## 2 ELM Contents  
To help in understanding the presented crack detecting algorithm, we briefly review the theory and concepts of ELM, as follows. For more detailed contents, the readers can refer to these works [30; 31; 32; 33]. The ELM was initially proposed for studying the single hidden layer feedforward neural network (SLFN) [30]. As shown in Figure 1, with \(L\) hidden nodes (here, \(L\) is an important parameter for ELM model), a SLFN can be expressed as  
\[f_{L}(x)=\sum\
olimits_{i=1}^{L}G\left(\mathbf{w}_{i},b_{i},x\right)\gamma_{i}= \sum\
olimits_{i=1}^{L}h_{i}(x)\gamma_{i}, \tag{1}\]  
where \(x\) is the input data of ELM network, \(b_{i}\) represents the bias of \(i\)-th hidden node, \(\mathbf{w}_{i}\) denotes the input weight linking the inputs \(x\) and the \(i\)-th hidden node, \(G(\cdot)\) is the variant sigmoid function, \(h_{i}(\cdot)\) denotes the output vector of \(i\)-th hidden node, and \(\mathbf{\gamma}\) is the ELM network output weight, which needs to be computed. Figure 1: Representative framework of extreme learning machine (ELM) model. Differing from other neural network frameworks, the ELM model shows that the hidden input parameters (i.e., \(b_{i}\) and \(\mathbf{w}_{i}\) in the \(G\left(\mathbf{w}_{i},b_{i},x\right)\) function) can be randomly chosen from a continuous probability distribution [30]. Thus, the ELM framework can obtain a much faster training performance than other learning models. Moreover, [PERSON] et al. have proven that ELM has both universal approximation capability and classification capability:  
**Theorem 1**. : _Universal approximation capability [31]: Given any bounded nonconstant piecewise continuous function as the activation function, if the SLFNs can approximate any target function \(f(x)\) via tuning the parameters of hidden neurons, then the sequence \(\left\{h_{i}(x)\right\}_{i=1}^{L}\) could be randomly generated based on any continuous sampling distribution, and \(\lim_{L\rightarrow\infty}\left\|\sum_{i=1}^{L}h_{i}(x)\gamma_{i}-f(x)\right\|=0\) holds with probability 1 with appropriate output weight \(\mathbf{\gamma}\)._  
**Theorem 2**. : _Classification capability [32]: Given any feature mapping \(h(x)\), if \(h(x)\mathbf{\gamma}\) is dense in \(C(\mathbb{R}^{d})\) or in \(C(M)\), where \(M\) is a compact set of \(\mathbb{R}^{d}\), then SLFNs with random hidden layer mapping \(h(x)\) can separate arbitrary disjoint regions of any shapes in \(\mathbb{R}^{d}\) or \(M\)._  
For simplicity, we can rewrite Equation (1) as \(f_{L}(x)=\sum_{i=1}^{L}h_{i}(x)\gamma_{i}=\mathbf{H}(x)\mathbf{\gamma}\). Here, \(\mathbf{H}(x)=[h_{1}(x),\ldots,h_{L}(x)]\) represents the row outputs of ELM network. If we randomly generate the input hidden parameters, \(\mathbf{H}(x)\) will be known and, then, the ELM learning function will be linear. In this case, finding the output weights \(\mathbf{\gamma}\) becomes the only objective goal. Suppose that the training sets are \(\left\{\mathbf{X},\mathbf{T}\right\}=\left\{x^{i},t_{i}\right\}_{i=1}^{N}\). Here, \(x^{i}\in\mathbb{R}^{d}\) denotes the \(i\)-th input data and \(t_{i}\in\mathbb{R}^{m}\) is the corresponding training label. The linear learning function can be expressed in the following matrix form  
\[\mathbf{H}\mathbf{\gamma}=\mathbf{T}, \tag{2}\]  
where \(\mathbf{H}\) represents the randomized matrix of ELM hidden layer, and can be computed as follows:  
\[\mathbf{H}=\left[ \begin{array}{c}h(x^{1})\\ \vdots\\ h(x^{N})\end{array} \right]=\left[ \begin{array}{ccc}h_{1}(x^{1})&\cdots&h_{L }(x^{1})\\ \vdots&\ddots&\vdots\\ h_{1}(x^{N})&\cdots&h_{L}(x^{N})\end{array} \right].