Morphological wavelet is a non-linear extension of traditional linear wavelet based on mathematical morphology. Mathematical morphology and its related operators will not be described here. Next, morphological wavelet is defined based on the basic concepts of dual wavelet decomposition and non dual wavelet decomposition.
Suppose that there are data sets VJ and WJ, let VJ represent the j-th signal space (J = 0, 1, 2,…) WJ represents the j-th level of detail space. The morphological wavelet decomposition of signal is to decompose the sampled signal along the increasing direction of the number of layers J with signal analysis operator ψ ↑ J: VJ → VJ + 1 and detail analysis operator ω ↑ J: VJ → WJ + 1. On the contrary, to reconstruct the signal means to reconstruct the sampled signal along the direction of decreasing the number of layers j by using the signal composition operator ψ ↓ J: VJ + 1 × WJ + 1 → VJ. Therefore, the above decomposition scheme is called dual wavelet decomposition (as shown in Figure 1).
Suppose that there are additive operators “+” and operators ψ ↓ J: VJ + 1 → VJ and ω ↓ J: WJ + 1 → VJ in signal space VJ
Then, ψ ↓ J and ω ↓ J are called signal composition operator and detail composition operator respectively, and this decomposition scheme is called non dual wavelet decomposition (as shown in Figure 2).
Due to the simple structure and morphological characteristics of Haar wavelet, the non dual wavelet decomposition scheme is used to construct the morphological Haar wavelet as follows:
Where “∨” – morphological expansion operator (maximum); “∧” – morphological erosion operator (minimum).
When the signal analysis operator in the formula adopts erosion operation, the morphological Haar wavelet constructed by the formula is the morphological Haar wavelet with erosion operator; on the contrary, the morphological Haar wavelet with inflation operator can be constructed by replacing the morphological erosion operator in the formula with morphological dilation operator. In the research, morphological Haar wavelet with expansion operator is used to denoise the transmission gear fault signal.