## 4 Key Enablers in the HDL Model
### Time Series Imaging of Cutting Forces  
Since the convolutional neural network is sensitive to the spatial characteristics of the signal, time series imaging is carried out on the collected time series to convert the one-dimensional signal into a two-dimensional image, so that the result can be completed by time series imaging. The method adopted in this paper is the Gramian angular summation fields (GASF) method proposed by [PERSON] et al. [26], which can encode time series. In addition, the original time series obtained by this method can be reconstructed to understand how features are classified during the coding process. The image data are input into the HDL model.  
The GASF method consists of two steps. First, the time series is represented in a polar coordinate system rather than in typical Cartesian coordinates. In this step, given  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Layer** & **Neurons** & **Activation Function** \\ \hline Input Layer & 7 & / \\ Hidden Layer 1 & 32 & ReLU \\ Hidden Layer 2 & 16 & ReLU \\ Hidden Layer 3 & 8 & ReLU \\ Hidden Layer 4 & 4 & ReLU \\ Output Layer & 1 & / \\ \hline \hline \end{tabular}
\end{table}
Table 1: Structure of the upper-level machine learning module for cutting force estimation.  
observed values of a time series \(\mathbf{x}=x_{1},x_{2},\dots,x_{n}\), and then normalized \(X\), all values fall on the interval \([-1,1]\):  
\[\widetilde{x}^{i}=\frac{x_{i}-\max(\mathbf{x})+(x_{i}-\min(\mathbf{x}))}{\max(\mathbf{x})- \min(\mathbf{x})}. \tag{6}\]  
The time series can be represented in polar coordinates by encoding the value as the angle of the corresponding cosine value, while the time is encoded as the radius in polar coordinates:  
\[\varphi=\arccos(\widetilde{x}_{i}),-1\leq\widetilde{x}_{i}\leq 1,\ \widetilde{x}_{i} \in\widetilde{\mathbf{x}}, \tag{7}\]  
\[s.t.:\ r=\frac{t_{i}}{N},\ t_{i}\in\mathbb{N}, \tag{8}\]  
where \(\mathbb{N}\) is a constant factor of the span of the regularized polar coordinate system. As time goes on, the corresponding values on the polar coordinate system move between different corners. This movement preserves the time relationship and makes it easy to identify time dependencies over different time intervals. This time correlation is expressed as follows:  
\[\mathbf{G}=\begin{bmatrix}\cos(\varphi_{1}+\varphi_{1})&\dots&\cos(\varphi_{1}+ \varphi_{n})\\ \cos(\varphi_{2}+\varphi_{1})&\dots&\cos(\varphi_{2}+\varphi_{n})\\ \vdots&\ddots&\vdots\\ \cos(\varphi_{n}+\varphi_{1})&\dots&\cos(\varphi_{n}+\varphi_{n})\end{bmatrix}. \tag{9}\]  
GASF images provide a time-dependent method of preservation. \(\mathbf{G}(i,j||i-j|=k)\) refers to the time interval \(k\) superposition in the direction of the relative correlation. When the length of the time series is \(n\), the dimension of the GASF image obtained is \(n\times n\). To reduce the size of the image, the piecewise aggregation approximation (PAA) technique can be applied to smooth the time series and reduce the number of sampling points. According to the sampling points of each group of experimental data, they were converted into 512 sampling points by the PAA method and then converted into two-dimensional images by the GASF method, each of whose size was 512 \(\times\) 512. After output in the form of pictures, the image can be obtained as shown in Figure 3.  
Figure 3 shows the typical examples of the GASF images of the cutting force signals of non-wear diamond tools. The images reflect the signal distribution of the cutting force within 0.6 s sampling time. The images are on the diagonal symmetry and the distribution of cutting force can be easily found. With the powerful computing power of CNN, more hidden features can be excavated.  
Figure 3: Cutting force GASF imaging of non-wear diamond tools.