## 3 Proposed Method
### Pseudocode of the Proposed Networks
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Image** & \(AP\) & \(AP_{50}\) & \(AP_{75}\) & \(AP_{S}\) & \(AP_{M}\) & \(AP_{L}\) \\ \hline Fisheye & 23.0 & **45.7** & 20.4 & 7.9 & **24.0** & 39.3 \\ Spherical & 22.3 & 44.4 & 19.6 & **8.6** & 23.4 & **39.9** \\ Expanded Spher. & **23.1** & 45.6 & 19.8 & **8.6** & 23.2 & 37.1 \\ \hline \hline \end{tabular}
\end{table} 
Table 11: Accuracy of different projections of LCat model on the WoodScape. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Image** & \(AP\) & \(AP_{50}\) & \(AP_{75}\) & \(AP_{S}\) & \(AP_{M}\) & \(AP_{L}\) \\ \hline Fisheye & 23.0 & **45.7** & 20.4 & 8.7 & **24.1** & 36.9 \\ Spherical & 22.5 & 45.0 & 19.2 & 8.6 & 23.8 & **40.8** \\ Expanded Spher. & **23.5** & **45.7** & **20.9** & **8.9** & 23.7 & 38.1 \\ \hline \hline \end{tabular}
\end{table} 
Table 12: Accuracy of different projections of SLCat model on the WoodScape. Since most of the objects are much smaller than the image size, we present cropped example images of the detection result on YOLOv3-SPP from differently projected images are shown in Figure 15.

#### 4.5.3 Comparison of Different Concatenated Models

The detection results of differently concatenated models on the fisheye, spherical, and expanded spherical images are reported in Table 14, Table 15 and Table 16, respectively. The accuracy \(AP\) in expanded projected images at SCat is the highest score compared to other types of image and models. Unlike other modified models, SCat can increase overall accuracy across the IoU thresholds and scales without using lower level features than the baseline model. Moreover, when the datasets contain more complex scenes such as many overlapped objects with diverse scales in one image, SCat mostly obtains better results. On the other hand, \(AP\) in LCat decreased than the baseline from our projection images, same as Fisheye-CityScape. We assume merging the features with too low-level details without a sufficient number of following convolutions can hinder the network from correctly extracting relevant features. ### Computation Time

Table 17 shows the computation time of our proposed projection from each training dataset. The undistortion process is divided into two steps: generating the rectification map and updistorting the fisheye image. We create the rectification map with mapping information between the cylinder and fisheye image, which can be run only one time at the beginning of the program. Then, the program reads the mapped points from the map and renders the undistorted cylinder image. Table 18 presents the inference time [sec] from each image size in Titan V GPU. When the size of the spherical image is around 600 \(\times\) 600 pixels and the size of test-image is \(512\times 512\), the total computation time will be 0.034 seconds, processing the 29.4 frames per second. The projected image will be the same size as the image of the inference in future work, assuming that the whole process can be accomplished by increased speed over 30 FPS. ## 5 Discussion

As shown in the experimental sections, the proposed expandable spherical image yields better performance than using the original fisheye image. In all cases of the feature concatenation methods, object detection performance is enhanced. In Table 19, the AP performance enhancement of the proposed method is summarized. The numbers in the table are the percentile (\(\%\)) AP enhancement of the proposed expandable spherical projection model in \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Image** & \(\mathbf{AP}\) & \(\mathbf{AP_{50}}\) & \(\mathbf{AP_{75}}\) & \(\mathbf{AP_{5}}\) & \(\mathbf{AP_{M}}\) & \(\mathbf{AP_{L}}\) \\ \hline Fisheye & 49.9 & 87.0 & 48.7 & 29.8 & 51.0 & **65.4** \\ Spherical & 46.7 & 85.5 & 44.0 & 38.5 & 55.6 & 61.5 \\ Expandable Spher. & **51.5** & **88.7** & **52.2** & **41.3** & **59.0** & 63.6 \\ \hline \hline \end{tabular}
\end{table} 
Table 16: Accuracy of different projections of SLCat model on the Fisheye-Dongseongro.