# Intra Prediction-Based Hologram Phase Component Coding Using Modified Phase Unwrapping
## 2 Phase Information of Full-Complex Hologram
### Phase Unwrapping  
If the delay in the phase response can be expressed as a multiple of the sinusoidal period, we can unwrap this phase response. The distinction between a true phase delay and a phase delay with discarded sinusoidal periods is the most important factor for successful phase unwrapping. The purpose of this research is not to find the original phase response for phase unwrapping, but to increase the compression efficiency by making phase linearity, so the difficulty of such distinction is not a big problem. Figure 2 shows an example of phase unwrapping. As shown in Figure 2a, the phase, which was limited to a value from \(-\pi\) to \(\pi\), was expanded to a range of \(-3\pi\) to \(6\pi\) as shown in Figure 2b through phase unwrapping. As a result, the phase has a linear shape. If we take a modulo operation with a value of 2 in the phase of Figure 2b, it becomes the same value as Figure 2a. We call this process rewrapping. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline  & \multicolumn{3}{c}{**PSNR (dB)**} & \multicolumn{3}{c}{**SSIM**} \\ \hline
**Real** & **Imaginary** & **Complex** & **Phase** & **Reconstruction** \\ \hline
27.43 & 25.27 & 26.35 & 16.39 & 23.29 & 0.82 \\
31.43 & 27.18 & 29.31 & 17.11 & 26.43 & 0.90 \\
31.71 & 31.71 & 31.71 & 16.99 & 26.83 & 0.91 \\
31.78 & 31.29 & 31.54 & 16.68 & 25.81 & 0.87 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Relationship between phase information of holograms and the reconstruction result. Figure 2: Example of phase unwrapping: (**a**) original phase response, (**b**) unwrapped phase response. Figure 3 shows the results of phase unwrapping for the phase component of a full-complex hologram. The unwrapping result of the original phase information in Figure 3a is the image in Figure 3b. Figure 3c,d are the results of plotting one row in the middle position. It can be seen that the phase changed very linearly after phase unwrapping. However, as described above, since the dynamic range is very wide due to phase unwrapping, when the result of phase unwrapping is normalized to a limited input width, the error per bit of the phase becomes very large. We consider the case of compressing a hologram using a standard codec (assuming the most common 8-bit input). If the original hologram phase component with a range of \(2\pi\) is normalized to 8-bit integer type, the error has \(2\pi/256\) per 1 bit. However, if the range is \(160\pi\) due to phase unwrapping and is normalized to 8 bits, the error is \(160\pi/256\). Next, if data with such an error rate are compressed, the error per bit becomes too large and phase unwrapping acts as a disadvantage. An experiment was performed on the effect of phase unwrapping using the Horse hologram [18] of JPEG Pleno. Figure 4 shows the PSNR value of the compression result while compressing up to 100:1 using the HEVC codec. It can be seen that almost similar compression trends are shown for all cases, and the results for the original phase values having a \(2\pi\) range are the best in all cases. In the case of \(32\pi\), which has higher linearity, the compression efficiency is lower than that of \(2\pi\). It can also be seen that the result of full unwrapping without phase limitation is the worst. From this result, it can be confirmed that the increase in the error rate per bit due to the expansion of the dynamic range described above has a greater influence than the linearity of the data. Phase unwrapping improves the correlation between data by increasing the continuity and linearity of the data. This characteristic makes it possible to expect various possibilities for the application of phase unwrapping. However, the disadvantage of phase unwrapping is that due to phase unwrapping, the data range of \(2\pi\) that the original data had is increased. The increase in data range due to phase unwrapping is proportional to the image size.