## 3 Proposed Crack Detection Model
### Multilayer Feature Representation  
Owing to the limited performances of hand-crafted features, most of the existing concrete crack detection methods are unable to achieve satisfactory results under the condition of complicated surroundings. Meanwhile, the DL-based feature learning models need expert knowledge, and require time-consuming parameter fine-tuning. To address the problems mentioned above, we present an efficient multi-layer feature learning scheme, based on an ELM-based auto-encoder (AE). An AE is usually utilized for unsupervised feature learning. In an auto-encoder, the input \(x\in\mathbb{R}^{d}\) is mapped into a latent representation \(\psi\in\mathbb{R}^{L_{f}}\), and the resulting \(\psi\) is used to recover the input \(x\) via minimizing the reconstruction errors. Here, \(L_{f}\) is an important parameter of AE, and its value setting will be discussed later. Differing from the traditional auto-encoders, ELM-AE randomly projects the inputs \(x\) by a non-linear function \(h(x)=G(\mathbf{w},\mathbf{b},\mathbf{x})\), and searches the way to recover the original data \(x\) by \(h(x)\mathbf{\gamma}=x\). As \(h(x)\) is given, ELM-AE can learn the image features faster than the existing methods (e.g., deep Boltzmann machine), and has achieved favorable performance in image classification [38]. The success of ELM-AE feature representation has inspired us to extend it to crack region feature extraction. The proposed single layer feature learning model mainly consists of two steps: ELM feature training and ELM-AE feature mapping. Suppose that, in an ELM-AE with \(L_{f}\) hidden nodes, the input data set is \(\mathbf{X}=\{\mathbf{x_{1}},\mathbf{x_{2}},\ldots,\mathbf{x_{N}}\}\), where \(x_{i}\in\mathbb{R}^{d}\) is the \(i\)-th input vectorizing the image region data. As mentioned above, we randomly choose the hidden parameters \(\mathbf{w}\) and \(b\) (see Equation (1)), which project the input data \(x_{i}\in\mathbb{R}^{d}\) into an \(L_{f}\) dimensional ELM random space, as follows:  
\[H_{i}=\left\{G(\mathbf{w}_{j},b_{j},x_{i})\right\}_{j=1}^{L_{f}},\ \ i=1,\ldots,N. \tag{5}\]  
Figure 2: Schematic diagram of the presented concrete crack region detecting model. For ELM feature training, the main objective is to reconstruct the inputs \(\mathbf{X}\) from ELM random space \(H\) by solving the following problem:  
\[\boldsymbol{\hat{\gamma}} =\operatorname*{argmin}_{\boldsymbol{\gamma}}\left\{\|\mathbf{X}- \mathbf{H}\boldsymbol{\gamma}\|_{2}^{2}+\mu\|\boldsymbol{\gamma}\|_{p}\right\}, \tag{6}\]  
where \(\boldsymbol{\gamma}\) is the output weight to be obtained and \(\mu\) is a regularization parameter. The setting of \(p\) controls the characteristics of the ELM learning. To obtain more sparse and meaningful hidden features, we set \(p\) to be the \(\ell_{1}\) norm. To efficiently solve Equation (6), a fast APG algorithm [39] is adopted for getting the optimum result within several iterations. The APG method can attain an \(O(1/k^{2})\) convergence rate with \(k\) iterations. In this case, the object function of Equation (6) can be rewritten as  
\[\left\{ \begin{array}{l}f(\boldsymbol{\gamma})=\|\mathbf{X}-H\boldsymbol{ \gamma}\|_{2}^{2}\\ g(\boldsymbol{\gamma})=\mu\|\boldsymbol{\gamma}\|_{1}\end{array} \right. \tag{7}\]  
where \(f(\boldsymbol{\gamma})\) is a differentiable smooth convex function with Lipschitz continuous gradient and \(g(\boldsymbol{\gamma})\) is a non-smooth but convex function. The gradient of the smooth convex function is \(\
abla f(\boldsymbol{\gamma})=2H^{\mathsf{T}}\left(H\boldsymbol{\gamma}- \mathbf{X}\right)\), and its corresponding Lipschitz constant is \(\xi=2\times\max(eig(H\cdot H^{\mathsf{T}}))\). The details for finding the output weight \(\boldsymbol{\gamma}\) can be found in Algorithm 1. ```
0: The image data \(\mathbf{X}\), number of hidden nodes \(L_{f}\), Lipschitz constant \(\xi\), regularization parameter \(\mu\).