Van Fleet P.J. Discrete Wavelet Transformations. An Elementary Approach with Applications

Издательство John Wiley, 2008, -571 pp.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 detection problem. I was surprised by the number of concepts from calculus and linear algebra that we had reviewed in the process of coding and applying the Haar wavelet transform. I was also impressed with the way the student embraced the coding portion of the work and connected to it ideas from linear algebra. In December 1998, I attended a wavelet workshop organized by Gilbert Strang and Truong Nguyen. They had just authored the book Wavelets and Filter Banks [73], and their presentation of the material focused a bit more on an engineering perspective than a mathematical one. As a result, they developed wavelet filters by using ideas from convolution theory and Fourier series.
I decided that the class I would prepare would adopt the approach of Strang and Nguyen and I planned accordingly. I would attempt to provide enough detail and background material to make the ideas accessible to undergraduates with backgrounds in calculus and linear algebra. I would concentrate only on the development of the discrete wavelet transform. The course would take an "applications first" approach. With minimal background, students would be immersed in applications and provide detailed solutions. Moreover, the students would make heavy use of the computer by writing their own code to apply wavelet transforms to digital audio or image files. Only after the students had a good understanding of the basic ideas and uses of discrete wavelet transforms would we frame general filter development using classical ideas from Fourier series. Finally, wherever possible, I would provide a discussion of how and why a result was obtained versus a statement of the result followed by a concise proof and example. The first course was enough of a success to try again. To date, I have taught the course seven times and developed course materials (lecture notes, software, and computer labs) to the point where colleagues can use them to offer the course at their home institutions.
As is often the case, this book evolved out of several years' worth of lecture notes prepared for the course. The goal of the text is to present a topic that is useful in several of today's applications involving digital data in such a way that it is accessible to students who have taken calculus and linear algebra. The ideas are motivated through applications — students learn the ideas behind discrete wavelet transforms and their applications by using them in image compression, image edge detection, and signal denoising. I have done my best to provide many of the details for these applications that my SHSU student and I had to discover on our own. In so doing, I found that the material strongly reinforces ideas learned in calculus and linear algebra, provides a natural arena for an introduction of complex numbers, convolution, and Fourier series, offers motivation for student enrollment in higher-level undergraduate courses such as real analysis or complex analysis, and establishes the computer as a legitimate learning tool. The book also introduces students to late-twentieth century mathematics. Students who have grown up in the digital age learn how mathematics is utilized to solve problems they understand and to which they can easily relate. And although students who read this book may not be ready to perform high-level mathematical research, they will be at a point where they can identify problems and open questions studied by researchers today.
Introduction: Why Wavelets?
Vectors and Matrices 1.
An Introduction to Digital Images.
Complex Numbers and Fourier Series.
The Haar Wavelet Transformation.
The Haar Wavelet Transformation.
Daubechies Wavelet Transformations.
Orthogonality and Fourier Series.
Wavelet Shrinkage: An Application to Denoising.
Biorthogonal Filters.
Computing Biorthogonal Wavelet Transformations.
The JPEG2000 Image Compression Standard.
A: Basic Statistics.