# Robust Visual Localization with Dynamic Uncertainty Management in Omnidirectional SLAM
## 2 Monocular Omnidirectional System
### Epipolarity Adaption Definitions  
The epipolar constraint [48] becomes a fundamental tool so as to design a localization technique, due to the fact that it represents a motion recovery tool. The nature of our omnidirectional system made us redesign the standard planar epipolar constraint to adapt it to the geometry of our catadioptric system. This allows us to gather visual information from a more reliable point of view, since an advanced feature detection and matching procedure between two omnidirectional images can be designed, in order to localize the robot in terms of poses. To that aim, it is necessary to introduce the essential matrix, \(E_{3x3}\)[49], and its relation with two matched points between images, \(x\) and \(x^{\prime}\). Given a particular calibration, the corresponding points may be normalized to \(\hat{x}\) and \(\hat{x^{\prime}}\):  
\[\hat{x}^{\prime T}E\hat{x}=0 \tag{1}\]  
The terms in \(E\) encode a decomposition: \(R_{3x3}\) and \(T=[t_{x},t_{y},t_{z}]\), as a general rotation and translation, associated with the skew symmetric \([T]_{x}\)[48]. Then, a 2D movement \(\in XY\), and angular movement \((\beta,\phi)\), between two poses of the robot can be recovered up to the scale factor, from:  
\[E=[T]_{x}R= \begin{bmatrix}0&0&\sin(\phi)\\ 0&0&-\cos(\phi)\\ \sin(\beta-\phi)&\cos(\beta-\phi)&0\end{bmatrix}  \tag{2}\]  
This adaption to the omnidirectional geometry is represented in Figure 2. The projection of the point \(X\) on two image references, \(x\) and \(x^{\prime}\), is determined by the epipolar plane, \(\pi\), and both camera centers \(C\) and \(C^{\prime}\). An essential aspect is related to \(l\) and \(l^{\prime}\), as the epipolar lines that determine the geometric location where feature points should be matched. This is a key aspect for us in order to implement an advanced visual matching, which ultimately returns potential data for a robust localization. In contrast to traditional stereo-planar applications [50; 51], here epipolar lines transform into ellipses as they are the intersection of the plane \(\pi\) with the surface of the hyperboloid of the mirror. Figure 1: Real equipment: (**a**) Pioneer P3-AT with omnidirectional camera mounted, internal odometer and SICK-LMS200 laser range finder. Omron Adept Technology, Inc. 10 Columbia Drive, Amherst, NH 03031, USA; (**b**) CCD FireWire DMR21 BF04. The Imaging Source Europe GmbH, Uberseetor 18, 28217 Bremen, Germany; (**c**) Eizho Wide 70 Mirror. Eizho Laboratory Ltd. Suminoue, Osaka, 559-0034, Japan. Then, using this adaption for a consistent matching process, some contributions are presented in the following sections, so as to come up with a robust visual localization, sustained by a dynamic uncertainty management and Bayesian techniques, on the grounds of this epipolar adaption. Finally, it is worth mentioning a last processing step on the image projection. In particular, we obtained better feature point detection with a post-processed projection conversion to a panoramic view. As mentioned above, omnidirectional cameras are very sensitive to non-linearities, due to the nature of the mirror, and this effect is especially evident at top areas of its surface, which consequently reflect 3D points on a pixel area over the image frame with lower resolution. These are the main reasons for computing such a planar conversion, together with a Gaussian filter over the top areas of the image:  
\[I_{pano}(u,v)=imfilter[I_{omni}(r\cdot cos(\alpha)+u_{c},r\cdot sen(\alpha)+v_{ c}),r_{max},r_{min},r_{step},u_{1},u_{2}] \tag{3}\]  
being \(I_{pano}\) and \(I_{omni}\) the panoramic and omnidirectional image frames in pixels, for which \((u_{c},v_{c})\) represent the center of the omnidirectional image and \(imfilter\) is a Gaussian filter applied between heights \(u_{1}\)-\(u_{2}\) in the panoramic view.