## 1 Introduction  
With the continuous development of industrial processes, rotating machinery has been widely applied in sectors such as aerospace, wind power generation, and rail transportation. As the core component of rotating machinery, the rotor system directly affects the performance of all of the equipment [1]. The axis trajectory can accurately characterize the operational state of the rotor system. For typical rotor systems in rotating machinery, such as turbines, compressors, fans, and rolling mills, the operational state can be directly assessed by observing the axis trajectory. This includes identifying conditions such as rotor imbalance, misalignment, oil film whirl, and rubbing. Analyzing the characteristics of axis motion trajectory is of significant importance in identifying the operational state of rotor systems. The axis trajectory is derived from two vibration signals collected in perpendicular directions at the same cross section, providing the most direct and comprehensive information about the rotor's operational state. However, the raw vibration signals collected in practical scenarios often suffer from various interferences and random noise, leading to non-linear and non-stationary characteristics. The intricate nature of the axis trajectory complicates its interpretation and significantly hampers the assessment of the rotor system's operational condition. Hence, it is crucial to process the axis trajectory by filtering out noise and extracting relevant features before conducting further analysis. Common methods for axis trajectory purification include morphological filtering [2], wavelet transform [3], Empirical Mode Decomposition (EMD) [4], and singular value decomposition (SVD) [5]. Morphological filtering is a non-linear filtering method successfully applied in power system relay protection for traveling wave noise reduction [6] and power quality detection [7]. It has been introduced into vibration signal denoising [8] and axis trajectory purification [9]. Its principle involves decomposing a complex axis trajectory signal into several parts through mathematical morphological transformations, thereby separating it from the background while preserving the signal's main shape characteristics. Even with strong noise accompanying the original signal, its basic shape can still be recognized, reconstructed, and enhanced. Wavelet transform is widely used in bearing fault diagnosis due to its multi-resolution characteristics, which allow for the extraction of the original signal from noise-contaminated signals [10]. Reference [11] introduced wavelet algorithms into axis trajectory purification, using harmonic wavelet algorithms to purify rotor displacement signals at specified frequency characteristics and synthesize axis trajectories. However, wavelet transform parameters and thresholds greatly influenced the results, making parameter optimization challenging and the method relatively complex. Time-frequency analysis based on Empirical Mode Decomposition (EMD) is highly suitable for processing non-linear and non-stationary signal sequences and is widely used in signal processing and bearing fault diagnosis [12]. EMD stabilizes non-stationary signals by decomposing their fluctuations and trends at different scales step by step, generating a series of data sequences with different characteristic scales known as Intrinsic Mode Functions (IMFs). In the literature [13], EMD is combined with a Probabilistic Neural Network (PNN) to construct an adaptive filter, achieving efficient noise reduction in bearing fault signals. This is followed by feature extraction and input to the PNN for bearing fault diagnosis. In another study [14], EMD is used to process acceleration vibration signals of mine hoists. The number of IMFs obtained sets the number of modes for Variational Mode Decomposition (VMD). Through correlation analysis, specific IMFs are selected from VMD for feature extraction, and a Support Vector Machine (SVM) is used for equipment monitoring.