly moving microorganisms [25, 26]. In addition, the single-shot multi-wavelengthincoherent digital holography was demonstrated by using the color polarization image sensor [27] or the polarization-sensitive phase-shifted array [28].  
The polarization image sensors consist of four groups of pixels corresponding to the four phase shifts. Pixels with the same phase shifts were extracted to synthesize four images required for the phase-shifting technique. The interpolation or demosaicing technique was used to obtain four pixel values at identical positions for the four images corresponding to the four phase shifts. The parallel phase-shifting technique can reduce the time taken for image capturing; however, the image interpolations require more time. This additional time required for the interpolations will become an obstacle for real-time operations, especially when the resolution of the image sensor increases. In general, interpolation algorithms with higher accuracies require a longer calculation time than those with lower accuracies. Thus, it is equally important to decrease the time required for image interpolations as it is to decrease the time required for calculating the diffraction.  
In this study, we propose a technique to reduce the calculation time required for the image interpolations in the parallel phase-shifting technique. In Section 2, the proposed technique is explained. The experimental verification of the proposed technique is shown in Section 3. The discussion is given in Section 4, which is followed by the conclusions in Section 5.