# Pixel Resolution Imaging in Parallel Phase-Shifting Digital Holography
## 1 Introduction  
Digital holography (DH) enables two-dimensional quantitative complex amplitude imaging or simultaneous amplitude and phase imaging, which is a very powerful method for metrology, analysis, and detection, with broad applications in the areas of biomedicine, instrument manufacturing, aquatic science, extraterrestrial life detection, 3D imaging and displays, security and defense, etc. [1; 2; 3; 4]. Zero-order and twin-image noise are serious obstacles to achieving high-quality imaging in in-line DH, and the phase-shifting (PS) technique is often required for noise suppression. Among the existing noise removal techniques, the traditional temporal PS technique can theoretically provide the best quality of reconstructed images [5; 6]. However, due to the nonlinear response and response time of practical PS devices such as piezoelectric transducers, temporal phase-shifting digital holography (PSDH) has the following drawbacks: (1) it is extremely sensitive to environmental vibration and turbulence; (2) the relative phase shifts between interferograms cannot be precisely controlled [7; 8]; and (3) real-time measurement of fast-moving targets is not possible [9; 10]. To exceed the limitations of PSDH, [PERSON] et al. proposed parallel phase-shifting digital holography (PPSDH) in 2004, using an optical-path-length-shifting array device with a pixelated periodic thickness distribution, or a micropolarizer array device with a pixelated periodic polarization direction distribution, to capture multiple digital holograms with different phase shifts in a single exposure [9; 11; 12; 13; 14; 15]. The spatial PS technique based on the pixelated polarization mask has the advantages of being essentially simple to align and compose, achromatic over a wide range, etc. [13; 16], but also has the drawback that there was no available commercial cost-effective polarization camera until the launch ofthe polarization-dedicated image sensor IMX250 MZR by Sony for snapshot polarization imaging in 2018 [17; 18; 19]. The IMX250 MZR sensor, with an image resolution of 2448 \(\times\) 2048 and a pixel pitch of 3.45 \(\upmu\)m, not only promotes the development of snapshot polarization imaging, but also solves the core device problem of PPSDH, and enables PPSDH to be more widely studied [20; 21; 22; 23; 24; 25; 26; 27]. Similar to polarization imaging, \"hologram demosaicing\" is also a crucial imaging pipeline of PPSDH [28; 29; 30]. There are currently two classical demosaicing methods in PPSDH: the first is the super-pixel method (SPM), which discards the blank pixels in the sparse sub-phase-shift holograms [16; 31; 32]; and the second is the hologram interpolation method (HIM), which uses adjacent pixel values instead of blank pixels [9]. Among them, the SPM, which regards a set of four neighboring pixels corresponding to the four phase shifts as a \"unit cell\", is simple to implement, but can only calculate the complex amplitude value of the optical field at one point, resulting in the maximum lateral spatial resolution and image resolution of the reconstructed image of PPSDH both being reduced to one-half of that of PSDH in the horizontal and vertical directions, respectively. Although the HIM is more complex than the SPM, it can avoid the loss of image resolution [23]. [PERSON], [PERSON] et al. compared the root-mean-square error of the reconstructed image by the HIM based on bilinear, bicubic, and B-spline through simulation in 2013 [30], and after that there has been no specific analysis of the differences between the SPM and the HIM in PPSDH. Lateral spatial resolution is one of the main parameters of PPSDH. To solve the problem of limited lateral resolution caused by the decrease in the sampling frequency of the traditional SPM in practical applications, we propose an oversampled super-pixel method (OSPM), referring to the sensor-shifting implementation of pixel super-resolution (PSR) for on-chip lensless digital holographic microscopy (DHM) [33; 34].