# Pixel Resolution Imaging in Parallel Phase-Shifting Digital Holography
## 3 Results and Discussion  
The proposed OSPM was verified experimentally based on the PPSDH system shown in Figure 1. A negative test target USAF1951 (Thorlabs R1 DS1N) was used as a recording object. The central 2048 \(\times\) 2048 pixels of the FLIR polarization camera based on the IMX250 MZR sensor were used for hologram recording. Figure 4a-e show the reconstructed enlarged amplitude images obtained using the conventional SPM, and Figure 4f-j show those obtained using the proposed OSPM at different recording distances z. Among them, the reconstructed image resolutions by the SPM and OSPM are 1024 \(\times\) 1024 and 2048 \(\times\) 2048, respectively, while the lateral spatial resolution of the reconstructed images increases with decreasing z. The critical distances z beyond where the lateral resolution of the SPM and OSPM is limited by the sampling interval are 77 mm and 38.5 mm in the case \begin{table}
\begin{tabular}{c c c} \hline \hline
**Method** & **Kernel Function**\(f(x,y)=hh^{T}\) & **Size** \\ \hline
0 elements & \(h=(\,1\,)^{T}\) & \(1\times 1\) \\ Nearest-neighbor & \(h=(\,1\,1\,)^{T}\) & \(2\times 2\) \\ Bilinear & \(h=\left(\,\frac{1}{2}\,1\,\frac{1}{2}\,\right)^{T}\) & \(3\times 3\) \\ Bicubic & \(h=\left(-\frac{1}{16}\,\frac{0}{16}\,\frac{9}{16}\,\frac{9}{16}\,0-\frac{1}{16} \right)^{T}\) & \(7\times 7\) \\ B-spline & \(\left(-\frac{3}{152}\,0-\frac{9}{76}\,0\,\frac{91}{152}\,1\,\frac{91}{152}\,0- \frac{9}{76}\,0-\frac{3}{152}\right)^{T}\) & \(11\times 11\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: The classical convolutional interpolation kernels [31]. Figure 4: Reconstructed enlarged amplitude images using (**a**–**e**) the conventional SPM and (**f**–**j**) proposed OSPM at different recording distances z.  
of \(B_{x}=B_{SPx}\) and \(B_{x}=B_{OSPMx}\), respectively. The maximum lateral spatial resolution of the reconstructed images at recording distance z = 29.059 mm corresponds to group 6 element 3 (80.6-line pairs/mm, 12.4 mm) and group 7 element 1 (128.0-line pairs/mm, 7.8 mm) of the target. Thus, the proposed OSPM can simultaneously improve the image resolution and the maximum lateral spatial resolution of the reconstructed image. Figure 5 shows the magnified amplitude images and Fourier spectral images of the normalized reconstructed images of the hologram with z = 29.059 mm, obtained using the HIM for comparing the effects of the interpolation algorithms. From the bottom to the top of Table 1, the sizes of the interpolation kernels are reduced from 11 \(\times\) 11 to 1 \(\times\) 1, and the attenuation effects of the corresponding low-pass filters on high-frequency information of the diffracted field \(o(x,y)\) are gradually weakened. Thus, the lateral resolutions of the reconstructed \(u(x,y)\) images based on B-spline, bicubic, bilinear, and nearest-neighbor interpolation are gradually improved, corresponding to group 6 element 4 (11.4 mm), element 5 (9.8 mm), and element 6 (8.8 mm), and group 7 element 1 (7.8 mm) of the target, respectively. However, the reconstructed image with zero interpolation is seriously degraded by the residual zero-order and twin-image noise. Thus, the proposed OSPM implemented by nearest-neighbor interpolation has the highest spatial lateral resolution of 7.8 mm. The maximum lateral resolution of the proposed OSPM is more than the theoretical resolution of 2\(\Delta\) = 6.9 mm, which may be an inherent error introduced by super-pixels. Figure 6 compares the reconstructed images of PSDH and PPSDH by numerical simulation. The simulation target was the processed reconstructed image of the R1 DS1N with z = 29.059 mm, and the simulated pixel pitch was 3.45 mm, corresponding to group 7 element 2 (6.9 mm). The simulation results were in good agreement with the experimental results shown in Figure 5. Group 7 element 2 can be resolved in PSDH reconstructed images, but not in PPSDH, which is influenced by the residual zero-order and twin images in PPSDH.