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.
This paper reviews the area of coherent optical techniques for performing either one-dimensional or two-dimensional linear, space-variant operations. These techniques are becoming increasingly important due to the variety of possible applications for space-variant optical processors. Examples are presented to illustrate the major categories of processors.
Berzelius failed to make use of Faraday's electrochemical laws in his laborious determination of equivalent weights.