## 3 Methodology
### Epipolar Geometry for Panoramic Cameras  
Epipolar geometry describes the relationship between positions of corresponding points in a pair of images acquired by different cameras [41; 43; 44]. Given a feature point in one image, its matching view in the other image must lie along the corresponding epipolar curve. This fact is known as the epipolar constraint. Once we know the stereo system's epipolar geometry, the epipolar constraint allows us to reject feature points that could otherwise lead to incorrect correspondences.  
In our system, the images are acquired by the catadioptric system formed by the upper camera-mirror (PMU) and the lower camera-mirror (PML), as seen in Figure 3. The projections of a 3D point \(\mathbf{X}_{W}\) over the mirrors PMU and PML are denoted by \(\mathbf{x}_{\text{PMU}}^{i}\) and \(\mathbf{x}_{\text{PML}}^{i}\) respectively. The coplanarity of the vectors \(\mathbf{x}_{\text{PMU}}^{i}\), \(\mathbf{x}_{\text{PML}}^{i}\), \(\mathbf{T}_{\text{PMU}\rightarrow\text{PML}}=[tx,ty,tz]^{T}\) is expressed by Equation (4)  
\[(\mathbf{x}_{\text{PML}}^{i})^{T}R(\mathbf{T}_{\text{PMU}\rightarrow\text{PML }}\times\mathbf{x}_{\text{PMU}}^{i})=0 \tag{4}\]  
where \(\times\) corresponds to the cross product and \(R\) and \(\mathbf{T}\) are the rotation matrix and translation vector between the upper and lower mirrors. The coplanarity restriction is simplified in Equation (5)  
\[(\mathbf{x}_{\text{PML}}^{i})^{T}E\mathbf{x}_{\text{PMU}}^{i}=0 \tag{5}\]  
where:  
\[E=RS \tag{6}\]  
\(E\) is the essential matrix. The skew symmetric matrix \(S\) is given by Equation (7).  
\[S=\left[\begin{array}{ccc}0&-t_{z}&t_{y}\\ t_{z}&0&-t_{x}\\ -t_{y}&t_{x}&0\end{array}\right] \tag{7}\]  
Figure 3: Epipolar geometry between two catadioptric cameras with parabolic mirrors. \(R\) is the rotation matrix and \(\mathbf{T}\) is the translation vector, between the upper and lower mirrors (PMU\(\rightarrow\)PML) expressed as a skew symmetric matrix. \(\Pi\) is the epipolar plane.  
For every point \(\mathbf{u}_{1}\) in the catadioptric image, an epipolar conic is uniquely assigned to the other image \(\mathbf{u}_{2}\), the search space between corresponding points is limited by Equation (8).  
\[\mathbf{u}_{2}{}^{T}A_{2}(E,\mathbf{u}_{1})\mathbf{u}_{2}=0 \tag{8}\]  
In the general case, the matrix \(A_{2}(E,\mathbf{u}_{1})\) is a non-linear function of the essential matrix \(E\), the points \(\mathbf{u}_{1}\), and the calibration parameters of the catadioptric camera.  
The vector \(\mathbf{T}_{\text{PMU}\rightarrow\text{PML}}\) defines the epipolar plane \(\Pi\). Equation (9) describes the vector normal to the plane \(\Pi\) in the camera coordinates system.  
\[n_{1}=\mathbf{T}_{\text{PMU}\rightarrow\text{PML}}\times\mathbf{x}_{\text{ PMU}}^{i} \tag{9}\]  
The normal vector \(n_{1}\) can be expressed in the second camera coordinate system using the essential matrix as follows:  
\[\begin{array}{rcl}n_{2}&=&Rn_{1}\\ &=&R(\mathbf{T}_{\text{PMU}\rightarrow\text{PML}}\times\mathbf{x}_{\text{ PMU}}^{i})\\ &=&RS\mathbf{x}_{\text{PMU}}^{i}\\ &=&Ex_{\text{PMU}}^{i}\end{array} \tag{10}\]  
Then Equation (10) is represented as:  
\[n_{2}=[p,q,s]^{T} \tag{11}\]  
The plane equation \(\Pi\) described in the second camera coordinate system is given by Equation (12).  
\[px+qy+sz=0 \tag{12}\]  
The derivation of \(A_{2}(E,\mathbf{u}_{1})\) from Equation (8) based on a parabolic mirror, \(z\) is substituted into Equation (12) resulting in Equation (13) that allows the calculation of the epipolar conics for this type of mirror:  
\[sx^{2}+2b_{2}px+sy^{2}+2b_{2}qy-sb_{2}^{2} \tag{13}\]  
where \(b_{2}\) is the mirror parameter and \(p,q,s\) are the components of the normal vector \(n_{2}\) computed in Equation (10).