# Viewport Rendering Algorithm with a Curved Surface for a Wide FOV in 360\({}^{\circ}\) Images
## 3 Limitation of the Conventional Methods
### Analysis of the Distortion Resulting from Conventional Methods  
As illustrated in Figure 2, the pixel values in the viewport are set to those with the corresponding \(P^{\prime}\) points on the unit sphere. When the pixel value at \((m,\ n)\) in the VP is denoted by \(I(m,\ n)\), the corresponding \(P^{\prime}\) point on the unit sphere is represented by \(P^{\prime}_{(m,n)}\), where the coordinate of \(P^{\prime}_{(m,n)}\) is denoted with the form \((X_{(m,n)},Y_{(m,n)},Z_{(m,n)})\). To analyze the distance between the positions of the corresponding \(P^{\prime}_{(m,n)}\) points, we calculated the vertical length of the arc between \(P^{\prime}_{(m,n)}\) and \(P^{\prime}_{(m,n+1)}\) and the horizontal length between \(P^{\prime}_{(m,n)}\) and \(P^{\prime}_{(m+1,n)}\), as demonstrated in Figure 7. Note that the length is the sampling interval that is used to generate a VP from a \(360^{\circ}\) image.  
In Figure 7, \(d^{h}(m,\ n)=\left|\overline{P^{\prime}_{(m,n)}P^{\prime}_{(m+1,n)}}\right|\) and \(l^{h}(m,\ n)=\left|P^{\prime}_{(m,n)}P^{\prime}_{(m+1,n)}\right|\) are the shortest direct distance and the length of the arc between the horizontally nearest \(P^{\prime}_{(m,n)}\) neighbors. Similarly, \(d^{v}(m,\ n)\) and \(l^{v}(m,\ n)\) are the values for the vertically nearest neighbors, respectively, as follows:  
Figure 6: Distortions resulting from the perspective projection [33] (**a–c**) and stereographic projection [33] (**d–f**).  
Figure 7: Cross-section of the sphere to explain the shorter direct distance and the distance following the curved surface between \(P^{\prime}_{(m,n)}\) and \(P^{\prime}_{(m+1,n)}\).  
\[d^{h}(m,\ n*)=\left|\overline{P^{\prime}_{(m,n*)}P^{\prime}_{(m+1,n*)}}\right|= \sqrt{\ (X_{(m,n*)}-X_{(m+1,n*)})^{2}+(Y_{(m,n*)}-Y_{(m+1,n*)})^{2}+(Z_{(m,n*)}-Z_{(m+1, n*)})^{2}} \tag{25}\]  
\[d^{v}(m*,\ n)=\left|\overline{P^{\prime}_{(m*,n)}P^{\prime}_{(m*,n+1)}}\right|= \sqrt{\ (X_{(m*,n)}-X_{(m*,n+1)})^{2}+(Y_{(m*,n)}-Y_{(m*,n+1)})^{2}+(Z_{(m*,n)}-Z_{(m*,n+1)})^{2}} \tag{26}\]  
\[l^{h}(m\,n*)=\left|P^{\prime}_{(m,n*)}P^{\prime}_{(m+1,n*)}\right|=2\sin^{-1} \left(\frac{d^{h}(m,n*)}{2}\right) \tag{27}\]  
\[l^{v}(m*,n)=\left|P^{\prime}_{(m*,n)}P^{\prime}_{(m*,n+1)}\right|=2\sin^{-1} \left(\frac{d^{v}(m*,n)}{2}\right) \tag{28}\]  
where \(0\leq\ m<W_{VP}-1\), \(0\leq\ n<H_{VP}-1\), \(n^{*}\) and \(m^{*}\) are the specific values that are used to calculate the lengths of the arcs between the horizontal and vertical positions. Figure 8 presents the normalized values of \(l^{h}(m,\ n*)\) according to the horizontal FOV (\(F_{h}\)) when the perspective and stereographic projections are used. In this figure, we can observe that the length between the locations of the consecutive sampling points on the sphere decreases as \(m\) moves away from the center (\(m=(W_{VP}/2)-1\)). This means that the pixels in the boundary region of a VP are made by sampling the \(P^{\prime}\) points with reduced sampling intervals on the sphere. This makes the pixels in the VP look like a stretched form of \(P^{\prime}\) points on the sphere. This tendency is more significant as \(F_{h}\) increases. As observed in Figure 8a,b, the amount of stretching in the perspective projection is greater than that in the stereographic projection. Note that the analysis for \(l^{v}(m^{*},n)\) is similar to those in Figure 8, although it is not presented in this paper.