## 3 Recognizing the Wear State of Milling Cutters under Variable Conditions Based on Optimized SDP
### Optimization of SDP Parameters Based on the Differential Evolution Algorithm  
The discrete sampling data of vibration signals, after undergoing the SDP transformation, maps the time series waveform onto the polar coordinate system to create symmetrical scatter points. It combines vibration signals from the X, Y, and Z axes according to their time series in a designated area, thus constructing a fused image of the three-axis vibration signals. The effect of the SDP transformation is primarily influenced by the lag coefficient \(l\) and the aperture gain \(\xi\), mainly reflected in the SDP image's spiral arm area, curvature, and geometric center shape features. The selection of parameters should avoid overlapping adjacent spiral arms from different mirrors in the SDP image, thereby masking features. The SDP image uses symmetrical spiral arms for representation because symmetry is conducive to the human visual perception system's recognition and memory of image features. However, with the development and application of computer vision, it is no longer necessary to identify image features manually. Therefore, improving the SDP method by replacing Equation (2) with Equation (3) can yield the desymmetrization dot pattern (DDP), which can reflect the characteristics of time series signals well and improve the computational efficiency of image processing. \[\Theta_{ij}=\Theta^{\prime}+\left(\frac{F_{j+l}-L}{H-L}\right)\xi, \tag{3}\]  
Taking a sine signal \(y=\sin(200\pi t)\) with a sampling frequency of 12,800 Hz as an example, with the number of mirrors \(m\) set to 3, lag coefficient \(l\) to 32, and aperture gain \(\xi\) to 90\({}^{\circ}\), the SDP and DDP images of this sine signal under the same parameters are shown in Figure 5. The figure shows that using DDP can effectively avoid overlapping adjacent spiral arms from different mirrors caused by parameter selection, aiding in finding optimal parameters. Using reasonable parameters in the DDP transformation can make the feature differences of signals under different wear states more apparent in model training, providing a basis for milling cutter wear state identification based on DDP images. Since different DDP images of the same wear state will have unique features, if these unique features are not weakened, image matching will tend to favor these unique features, ultimately leading to inaccurate wear state identification. Therefore, using cluster analysis to establish template images for each wear state can strengthen common features and weaken unique features within the same wear state, helping to improve the accuracy of wear state identification. Describing the template images of each wear state with cluster centers and converting DDP images into binary images, if more than 60% of the pixels at the same position in all binary  
Figure 5: SDP and DDP images under the same parameters. images of the same wear state are 0, the corresponding pixel in the template image of that wear state is marked as 0, considered as an inherent feature of that template image. Optimizing parameters with the maximization of the differences between template images of each wear state as an indicator can narrow the intra-class distance and expand the inter-class distance, thus facilitating the construction of a generalized decision boundary and reducing the impact of working condition changes on the milling cutter wear state identification model. The Euclidean distance is a distance metric commonly used to evaluate the similarity between images. The Euclidean distance \(d\) can assess the difference in template images of different wear states, expressed as:  
\[d(x,y)=\sqrt{\left(x_{1}-y_{1}\right)^{2}+\left(x_{2}-y_{2}\right)^{2}+\cdots +\left(x_{n}-y_{n}\right)^{2}} \tag{4}\]  
where \(\{x_{1},x_{2},\cdots,x_{n}\}\) and \(\{y_{1},y_{2},\cdots,y_{n}\}\) are the grayscale values of corresponding pixels in template images \(x\) and \(y\), respectively.