## 1 Introduction
However, EMD decomposition faces issues such as mode mixing and end effects, which significantly limit its application in the purification and denoising of axis trajectories. Singular value decomposition is also widely used in the field of fault diagnosis and the feature extraction of rotating machinery [15]. SVD denoising maps a one-dimensional signal into a two-dimensional Hankel matrix, using effective rank to define boundaries between signal and noise spaces. It reduces the dimensionality of the signal space to obtain denoised signals [16]. The dimension of the Hankel matrix directly affects its denoising effect [17], with the maximum dimension method commonly used to determine the dimension [18; 19]. References [20; 21] applied the maximum dimension method to construct Hankel matrices for SVD, achieving good denoising effects. Reference [5] utilized an improved SVD algorithm to optimize the matrix structure and effective component selection mechanisms, achieving axis trajectory purification. It has been proven that using SVD and determining the Hankel matrix structure with the maximum dimension method enables maximum signal decomposition, resulting in excellent denoising and feature extraction effects suitable for axis trajectory purification. Common methods for extracting axis trajectory characteristics include invariant moments [22; 23; 24] and Fourier descriptors [25; 26], with the invariant moment method being more widely used. Invariant moments primarily characterize the geometric features of image regions and possess the mathematical properties of invariance to rotation, translation, and scale. The two-dimensional shape invariant moment method identifies shapes by extracting invariant moments from axis trajectory images. Reference [27] proposed a fault diagnosis model that integrates Hu invariant moments and deep convolutional autoencoder features, achieving high accuracy in axis trajectory recognition. Reference [28] utilized the affine invariant moments of axis trajectories combined with artificial neural networks for axis trajectory recognition. Reference [29] introduced a combination of Hu moments and affine moments as feature vectors and used an improved particle swarm algorithm and BP neural network to recognize axis trajectories. However, the invariant moment method is sensitive to noise and is computationally complex. Currently, scholars have invested considerable effort in the purification and noise reduction of axis motion trajectories, as well as in feature extraction. Many researchers have applied methodologies from related fields to this area, achieving notable results. However, challenges such as poor noise reduction effects and large data volumes for feature extraction still persist. This paper focuses on analyzing the characteristics of axis motion trajectories to address the issue of identifying the operational state of rotor systems. By constructing axis trajectory sample signals from the original vibration signals and employing singular value decomposition, the axis trajectory sample signals were purified and denoised. The distance sequence representation method was used to achieve the curvature region segmentation of the axis trajectory images, further enabling the accurate extraction of axis motion trajectory features. The experimental data from an imbalance condition were verified using an eddy current displacement sensor on a multi-disk rotor test rig, with results indicating that this method is effective.