# On Training Road Surface Classifiers by Data Augmentation
## 4 Experiments
\right] \tag{2}\] \[i=1\dots TSS\]  
where \(corr(\mathbf{v}\mathbf{,w})=\frac{\mathbf{v}^{T}\mathbf{w}}{\|\mathbf{v}\| \|\mathbf{w}\|}\) is the normalized correlation between the two vectors \(\mathbf{v}\) and \(\mathbf{w}\) and \(\|\cdot\|\) stands for the Euclidean norm. Thus, two sets of size \(TSS\) of 4-dimensional feature vectors are obtained. Let us assume that every set corresponds to a different class of a two-class problem. Then, the discriminator block implements a linear discriminant in an effort to separate these two classes. Let us call \(\mathbf{d}\) the linear discriminant vector learned from the two mentioned sets. Assuming that the overall mean vector has been extracted from the two sets, perfect separability would imply that  
\[\mathbf{d}^{T}\mathbf{s}_{i}>0\qquad\mathbf{d}^{T}\mathbf{s}_{i}^{\text{ \tiny{$\mathsf{S}$}}}<0\qquad i=1\dots TSS \tag{3}\]  
In practice, perfect separability is not possible, but we can impose a less strict condition to decide if both classes are separable or not. For example, we could decide that both classes are non-separable if (3) is not satisfied in more than half of the cases. If both classes turn out to be non-separable, the candidate is accepted to augment the set of real feature vectors. Otherwise, the candidate is rejected, and a new one must be provided by the generative block. To this aim, the generator is \"informed\" by the discriminator about the learned discriminant \(\mathbf{d}\), and then the generator changes the signs of the partitions of \(\mathbf{f}^{\text{\tiny{$\mathsf{S}$}}}\) contributingto a decrease in \(\mathbf{d}^{T}\mathbf{s}_{i}^{S}\;\;i=1\ldots TSS\) in more than \(TSS/2\) cases, in an effort to reduce the number of cases in (3) in which \(\mathbf{d}^{T}\mathbf{s}_{i}^{S}<0\), thus decrementing the separability between both sets. The modified candidate is provided to the discriminator and a new iteration is repeated until the candidate is finally accepted. In our experiments, two or three iterations have been enough most of the time. The curves in Figure 3 show that, in all cases, accuracy improves by increasing \(TSS\). Notice that for a given \(TSS\), the accuracy improves both in LDA and RDF by using GANSO for data augmentation, with respect to not using data augmentation. However, data augmentation with SMOTE does not show a significant improvement, neither with LDA nor with RDF. Note that SMOTE is a method normally used for imbalance problems, where only the minority class is oversampled, but, here, we consider a problem where all classes are oversampled. Moreover, incorporating structural information seems to be crucial to obtain realistic replicas of the real feature vectors. On the other hand, as expected, RDF achieves higher accuracy than LDA both with and without data augmentation. However, the complexity of training, the implementation, and the overfitting problems make LDA a better option. Notice that the improvement with GANSO decreases as \(TSS\) increases; it is because, for large \(TSS\), we are approximating the Bayes error rate (\(BER\)) [36,37]. \(BER\) is the probability of error (\(P_{e}\)) that could be achieved if we could have perfect knowledge of the class conditional probability densities in the feature space. In a practical setting, we never have this knowledge; instead, we have a finite training dataset. Thus, \(BER\) cannot be achieved but, assuming consistent estimates of the model, \(BER\) can be approximated as \(TSS\) increases to infinity. Remember that, in Figure 3, we showed the accuracy, i.e., \((1-P_{e})\).100; thus, all curves converge to (1-\(BER\)).100 for increasing \(TSS\). However, this limit is reached before with data augmentation by GANSO. For a better quantification of the improvement provided by data augmentation, we have defined the concept of effective training set size (\(ETSS\)), which is the \(TSS\) with no data  
Figure 4: Graph model and structured correlation between OFV or SFV and the OFV reference set.