New Convolution and Correlation Theorems for the Linear Canonical Wavelet Transform

The linear canonical wavelet transform (LCWT) is the generalization of the classical wavelet transform (WT) and the linear canonical transform (LCT). It has been proven to be a powerful mathematical tool and is widely used in signal processing, image processing, optics, and other fields. However, some basic results of this transform are not yet mature, such as convolution and correlation theorems. Therefore, this paper discusses the convolution and correlation theorems of the LCWT. Firstly, we review the basic theory of the WT, the LCT, and the LCWT. Secondly, we define the new convolution and correlation operators, and deduce the convolution and correlation theorems of the LCWT. The results show that they are similar in other joint space/spatial-frequency or time/frequency representations. Finally, we give the filter design method of the proposed convolution theorem in the LCWT domain, which provides us with more possibilities to consider performing spatially varying filtering operations in the LCWT domain.