# Pixel Resolution Imaging in Parallel Phase-Shifting Digital Holography
## 2 Principles and Methods
### Hologram Reconstruction  
Figure 2 illustrates the SPM calculation procedure of image reconstruction for PPSDH [16, 31, 32]. Pixels with the same phase shifts are extracted to obtain four sparse sub-phase-shift holograms: \(I_{0}(x,y;0)\), \(I_{0}(x,y;\pi/2)\), \(I_{0}(x,y;\pi)\), and \(I_{0}(x,y;3\pi/2)\). The SPM obtains four aligned phase-shift holograms--\(I(x,y;0)\), \(I(x,y;\pi/2)\), \(I(x,y;\pi)\), and \(I(x,y;3\pi/2)\)--by downsampling the sparse sub-holograms and discarding the blank pixels. The PS technique provides complex amplitude distribution of the object diffracted beam \(o(x,y)\) at the plane of the image sensor, and the diffraction calculation of \(o(x,y)\) is performed by the angular spectrum method (ASM) to obtain the reconstructed image of the object beam \(u(x,y)\), using the following equations [26, 37]:  
\[o(x,y)=[I(x,y;0)-I(x,y;\pi)]+j[I(x,y;\pi/2)-I(x,y;3\pi/2)] \tag{1}\]  
\[u(x,y)=\mathcal{F}^{-1}\bigg{\{}\mathcal{F}[o(x,y)]exp\bigg{[}-jkz\sqrt{1- \left(\lambda f_{x}\right)^{2}-\left(\lambda f_{y}\right)^{2}}\bigg{]}\bigg{\}} \tag{2}\]  
where \(\mathcal{F}\) and \(\mathcal{F}^{-1}\) denote the Fourier transform and inverse Fourier transform, respectively; k = \(2\pi/\lambda\) is the wave number; \(\lambda\) is the wavelength of the illumination light source; z is the numerical diffraction propagation distance, which is equal to the recorded distance from the object to the sensor; and \(\left(f_{x},f_{y}\right)\) represents the two-dimensional spatial frequency components. The object diffracted field \(o(x,y)\) calculated by the SPM can be equivalently regarded as the result of direct sampling of \(o(x,y)\) by an imaginary light-field image sensor with  
Figure 1: Experimental geometry of PPSDH. He–Ne laser: helium–neon laser source; P: polarizer; SF: spatial filter assembly: MO: microscope objective, Pi: pinhole, CL: collimating lens; M: mirror; HWP: half-wave plate; PBS: polarizing beam splitter; BS: non-polarizing beam splitter; QWP: quarter-wave plate; PPC: pixelated polarization camera. Figure 2: Conventional super-pixel method (SPM) calculation procedure of image reconstruction for PPSDH. SMH: spatially multiplexed hologram; PS: phase-shifting technique; ASM: angular spectrum method. a pixel pitch of \(2\Delta\) and an image resolution of \(N_{x}\times N_{y}/4\), where \(N_{x},N_{y}\) and \(\Delta\) represent the image resolution and the pixel pitch of the actual sensor, respectively. Considering the horizontal direction, a frequency sampling bandwidth of \(B_{SPx}=1/2\Delta\) corresponding to the \(2\Delta\) super-pixel (SP) sampling interval of the light-field sensor filters out high-frequency information beyond \(B_{SPx}\) in the diffraction field, with a bandwidth of \(B_{x}=N_{x}\Delta/\lambda z\): (1) when \(B_{SPx}>B_{x}\), the imaging lateral resolution mainly depends on \(B_{x}\); (2) when \(B_{SPx}<B_{x}\), the lateral resolution mainly depends on \(B_{SPx}\), and is not more than twice the sampling interval of the light-field sensor, which equals \(4\Delta\) in the SPM. Therefore, the lateral resolution can be improved by reducing the sampling interval or increasing the sampling bandwidth of the imaginary light-field sensor. To solve the above problem of the SPM, we propose an OSPM for PPSDH, referring to PSR for on-chip DHM [33,34]; Figure 3 depicts the proposed technique. As shown in Figure 3a, the OSPM calculates four object light diffraction field images by overlapping each independent SP, which is equivalent to four images captured after moving the imaginary light-field sensor in the horizontal and vertical directions by half an SP offset. After interpolating and merging four low-resolution images, a light-field image with an SP interval of \(\Delta\) and an image resolution of \(N_{x}\times N_{y}\) can be obtained.