Discrete Wavelet Transformations: An Elementary Approach with Applications

CHAPTER 9

WAVELET SHRINKAGE: AN APPLICATION TO DENOISING

We now consider the problem of denoising a digital image or audio sample. We present a method for denoising called wavelet shrinkage. This method was developed largely by Stanford statistician David Donoho and his collaborators. In a straightforward argument, Donoho [28] explains why wavelet shrinkage works well for denoising problems, and the advantages and disadvantages of wavelet shrinkage have been discussed by Taswell [76]. Vidakovic [81] has authored a nice book that discusses wavelet shrinkage in detail and also covers several other applications of wavelets in the area of statistics.

Note: The material developed in this section makes heavy use of ideas from statistics. If you are not familiar with concepts such as random variables, distributions, and expected values, you might first want to read Appendix A. You might also wish to work through the problems at the end of each section of the appendix. Two good sources for the material we use in this section are DeGroot and Schervish [27] and Wackerly et al. [82].

For ease of presentation, we develop the wavelet shrinkage method for vectors (signals and audio samples), but it is quite simple to adapt wavelet shrinkage to work on matrices (images) as well. After we have developed the wavelet shrinkage method, we use it to denoise some signals and digital images. Since we can’t hear the effects of the wavelet shrinkage method on digital audio files, we present these applications ...