Sparse representations for fault signatures via hybrid regularization in adaptive undecimated fractional spline wavelet transform domain

Recent studies on vibration-based machine diagnostics have highlighted the role played by the wavelet transform (WT). However, common WT-based denoising methods (e.g. wavelet thresholding and non-penalty regularization) are often challenging in attaining accurate sparse representations of fault signatures in practice due to artifacts. In this paper, a method via hybrid regularization in the adaptive undecimated fractional spline WT (AUFrSWT) domain is introduced to achieve the accurate sparse representations of fault signatures from strong noise environments. The method promotes wavelet sparsity by two aspects: wavelet basis design and wavelet coefficient processing. For the former, a new wavelet family, i.e. a fractional spline wavelet, is used, and the AUFrSWT is originally proposed to customize the optimal wavelet basis and meanwhile address the translation-invariant issue. For the latter, the wavelet coefficients are estimated by minimizing a single convex model function with hybrid regularization, where nonconvex arctangent penalty is employed for wavelet sparsity promoting and total variation is used to improve the reconstruction performance of the shrunken coefficients. The method is validated by the analysis of actual bearing and gear vibration data from fault-injection experiments. Analysis and comparison results show great potential for signal sparse representations and machinery fault diagnostics.