Discrete Wavelet Transformations: An Elementary Approach with Applications

PREFACE

Why This Book?

How do you apply wavelets to images? This question was asked of me by a bright undergraduate student while I was a professor in the mid-1990s at Sam Houston State University. I was part of a research group there and we had written papers in the area of multiwavelets, obtained external funding to support our research, and hosted an international conference on multiwavelets. So I fancied myself as somewhat knowledgeable on the topic. But this student wanted to know how they were actually used in the applications mentioned in articles she had read. It was quite humbling to admit to her that I could not exactly answer her question. Like most mathematicians, I had a cursory understanding of the applications, but I had never written code that would apply a wavelet transformation to a digital image for the purposes of processing it in some way. Together, we worked out the details of applying a discrete Haar wavelet transform to a digital image, learned how to use the output to identify the edges in the image (much like what is done in Section 6.4), and wrote software to implement our work.

My first year at the University of St. Thomas was 1998-1999 and I was scheduled to teach Applied Mathematical Modeling II during the spring semester. I wanted to select a current topic that students could immediately grasp and use in concrete applications. I kept returning to my positive experience working with the undergraduate student at Sam Houston State University on the edge ...