# Research on Panorama Generation from a Multi-Camera System by Object-Distance Estimation
## 3 Proposed Method
### Object-Distance Estimation Algorithm
#### 3.3.2 Estimation of Object-Distance Pyramid Maps in Overlapping Regions  
The traditional object space-based matching method, which was called the Vertical Locus Line Method (VLL) [29], is often used to estimate depth information by accurately matching a pair of image patches without directly calculating the three-dimensional computation for object points. Previous approaches of the VLL matching algorithm are mainly constrained to perspective projection images and require priori depth information. As a result, the VLL matching algorithm may exhibit reduced efficiency or failure due to the absence of prior object-space information. This paper proposes the Panoramic Vertical Locus Line (PVLL) matching algorithm to estimate the object-distance of panoramic surroundings based on pyramid grids ranging from coarse to fine. Estimation of Candidate Object-Distance  
An object-distance set is estimated priorly to increase the efficiency of the search-space size for PVLL matching. Supposing that the minimum and maximum object-distances of surroundings are defined as \(r_{min}\) and \(\infty\), the reprojected locations of the object point \(Q_{ij}\) (where \(r_{min}\leq r\leq\infty\)) will be restricted between point \(p_{i,min}\) to point \(p_{i,max}\) in image \(I_{i}\), and between point \(p_{j,min}\) and point \(p_{j,min}\) in \(I_{j}\), respectively, as shown in Figure 5.  
Given the object-distance of point \(Q_{ij}\) is \(r\), the horizontal displacement \(h(r)\) between the projection location and point \(p_{i,max}\) in image \(I_{i}\) can be approximately computed by Equation (8). \[h(r)\approx\frac{L}{\alpha}\text{arccos}\left(\frac{r-dcos\left(\frac{\pi}{2 n}\right)}{\sqrt{d^{2}+r^{2}-2 droso\left(\frac{\pi}{2n}\right)}}\right) \tag{8}\]  
where \(L\) is the width of the image of the sub-camera, \(\alpha\) is the horizontal field of view of the sub-camera, and \(d\) is the displacement between the optical center \(C_{i}\) and the origin \(O\) of the cylindrical projection coordinate system. Further, the list of candidate object-distances, set as \(r_{list}=\{r_{0},r_{1},r_{2},\cdots,r_{T-1}\}\), can be acquired using Algorithm 1 based on Equation (8), where \(\varepsilon\) is the displacement steps in pixels. As the object-distance changes continuously from \(r_{0}\) to \(r_{T-1}\), the back-projection location of an object point imaged in an overlapping region approximately moves along a straight line by \(\varepsilon\) pixels in a real image, as shown in Figure 6. Here, \(\varepsilon\) has been set to 1 pixel to provide an optimal balance between the object-distance resolution and computational efficiency. Figure 5: Horizontal disparity caused by different object-distances. ```
0:\(L\), \(\alpha\), \(d\), \(n\), \(r_{min}\), \(\varepsilon\)
0:\(r_{list}=\{r_{0},r_{1},r_{2},\cdots,r_{T-1}\}\)
1:\(r\gets r_{min}+1\), \(r_{0}\gets r_{min}\), \(m\gets 1\)
2:\(h\leftarrow\frac{l}{\alpha}\arccos(\frac{r-d\cos(\frac{\pi}{\alpha})}{\sqrt{d ^{2}+r^{2}-2d\cos(\frac{\pi}{\alpha})}})\)
3:repeat
4:\(h_{b}\leftarrow\frac{l}{\alpha}\arccos(\frac{r-d\cos(\frac{\pi}{\alpha})}{\sqrt{d ^{2}+r^{2}-2d\cos(\frac{\pi}{\alpha})}})\)
5:if\(abs(h-h_{b})>\varepsilon\)then
6:\(r_{m}\gets r,m\gets m+1\), \(h\gets h_{b}\)
7:endif
8:\(r\gets r+1\)
9:until\(h<\varepsilon\)
0:\(\{r_{0},r_{1},r_{2},\cdots,r_{T-1}\}\)
```  
**Algorithm 1**Solution of Projected Radius List  
(2) Panoramic Vertical Line Locus Algorithm  
**Definition 1**.