## 3 Proposed Method
### Expandable Spherical Projection  
The idea of spherical projection is to project the distorted fisheye image into the undistorted cylinder image, whose equal distances measured from the center correspond to equal steps in the angle of 3D scene. The mathematics of this projection can be expressed in longitude and latitude coordinate systems. \[ \begin{split} x_{n}&=\frac{x_{cy}-x_{c}}{W_{cy}},\\ y_{n}&=\frac{y_{cy}-y_{c}}{H_{cy}},\end{split}  \tag{5}\]  
Figure 5: Fisheye camera installed on the front window of a test car. where \(x_{n}\) and \(y_{n}\) are normalized cylinder points, presented in (5) and \(x_{cy}\) and \(y_{cy}\) are pixel points of cylinder image. \(x_{c}\) and \(y_{c}\) indicate the center point of the image and \(W_{cy}\) and \(H_{cy}\) means the width and height of the image, respectively. The cylinder points are directly mapped to the normalized longitude \(\theta_{n}\) and latitude \(\phi_{n}\). \[\theta_{n} =x_{n}, \tag{6}\] \[\phi_{n} =y_{n}.\]  
Then, the coordinates multiplied by \(\theta_{max}\) and \(\phi_{max}\) of the sphere are transformed into 3D sphere point \(P\left(P_{x},P_{y},P_{z}\right)\) in (7) and ( 8). \[\theta =\theta_{max}\phi_{n}, \tag{7}\] \[\phi =\phi_{max}\phi_{n},\]  
where \(\theta_{max}\) is half of the horizontal field of view and \(\phi_{max}\) is half of the vertical field of view. In this paper, both horizontal and vertical FOV are set as \(185^{\circ}\). \[P_{x} =\cos\phi\cos\theta, \tag{8}\] \[P_{y} =\cos\phi\sin\theta,\] \[P_{z} =\sin\phi.\]  
Using real spatial coordinate \(P\), we calculate \(r\) and \(\theta^{{}^{\prime}}\) for the fisheye pixel points, as shown in (9). The \(F\) is the field of view of the fisheye lens. \[r= \frac{2\tan^{-1}(\sqrt{{P_{x}}^{2}+{P_{z}}^{2}},P_{y})}{F}, \tag{9}\] \[\theta^{{}^{\prime}} =\tan^{-1}(P_{z},P_{x}).\]  
Finally, the transformation of the pixel coordinate (\(x_{fe},y_{fe}\)) in the fisheye image is as follows in (10). \(R\) and (\(c_{x},c_{y}\)) are the radius and center of circular image, respectively. \[\theta_{n} =x_{n}, \tag{6}\]0 \[\theta_{n} =x_{n}, \tag{6}\]1  
The relationship between pixel point \(x\) of the cylinder image and longitude \(\theta\) in basic spherical projection is linearly mapped. Instead of using \(\theta\) in (7), we suggest the expression of \(\theta_{proposed}\) which is the multiplication of expansion weight \(w\) and \(\theta\) in expandable spherical projection, as shown in (11). Compared to our previous study [41], we described the equation of the \(\theta\) in more details of normalization and scaling to the field of view. The weight \(w\) is a non-negative parameter for increasing the marginal or central area of the image, consisting of scale factor \(\alpha\) for determining the expansion, and \(\beta\) for balancing the effect of the edge areas, as shown in (12). \[\theta_{n} =x_{n}, \tag{6}\]2  
\[\theta_{n} =x_{n}, \tag{6}\]3  
expanding effect to the center area. If \(\alpha\) is higher than one, the center areas are projected as narrower in the cylinder image. On the other hand, \(\frac{|\beta|}{g_{\mathrm{m}xx}}\) becomes one around the margin regions, where \(w\cong(\alpha+\beta)\). When value of \((\alpha+\beta)\) is lower than one, the edge areas are stretched more. When \((\alpha+\beta)\) is higher than one, projected areas are narrower than original spherical image. For synthetic fisheye images, objects near the edge areas appear smaller and narrower, and objects near the center appear larger than their actual size. Therefore we set higher expansion to margin, and lower expansion to center areas, where \(\alpha\) and \(\beta\) are set as 1.2 and \(-0.3\), respectively. In the case of a real fisheye image, smaller objects are frequently around the center area. distant objects are more often located at the center and appear much smaller than their real size. On the other hand, objects appearing near the margin are located closer to the test vehicle and do not need much expansion.