## 3 Proposed Method
### Pseudocode of the Proposed Networks
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Dataset** & **Image Size** & **Rectification Map** & **De-Warping** \\ \hline KITTI & (600, 600) & 0.23 & 0.012 \\ CityScape & (640, 640) & 0.26 & 0.015 \\ WoodScape & (1480, 966) & 0.89 & 0.058 \\ Dongseongro & (1205, 905) & 0.67 & 0.054 \\ \hline \hline \end{tabular}
\end{table} 
Table 17: Computation time of Expandable Spherical Projection. Rectification map is generated only one time at the start of the program. \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 & 49.4 & 86.4 & 47.8 & 28.8 & 50.7 & **63.9** \\ Spherical & 47.8 & 86.6 & 44.8 & 38.6 & 58.1 & 62.8 \\ Expandable Spher. & **50.5** & **88.0** & **51.5** & **40.4** & **58.2** & 61.8 \\ \hline \hline \end{tabular}
\end{table} 
Table 15: Accuracy of different projections of LCat model on the Fisheye-Dongseongro. each concatenation method. All percentile performance is positive, which means that the proposed method is always better than using the original fisheye image. The highest enhancement, 4.7%, is achieved when using the KITTI dataset. Overall performance of the proposed method is better when using the two synthetic datasets. However, when using the real image datasets, the amount of enhancement is low. This is due to the fact that the proposed expandable projection model uses the ideal fisheye lens parameters. The lens parameters for generating the synthetic fisheye datasets follows the ideal fisheye lens model. Thus, the transformation of the fisheye image to the expandable spherical image can be also considered as an ideal, without any inherent optical distortion. As the future study, more sophisticated calibration of the real fisheye image can generate more ideal spherical projection image. ## 6 Conclusions

This paper proposes a deep neural network for detecting small and tiny objects in fisheye images. Nowadays, the use of the fisheye image is increasing because of the unique advantage of obtaining ultra-wide field of views. However, object detection in the fisheye image suffer from too small object size, curving and tilting in the image boundary. In this paper, we propose to transform the original fisheye image to an effective spherical projection image using the expansion weight. Using two scale parameters, central or marginal areas of spherical images are expandable for reducing the effect of radial and overall size distortions to the objects. Additionally, we propose three multi-level feature concatenation methods and analyze the effect of small object detection: SCat, LCat, SLCat. With short-skip concatenated layers and additional convolutions, the SCat achieves higher accuracy on complex urban scene datasets. From the LCat model, we have shown that the feature concatenation with a too shallow layer without sufficient convolution layers increases the difficulties to extract important features for the prediction layers. The SLCat network, combining short and long skip-layers, mostly presents better performance compared to the baseline model. Finally, we provide a fisheye dataset from one front view camera for autonomous driving with 2D bounding box annotation files, hoping the release of this dataset can help the development of fisheye lens related research. Conceptualization, S.K. and S.-Y.P. ; methodology, software, validation, formal analysis, S.K. ; investigation, S.-Y.P. ; writing--original draft preparation, S.K. ; writing--review and editing, S.-Y.P. ; supervision, S.-Y.P. ; project administration, S.-Y.P. ; funding acquisition, S.-Y.P. All authors have read and agreed to the published version of the manuscript. This research was funded partly by Basic Science Research Program through the National Research Foundation of Korea funded by the Ministry of Education (No. 2021R1A6A1A03043144) and partly by BK21 FOUR project funded by the Ministry of Education, Korea (4199990113966).