## 1 Introduction
According to the characteristics of the signal itself, the method adaptively decomposes the signal into a series of intrinsic mode components and sums these components to obtain the filtered signal. EMD has good adaptability and does not rely on pre-defined basis functions, making it suitable for the decomposition and analysis of nonlinear and non-stationary signals, such as MFL signals. To improve EMD, many methods have been proposed. [PERSON] [11] proposed local mean decomposition (LMD), which suppressed end-point effects and modal aliasing in EMD while improving the decomposition accuracy. [PERSON] et al. [12] proposed ensemble empirical mode decomposition (EEMD). By adding white noise multiple times to the signal to weaken the modal aliasing caused by impact interference, more precise upper and lower envelopes were obtained, improving the decomposition accuracy. [PERSON] et al. [13] proposed complementary ensemble empirical mode decomposition (CEEMD) based on EEMD. By adding a pair of white noise sources, opposite to each other, to the signal, the improved method solves the modal aliasing problem while ensuring that white noise is eliminated from the reconstructed signal. [PERSON] et al. [14] transformed the signal decomposition problem into a variational solution problem and proposed variational mode decomposition (VMD), which can suppress modal aliasing and has good noise robustness. In addition, [PERSON] et al. [15] proposed a filtering method for the MFL signal based on EMD and wavelet filtering. The decomposed components of the MFL signal were wavelet-filtered and reconstructed, achieving good SNR. [PERSON] et al. [16] proposed an MFL signal image-filtering method based on the combination of mean filtering and bidimensional empirical mode decomposition (BEMD). The bidimensional intrinsic mode functions decomposed by BEMD are then mean-filtered and reconstructed to obtain high-SNR MFL images. Combining the tightly supported framework of WT with the adaptive concept of EMD, [PERSON]. [17] proposed empirical wavelet transform (EWT) based on narrowband signal analysis theory and WT. Non-iterative methods are used for the signal decomposition. EWT provides a time-frequency analysis approach for constructing adaptive wavelets for signal processing, continuously used and improved, and gradually applied in signal processing. Basing on spectral kurtosis and EWT, [PERSON] et al. [18] proposed an adaptive partial discharge fluorescence signal-filtering algorithm. By using a fast spectral kurtosis graph to merge the tightly supported regions of the noisy signal, the Fourier spectrum of the noisy signal is re-divided, reducing the division of invalid noise components and thus reducing the computational cost of the algorithm. [PERSON] et al. [19] proposed an improved adaptive parameterless EWT filtering method that combined mutual information (MI) and spline interpolation fitting to optimize spectrum division. The method effectively suppressed noise in the high-frequency partial-discharge signals of transformers. At present, MFL signal processing has achieved fruitful results, but its application mainly focuses on steel wire ropes, oil and gas pipelines, steel pipes, etc. There is less discussion in the field around the non-destructive detection of rails [20]. EWT combines the advantages of EMD and WT, and is suitable for processing nonlinear and non-stationary signals. Due to rail-surface-defect MFL signals being nonlinear and non-stationary, EWT is suitable for the analysis, and it may also result in a better processing efficiency by improving EWT. However, the rail is long, and the detection speed is always faster than that of a rope or pipeline, so the interferences are different. In this work, the suppression of the vibration noise and random interference while maintaining the same inspection speed is studied, and a method for rail-head surface-defect MFL signal filtering based on improved EWT via MI and kurtosis is proposed to suppress the defect signal background noise and improve the SNR.