# Robust Visual Localization with Dynamic Uncertainty Management in Omnidirectional SLAM
## 3 Robust Visual Localization
### Bayesian Regression  
The dynamic uncertainty management strategy presented in the previous subsection is based on an advanced matching with uncertainty propagation. Now, a further contribution to the robustness of the procedure is presented: a Bayesian regression technique, that is mainly comprised by Gaussian Processes (GP) [40]. We pursue the development of a sensor data distribution for the feature matching on the omnidirectional frame. GP is a powerful regression technique since it does not require the extraction of conventional associations between inputs and outputs, in comparison to traditional inference techniques. We have customized the implementation of a GP in order to return a matching data distribution in terms of probability, which can be mapped onto a global reference system. This information is advantageous in order to refine the search area for matching points. A GP, denoted as \(f(x)\), is constituted by its mean, \(m(x)\), covariance \(k(x,x^{\prime})\), and training and test input vectors, \(x\) and \(x^{\prime}\), respectively. \[f(x)\sim\mathcal{GP}[m(x),k(x,x^{\prime})] \tag{10}\]  
Considering that the implemented GP provides a probability distribution for our matching, now we can exploit the fusion of this probability with an information metric, in order to reduce even more the search area for matching, thus to enhance the overall system operation and its robustness. This represents valuable feedback data for refinement, which basically represents a second weighting  
Figure 5: Dynamic uncertainty management: advanced matching results. \(p_{1}\) represents a multi-scaled distribution, \(\lambda_{i}p_{1}\). \(q_{i}\) projects onto the second image as per \(R\)\(\sim\)N\((\beta,\sigma_{\beta})\) and \(T\)\(\sim\)N\((\hat{\phi},\sigma_{\phi})\), in Equation (8). A reduced search area is obtained (green). Epipolarity transforms into an elliptic area (blue) due to the uncertainty propagation on Equation (7). stage before the final match is returned. As far as the selected metric is concerned, we adapted the information-based Kullback-Leibler divergence (KL) [54]:  
\[KL(F_{t})=-\sum_{i}F_{t}(i)\log F_{t}(i) \tag{11}\] \[KL(F_{t-1}\parallel F_{t})=H(F_{t-1},F_{t})-H(F_{t-1})=\sum_{i=1} ^{k}F_{t-1}(i)\log\frac{F_{t-1}(i)}{F_{t}(i)} \tag{12}\]  
where \(H(F_{t})\) is the entropy [55] of the information distribution of the current feature point set at time \(t\). \(F_{t}(i)\) represents the probability of the existence of the current \(i\)-th feature points at time \(t\) (system step time), that is on the image acquired at the current robot's pose at \(t\), and also \(F_{t-1}(i)\) for the previous \(i\)-feature points detected at a previous image at \(t-1\). We seek the evaluation of the variations and fluctuations in the appearance of the matching points and their location. In these terms, KL allows one to determine such variations between matching points detected on a pair of images, \(F_{t-1}\), and the next matching points on the following pair \(F_{t}\). Then, KL encodes substantial variations on the matching points' location while the robot navigates. Higher values for KL imply newer visual information observed by the robot. Such values can be used for determining a specific metric that finally specifies the weighting coefficients to get robust candidate matchings. This ultimately establishes the probability areas where a corresponding point is more likely to be found, that is distribution sensor data. An illustrative example may be observed in Figure 6. Here, the robot navigates through a scenario passing through three different poses (\(A\), \(B\) and \(C\)). Likewise, Figure 7a,b presents the motion transformation between these poses and the same relation expressed in terms of the image reference. Finally, Figure 8a-c shows the returned probability data for the feature points found at each pose, fused with an accumulated probability, while the robot moves and discovers new scenes within this scenario.