# Latitude-Adaptive Integer Bit Allocation for Quantization of Omnidirectional Images
## 6 Experiments
### Results and Analysis for JPEG-Based Methods  
In this part, our bit allocation methods are applied to adjust JPEG quantization tables, and a comparative analysis is conducted with other JPEG-based methods, including traditional codec JPEG [23], and two improved methods designed for omnidirectional images by [PERSON] et al. [20] and [PERSON] et al. [10]. This comparison is performed on images with a resolution of \(512\times 1024\) pixels. Within our methods, bit allocation is computed according to the proposed greedy algorithm. We denote the bits allocated to blocks at latitude \(k\) by \(b_{k}\) and denote the block length by \(L\) where \(L\) is set to \(L=8\). The quality ratio is then calculated as \(\frac{b_{k}}{L^{2}}\). We adjust the quality factor of the JPEG quantization table according to the following formula, where a larger quality factor indicates better image quality. \[Q_{\text{cur}}=\frac{b_{k}}{L^{2}}\times Q_{\text{base}}, \tag{17}\]where the base quality factor \(Q_{\text{base}}\) is set to 50, and \(Q_{\text{cur}}\) represent the quantization quality factor of the current block. [PERSON] et al. [10] proposed that each block on the omnidirectional image is associated with an elevation angle varying from 0 to \(\pi\), corresponding to the elevation of the block projected on the sphere. Similar to our method, its quantization table also varies with the latitude. In the following examples, we illustrate some instances of the adjusted JPEG table in our methods. The quantization table at the elevation angle \(\frac{\pi}{4}\) with \(ratio=0.67\) is presented in Table 2. To demonstrate the variation of our quantization tables across different ratios and latitudes, we change the elevation angle to \(\frac{\pi}{2}\) in Table 3, where the block is at the center of the omnidirectional image. We increase the bits allocated to the whole image, resulting in a higher quality ratio at the elevation angle \(\frac{\pi}{4  
A comparison of the quantization tables in Tables 2 and 3 reveals that our method employs more refined quantization tables near the equator, which significantly contributes to WS-PSNR. This implies that our quantization strategy places heightened importance on regions near the equator. Furthermore, a comparative analysis between Tables 2 and 4 reveals a discernible inverse relationship between the allocated number of bits and the quantization step size. As the bit allocation increases, there is a corresponding reduction in the quantization step size, leading to a more refined quantization process. This observed trend aligns consistently with theoretical expectations and intuitive understanding. We then execute a comprehensive rate-distortion analysis, wherein our approach is benchmarked against alternative methods. In each experiment, 100 images are randomly selected from the LIC360 dataset [5]. We repeat the experiment 10 times and calculate the average value. As a result, the rate-distortion curves are shown in Figure 5. \begin{table}
\begin{tabular}{c c c c c c c c} \hline
16 & 11 & 10 & 16 & 23 & 39 & 49 & 59 \\ \hline
12 & 12 & 14 & 18 & 25 & 56 & 58 & 53 \\
14 & 13 & 16 & 23 & 39 & 55 & 67 & 54 \\
14 & 16 & 21 & 28 & 49 & 84 & 78 & 60 \\
17 & 21 & 36 & 54 & 66 & 106 & 100 & 75 \\
23 & 34 & 53 & 62 & 78 & 101 & 109 & 89 \\
47 & 62 & 76 & 84 & 100 & 117 & 116 & 98 \\
70 & 89 & 92 & 95 & 109 & 97 & 100 & 96 \\ \hline \end{tabular}
\end{table} 
Table 4: The quantization table with \(ratio=1.03\) used in our method at the elevation angle \(\frac{\pi}{4}\). Figure 4: Coefficients frequencies at latitude 17. (**a**) DC coefficients \(\mathfrak{c}_{17,1}\). (**b**) AC coefficients \(\mathfrak{c}_{17,2}\). (**c**) AC coefficients \(\mathfrak{c}_{17,30}\). (**d**) AC coefficients \(\mathfrak{c}_{17,64}\).