## 3 Proposed Crack Detection Model
### Incremental Crack Region Classification  
In this section, as for the crack region detection, we utilize a regularized ELM-based classification model, and its stability and generalization performance have been studied, in particular, in [32]. Specifically, suppose that the hidden-node number of ELM classifier is \(L_{c}\), and we randomly generate the input-hidden parameters \(\mathbf{w}_{j}\) and \(b_{j}\) (\(j=1,\cdots,L_{c}\)). Here, \(L_{c}\) is an important parameter for ELM-based classification network, and its value will be discussed later. With these generated hidden parameters, the hidden layer matrix can be easily computed via \(H_{i}=\left\{h_{j}(y_{i})\right\}_{j=1}^{L_{c}}\) (\(i=1,\ldots,N\)). Here, \(h_{j}(y)=G(w_{j},b_{j},y)\) denotes the \(j\)-th hidden-node output of ELM network and \(y_{i}\) represents the \(i\)-th learned image region feature. The training process of crack region classification needs to solve the following problem [41]:  
\[ \begin{array}{ll}\min\limits_{\mathbf{\beta}}&\frac{1}{2}\|\mathbf{\beta}\|^{2}+ \frac{\lambda}{2}\sum\limits_{i=1}^{N}\|e_{i}\|^{2}\\ \text{s.t.}&h(y_{i})\mathbf{\beta}=t_{i}-e_{i},\;\;i=1,\cdots,N,\end{array}  \tag{10}\]  
where \(\mathbf{\beta}\) denotes the required output weights of the ELM classifier, \(t_{i}\) is the training label of inputs \(y_{i}\), and \(\lambda\) is the regularized parameter for penalizing the training error term \(e_{i}\). By substituting the constraints into the objective function, For Equation (10), solving the following problem is equivalent:  
\[\min\limits_{\mathbf{\beta}}\;\;\frac{1}{2}\|\mathbf{\beta}\|^{2}+\frac{\lambda}{2}\| H\mathbf{\beta}-T\|^{2}. \tag{11}\]  
For solving the problem Equation (11), the gradient, with respect to \(\mathbf{\beta}\), can easily be calculated. The optimal solution \(\mathbf{\beta}^{*}\) can then be obtained by setting this gradient function to be 0. \[\mathbf{\beta}^{*}=(I/\lambda+H^{\mathrm{T}}H)^{-1}H^{\mathrm{T}}T, \tag{12}\]  
Figure 3: Illustration of ELM auto-encoder (ELM-AE) feature learning: (**a**) The input crack region instances, and (**b**) the contents of the output weights \(\mathbf{\gamma}\). where **I** represents an identity matrix whose dimension is the hidden node number \(L_{c}\) of ELM classification model. The required crack damage identification decision function, \(F_{2}(\cdot)\), is  
\[F_{2}(y)=\textbf{H}(y)\cdot\mathbf{\beta}^{*}. \tag{13}\]  
Along with the continuous concrete crack damage detecting tasks, some new non-crack and crack concrete images will be available. To adapt to the changing concrete surroundings, it is necessary to renew the old crack damage detection network. As is known to us, retraining the initial ELM classification model using the old and new training samples is a straightforward way. Despite the fact that this motivation is simple, it will be faced with the burden of storage and computation time with an increasing number of training data. In this section, for solving this problem above, the online incremental updating method is adopted for updating the advocated crack damage identification network. It should be mentioned that the randomly generated hidden parameters (i.e., \(\textbf{w}_{j}\) and \(b_{j}\)) will be not changed in the ELM network retraining. In this case, the output weight \(\mathbf{\beta}\) of ELM network becomes the only option parameter to be updated. If we already have \(Z_{0}\) training sample features, including crack and non-crack image patches, the primeval ELM hidden output matrix \(\textbf{H}_{0}\) can be easily calculated, and the corresponding training labels are \(\textbf{T}_{0}\). According to Equation (12), the initial crack damage identification function can be computed, as follows:  
\[\mathbf{\beta}_{0}=(I/\lambda+H_{0}^{\text{T}}H_{0})^{-1}H_{0}^{\text{T}}T_{0}. \tag{14}\]  
For simplicity, we rewrite \(P_{0}=I/\lambda+H_{0}^{\text{T}}H_{0}\), and \(Q_{0}=H_{0}^{\text{T}}T_{0}\). Then, we have \(\mathbf{\beta}_{0}=\textbf{P}_{0}^{-1}\textbf{Q}_{0}\).