The convolution theorem for the continuous wavelet tranform

We study the application of the continuous wavelet transform to perform signal filtering processes. We first show that the convolution and correlation of two wavelet functions satisfy the required admissibility and regularity conditions. By using these new wavelet functions to analyze both convolutions and correlations, respectively, we derive convolution and correlation theorems for the continuous wavelet transform and show them to be similar to that of other joint spatial/spatial-frequency or time/frequency representations. We then investigate the effect of multiplying the continuous wavelet transform of a given signal by a related transfer function and show how to perform spatially variant filtering operations in the wavelet domain. Finally, we present numerical examples showing the usefulness of applying the convolution theorem for the continuous wavelet transform to perform signal restoration in the presence of additive noise. (C) 2003 Elsevier B.V. All rights reserved.