# Unbalance Detection in Induction Motors through Vibration Signals Using Texture Features
## 2 Methodology
### Texture Feature Extraction  
In image processing, texture is a property used to measure the variability of gray levels through an image. Such measures can be used as image descriptors. These descriptors are used in image processing to classify different elements contained in an image. There are different statistical features used to describe texture in an image. Although these properties are commonly used in 2-D signal analysis, they can be computed in 1-D. The texture as a statistical concept has its origin in the gray-level co-occurrence matrix proposed by [PERSON] et al. [29]. However, since the computational cost is expensive, [PERSON] et al. [26] proposed the sum and difference histograms as an alternative to the co-occurrence matrix. According to [PERSON], considering a digital image \(I\) with size \(M\times N\) denoted by \(I(i,j),\{i=1,\ldots,M;j=1,\ldots,N\}\) being \(G=\{1,2,\ldots,N_{g}\}\) the set of the \(N_{g}\) quantized grey levels, taking two points in a relative position fixed by \(u\) and \(v\), \(I(i,j)\), and  
Figure 1: Overall VTA model design to infer the unbalance fault in IM. from a feature vector [30]. According to [PERSON] et al. [30], in a dataset \(F^{l}=\{(x_{i}^{l},y_{i}^{l})\}_{i=1}^{n}\) with \(x_{i}^{l}=(x_{1,i}^{l},\ldots,x_{v,i}^{l})^{T}\in\mathcal{R}^{v}\), the method is used to learn a function \(z\in Z:x^{(l)}\mapsto y\). This relation can generate a decision boundary (hyperplane) by maximizing the separation between two or more classes. In the function \(F^{l}\), \(y_{i}^{l}\) represents the labels from the vector \(x_{i}^{l}\). According to [PERSON] et al. and [PERSON] et al. [30; 31], it is assumed that \(\{(x_{i}^{l},y_{i}^{l})\}_{i=1}^{n}\) are independents and they are distributed in a range \(\mathcal{M}\times\mathcal{N}\). Here, \(\mathcal{M}\) represents the set of samples in the dataset and \(\mathcal{N}\) represents the set of labels [30]. Thus, the SVM is considered a regularization problem and can be described by the Equation (13):  
\[\inf_{z\in\mathcal{H},z\in Z}\left\{\frac{1}{n}\sum_{i=1}^{n}\mathcal{L}(y_{i }^{\prime},z(x_{i}^{l}))+\lambda||z||_{\mathcal{H}}^{2}\right\}, \tag{13}\]  
where \(\lambda\in\mathbb{R}^{+}\), \(\mathcal{L}:\mathbb{R}\mapsto[0,\infty)\) is the lost function, and \(||z||_{\mathcal{H}}^{2}\) is the norm Hilbert space \(\mathcal{H}\subseteq\mathbb{R}^{\mathcal{M}}=z:\mathcal{M}\mapsto\mathbb{R}\). However, the unbalance set has a non-lineal behavior, hence an efficient way to deal with this problem is to map the input space \(\mathcal{R}^{v}\) to \(\mathcal{H}\) using a kernel [32]. Different kernels are available in the literature (linear kernel, polynomial kernel, and radial basis function kernel), and selecting the appropriate kernel plays an important role in the classification process. In this study, an empirical process was conducted, and the radial basis function kernel [33] was selected since it yielded better results. Thus, the Gaussian kernel is defined according to the Equation (14):  
\[k_{\sigma},\mathcal{C}^{d}=exp\left(-\sigma^{2}\sum_{j=1}^{d}(x_{j}-x_{j}^{ \prime})^{2}\right), \tag{14}\]  
where \(d\in\mathcal{N}  
can be observed that although there are differences between the signals, it can be difficult for an expert to detect this fault. Therefore, signal processing is required. For this study, a set composed of 5000 signals was used to extract the texture features. This set was formed by 2500 signals which represented the healthy class and the rest formed the unbalance class. Each signal contained 4096 samples normalized between 0 and 255, according to those established in the image analysis. The final set was formed by applying the sum and difference histograms and the variance and correlation texture features. Thus, the final dataset was composed of 5000 \(\times\) 2 values, where 5000 represented the samples in the final dataset and 2 represented the dimensionality. In order to generate the training-validation and testing sets, two subsets of the final set were formed.