# Robust Visual Localization with Dynamic Uncertainty Management in Omnidirectional SLAM
## 3 Robust Visual Localization
### Dynamic Uncertainty Management  
After introducing the definition of the fundamentals of the proposed omnidirectional robust localization, now a detailed description for one of the main contributions of this work is addressed: the dynamic uncertainty management. The final goal is to improve the matching by allowing this procedure to adapt dynamically to the instantaneous uncertainties of the system. That is the main reason for considering the instantaneous uncertainty of the system, at each \(t\), being a magnitude closely related to those non-linear effects associated with noise, which is ultimately by the uncertainty matrix, as follows. Figure 4: Omnidirectional visual localization between poses \(A\) and \(B\). (**a**) Robot reference system; (**b**) camera reference system. Projections \(p_{A}(u,v)\) and \(p_{B}(u,v)\) are indicated. The final outcome implies a reduction of the search area for matching points on the pixel side. For that purpose, the epipolar constraint is again evaluated from Equation (1). In such a context, the current inconsistencies in the system (mainly non-linear noise), represented by uncertainty, are propagated through such a constraint, which now accepts a dynamic threshold, \(\delta(\hat{z}_{t})\):  
\[x^{\prime T}\hat{E}x<\delta(\hat{z}_{t}) \tag{7}\]  
Note that this threshold depends on a predicted variable, denoted as the observation movement, \(\hat{z}_{t}=(\hat{\beta},\hat{\phi})\), which is extracted from the stages of a general filter-based SLAM approach, and it is implicitly associated with the uncertainty of the system. More specifically, Table 1 comprises all the interdependencies extracted from the stage structure of the filter-based SLAM. Then, we can take advantage of the innovation measured between consecutive states of the visual SLAM system, \(S_{t}\), in order to define a powerful tool to establish predicted \(\sigma\) values for \(\hat{z}_{t}\) and, thus, for a rotation and translation, \(R\) and \(T\), respectively:  
\[S_{t}= \begin{bmatrix}\sigma_{\phi}^{2}&\sigma_{\phi\beta}\\ \sigma_{\phi\phi}&\sigma_{\beta}^{2}\end{bmatrix} ;\quad R\sim N(\hat{\beta}, \sigma_{\beta});\quad T\sim N(\hat{\phi},\sigma_{\phi}) \tag{8}\]  
This aspect entails a predicted matching, which can be found on the second image reference, even when no feature detection is previously run on the second image at current \(t\). Then, a Gaussian multi-scale distribution, \(\lambda_{i}p_{1}\), is generated on the first image frame and next propagated on the second, \(q_{i}\), by introducing the current uncertainty in Equation (7) and also applying the predicted movement in Equation (8). In this sense, Figure 5 breaks down the complete procedure for this advanced matching, which finally returns a reduced search area where candidate matching points are expected, rather than a global search over the entire image. The construction of this new search area comes from the reformulation of the epipolar constraint, which now transforms into an elliptical surface due to the effects of the omnidirectional adoption and to the drifts induced by \(\delta(\hat{z}_{t})\). The final stage includes a visual descriptor comparison, \(d_{i}(x)\) and \(d_{j}^{\prime}(x^{\prime})\) being the visual descriptors of two candidate matching points in the first and second image, respectively. In the end, a Mahalanobis metric [53], (\(\chi\)), is evaluated over them, so as to get an acceptable visual distance in order to assume two feature points as a valid pair of matching points:  
\[||d_{i}(x)-d_{j}^{\prime}(x^{\prime})||\leq\chi[dim(z_{t})] \tag{9}\]