## 5 Discussion  
Hough Transform is a robust method used in Image Analysis and Computer Vision. The core idea is to map data onto the dual parameter space and then interpret it through classification and clustering. The major role of the Hough Transform is to detect straight lines and compute their representatives, that is, dual points. The term \"representative\" is strictly connected with Clustering Using Representatives (CURE), an efficient data-clustering algorithm for large-scale databases. Compared with \(K\)-means clustering, which has already been addressed in our previous work, it is more robust to outliers and able to identify clusters having non-spherical shapes and size variances. Trajectory Clustering (TRACLUS) [18; 19] is a characteristic algorithm which has been designed for partitioning and applying clustering among partitions of different trajectories, and finally finding the representative sub-trajectory for each cluster, as presented in Figure 8. Several representative spatio-temporal clustering methods have been reviewed in a more recent work [20]. Nevertheless, in our work, we made use of this term with a different meaning. The notion of the representative points of a trajectory sample points is defined. Thus, the representative point of a number of spatial points that belong to the same line segment is the dual transformation point. Our framework can be combined with any existing clustering algorithm. As a preliminary approach, we chose \(K\)-Means, which applied on-line clustering on dual-point data for a number of mobile objects' trajectories at specific time-stamps.  
Figure 7: (**a**) Initial points per trajectory and (**b**) compression ratio for \(N=87\) trajectories, \(L=100\) time-stamps. Clustering with \((x,y)\) and \(k\)-NN: (**c**) \(x\) and \((U_{x},a_{x})\) and (**d**) \((x,y)\) and \((U_{x},a_{x},U_{y},a_{y})\) for \(N=87\) trajectories, \(L=100\) time-stamps.  
This work investigated the impact of Hough-X, which has already been applied in range queries, to the robustness of the methods proposed in [9] for addressing secure \(k\)-NN queries. The experimentation with the number of clusters \(K\), which should be known in advance, obviously only affected the method utilizing clustering; there, it is obvious that vulnerability behaves a little worse as the cluster number increases. Indeed, when adding features, the data cluster density decreases, where the model becomes more sparse, and hence the clustering task becomes even more difficult. A usual phenomenon and important part in Machine Learning is the reduction of a higher dimensional space into a lower dimensional one in order to avoid the Curse of Dimensionality.  
An important property of Hough is its robustness to low quality or uncertain data (either due to non-uniform sampling or noise) [21]. Therefore, even if a trajectory is represented by different sample points in 2D Euclidean space, in Hough space it may have the same dual points. Under this condition, Hough space reflects mobility patterns better than the original trajectory spatial data \((x,y)\), leading to more homogeneous clusters and improving the \(k\)-NN performance. As experimental results verify in Figure 6a,b, the above properties have a positive impact on vulnerability which is pair-wise preserved, showing that the clusters occur within cluster space-time similarity. The authors in [22] provide an efficient scheme for representative clustering on uncertain data. Finally, assuming feature suppression, the method with clustering demonstrates higher robustness or lower vulnerability, which is the main issue in \(k\)-anonymity, and thus in privacy preservation. It is a case which shows the superiority of the method with clustering in terms of vulnerability. Indeed, when the mobile users are protected by \(k\) nearest neighbors based on lower dimensionality data than the ones used in clustering, it is more difficult for an attacker, who has access to history data, to link the public information of the \(k-1\) nearest neighbors (that is, unlinkability).