## 3 Proposed Method
### Pseudocode of the Proposed Networks
### Experiments on Synthetic Fisheye-KitTII

#### 4.3.1 Implementation Details

We set 24 as the mini-batch size, 512 as the size of the image, 350 epochs, 0.00019 initial learning rates, and apply the cosine decay scheduling strategy. The momentum and weight decay are, respectively, set as 0.843 and 0.00036. Finally, we use total of 4075 images and evaluate the detection accuracy from the validation dataset of CityScape. #### 4.3.2 Comparison of the Projection

Similar to Fisheye-KitTII, the expandable spherical image shows an expansion effect on the margin area and a narrowing effect on the center. From Table 5, our proposed approach achieves the highest score in overall \(AP\), while basic spherical projection shows mostly fewer scores than the original fisheye image dataset. Figure 12 shows the detection result from fisheye and each spherical image. #### 4.3.3 Comparison of Concatenated Models

As reflected in Tables 6-8, SCat and SLCat mostly outperform than baseline model, while the accuracy from LCat is less in fisheye and expanded projected image. SCat at the proposed projection achieves the highest score, showing 3.4% \(AP\) improvement compared to the base model at the fisheye dataset. \begin{table}
\begin{tabular}{c c c c c c c} \hline
**Image** & \(AP\) & \(AP_{50}\) & \(AP_{75}\) & \(AP_{5}\) & \(AP_{M}\) & \(AP_{L}\) \\ \hline Fisheye & 56.9 & 85.5 & 63.7 & 48.8 & 66.4 & 76.2 \\ Spherical & 59.5 & 86.0 & 67.6 & 46.9 & 65.4 & 75.4 \\ Expandable Spher. & **61.6** & **88.2** & **71.0** & **48.0** & **65.9** & **76.0** \\ \hline \end{tabular}
\end{table} 
Table 4: Accuracy of different projections of SLCat model on the Fisheye-KitTII. \begin{table}
\begin{tabular}{c c c c c c c} \hline
**Image** & \(AP\) & \(AP_{50}\) & \(AP_{75}\) & \(AP_{5}\) & \(AP_{M}\) & \(AP_{L}\) \\ \hline Fisheye & 57.0 & 86.4 & 63.1 & 48.7 & **66.4** & **76.9** \\ Spherical & 58.8 & 86.4 & 65.1 & 44.5 & 65.0 & **76.9** \\ Expandable Spher. & **61.4** & **88.9** & **70.0** & **47.8** & 65.6 & 75.6 \\ \hline \end{tabular}
\end{table} 
Table 3: Accuracy of different projections of LCat model on the Fisheye-KitTII. \begin{table}
\begin{tabular}{c c c c c c c} \hline
**Image** & \(AP\) & \(AP_{50}\) & \(AP_{75}\) & \(AP_{5}\) & \(AP_{M}\) & \(AP_{L}\) \\ \hline Fisheye & 56.9 & 85.5 & 63.7 & 48.8 & 66.4 & 76.2 \\ Spherical & 59.5 & 86.0 & 67.6 & 46.9 & 65.4 & 75.4 \\ Expandable Spher. & **61.6** & **88.2** & **71.0** & **48.0** & **65.9** & **76.0** \\ \hline \end{tabular}
\end{table} 
Table 4: Accuracy of different projections of SLCat model on the Fisheye-KitTII. \begin{table}
\begin{tabular}{c c c c c c c} \hline
**Image** & \(AP\) & \(AP_{50}\) & \(AP_{75}\) & \(AP_{5}\) & \(AP_{M}\) & \(AP_{L}\) \\ \hline Fisheye & 22.4 & 42.5 & 20.7 & 5.6 & 28.0 & 58.7 \\ Spherical & 21.2 & 40.0 & 18.1 & 6.2 & 28.2 & 56.8 \\ Expandable Spher. & **24.3** & **45.5** & **22.3** & **6.9** & **31.0** & **66.9** \\ \hline \end{tabular}
\end{table} 
Table 5: Accuracy of different projections of YOLOv3-SPP model on Fisheye-CityScape. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Image** & \(AP\) & \(AP_{50}\) & \(AP_{75}\) & \(AP_{5}\) & \(AP_{M}\) & \(AP_{L}\) \\ \hline Fisheye & 22.4 & 42.3 & 20.3 & 5.9 & 29.0 & 57.6 \\ Spherical & 22.9 & 42.9 & 20.5 & 6.2 & 30.8 & **63.0** \\ Expandable Spher. & **25.8** & **47.2** & **23.9** & **7.3** & **34.3** & 58.5 \\ \hline \hline \end{tabular}
\end{table} 
Table 6: Accuracy of different projections of SCat model on the Fisheye-CityScape. Figure 12: Detection results: (**a**) Synthetic fisheye image; (**b**) Spherical-projected image; (**c**) Expandable Spherical-projected image from Fisheye-CityScape. ### Experiments on WoodScape

#### 4.4.1 Implementation Details

We set 24 as the mini-batch size, 512 as the size of the image, 350 epochs, 0.00019 initial learning rates, and apply the cosine decay learning rate scheduling strategy. The momentum and weight decay are, respectively, set as 0.843 and 0.00036.