## 3 Void Detection Based on Deep SVDD
### Deep Support Vector Data Description  
Support vector data description (SVDD) [34] uses a support vector machine (SVM) [35]-like kernel function to represent the feature space of the input. The hypersphere's minimized boundary, which may identify the classes from the distribution of this feature space, is then determined. Therefore, SVDD is mapped so that normal data in the feature space are closer to the center point of the sphere, and defect data are further away by adding a penalty. Equation (2) describes the SVDD's objective function.  
\[\min_{R,A,\xi_{i}}R^{2}+C\sum_{i=1}^{l}\xi_{i}, \tag{2}\]  
\[s.t.\left\|\Phi(x_{i})-a\right\|^{2}\leq R^{2}+\xi_{i},\ \xi_{i}\geq 0,\forall i\]  
where \(\xi_{i}\) is the penalty imposed on the defect data, \(R\) is the sphere's radius, \(a\) is the sphere's center, and \(C\) is the penalty constant. The fact that the data are mapped to the feature space by the kernel function (\(\Phi\)), and that the distance of the data from the sphere's center must be located inside the \(sphere+\xi\) area, is also can be verified by \(s.t.\) Equation (4) determines whether the data are normal or defects using the values of \(\xi_{i}\), \(R\), and \(a\) obtained using this  
Figure 6: The overall flow of our proposed model for void detection inside duct in PSC bridge.  
objective function. The input data \(x_{i}\) are calculated to be positive if they are normal when inside the sphere, and negative if they are abnormal when outside the sphere.  
\[f(x)=sign\Big{(}R^{2}-\|\Phi(x_{i})-a\|^{2}\Big{)}, \tag{3}\]  
In contrast, Deep SVDD [29] replaced the kernel function with an autoencoder which is the deep learning approach. In other words, the input's feature space, or latent vector, is represented by the autoencoder. To distinguish between the feature spaces of normal and defective data provided by the autoencoder, Deep SVDD determines the minimum boundary of the hypersphere as follows: Deep SVDD trains data to be gathered more densely to the center point inside the hypersphere with a radius of \(R\). On the other hand, data outside of the hypersphere train to move further from the center point of the hypersphere, and Deep SVDD classifies data outside of the hypersphere as abnormal. Therefore, Deep SVDD determines the minimized boundary of the hypersphere that can distinguish the feature space of normal and defective data represented by the autoencoder as shown in Figure 7. Equation (4) is the object function that determines the hypersphere's minimized boundary.  
\[\underset{w}{\min}\frac{1}{n}\sum_{i=1}^{n}\|\phi(x_{i};\mathcal{W})-c\|^{2}+ \frac{1}{2}\sum_{\ell=1}^{L}\|\mathcal{W}^{\ell}\|_{F}^{2}, \tag{4}\]  
where \(c\) is the hypersphere's center and \(\phi(x_{i};\mathcal{W})\) is the feature space that the autoencoder transformed from the input data. The left term calculates to minimize the distance between the feature space and \(c\), the hypersphere's center. The right term is a weight decay regularizer, which prevents a particular weight from increasing excessively and trains it through optimization to map a normal data feature space close to \(c\). This allows Deep SVDD to define the anomaly score for input's feature space as in Equation (5).  
\[s(x)=\|\phi(x_{i};\mathcal{W})-c\|^{2}, \tag{5}\]  
To produce a feature space that suitably represents IE data, the autoencoder developed an ELMo-like structure, which will be covered in more detail in Section 3.3.