PREFACE TO THE FIRST EDITION

(Abridged and edited)

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 transformation to a digital image, learned how to use the output to identify the edges in the image (much like what is done in Section 4.4), 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 transformation. 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 [88], 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 transformation. 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 transformations to digital audio or image files. Only after the students had a good understanding of the basic ideas and uses of discrete wavelet transformations 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 transformations 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.

To the Student

Many of us have learned a foreign language. Do you remember how you formulated an answer when your instructor asked you a question in the foreign language? If you were like me, you mentally translated the question to English, formulated an answer to the question, and then translated the answer back to the foreign language. Ultimately, the goal is to omit the translation steps from the process, but the language analogy perfectly describes an important mathematical technique.

Mathematicians are often faced with a problem that is difficult to solve. In many instances, mathematicians will transform the problem to a different setting, solve the problem in the new setting, and then transform the answer back to the original domain. This is exactly the approach you use in calculus when you learn about u-substitutions or integration by parts. What you might not realize is that for applications involving discrete data (lists or tables of numbers), matrix multiplication is often used to transform the data to a setting more conducive to solving the problem. The key, of course, is to choose the correct matrix for the task.

In this book, you will learn about discrete wavelet transformations and their applications. For now, think of the transformation as a matrix that we multiply with a vector (audio) or another matrix (image). The resulting product is much better suited than the original image for performing tasks such as compression, denoising, or edge detection. A wavelet filter is simply a list of numbers that is used to construct the wavelet matrix. Of course, this matrix is very special, and as you might guess, some thought must go into its construction. What you will learn is that the ideas used to construct wavelet filters and wavelet transformation matrices draw largely from calculus and linear algebra.

The first wavelet filters will be easy to construct and the ad hoc approached can be mimicked, to a point, to construct other filters. But then you will need to learn more about convolution and Fourier series in order to systematically construct wavelet filters popular in many applications. The hope is that by this point, you will understand the applications sufficiently to be highly motivated to learn the theory. If your instructor covers the material in Chapters 8 and 9, work hard to master it. In all likelihood, it will be new mathematics for you but the skill you gain from this approach to filter construction will greatly enhance your problem-solving skills.

Questions you are often asked in mathematics courses start with phrases such as "solve this equation," "differentiate/integrate this function," or "invert this matrix." In this book, you will see why you need to perform these tasks since the questions you will be asked often start with "denoise this signal," "compress this image," or "build this filter." At first you might find it difficult to solve problems without clear-cut instructions, but understand that this is exactly the approach used to solve problems in mathematical research or industry.

Finally, if your instructor asks you to write software to implement wavelet transformations and their inverses, understand that learning to write good mathematical programs takes time. In many cases, you can simply translate the pseudocode from the book to the programming language you are using. Resist this temptation and take the extra time necessary to deeply understand how the algorithm works. You will develop good programming skills and you will be surprised at the amount of mathematics you can learn in the process.

To the Instructor

The technique of solving problems in the transform domain is common in applied mathematics as used in research and industry, but we do not devote as much time to it as we should in the undergraduate curriculum. It is my hope that faculty can use this book to create a course that can be offered early in the curriculum and fill this void.

I have found that it is entirely tractable to offer this course to students who have completed calculus I and II, a computer programming course, and sophomore linear algebra. I view the course as a post-sophomore capstone course that strengthens student knowledge in the prerequisite courses and provides some rationale and motivation for the mathematics they will see in courses such as real analysis.

The aim is to make the presentation as elementary as possible. Toward this end, explanations are not quite as terse as they could be, applications are...