Discrete Wavelet Transformations E-Book

Patrick Van Fleet

• Herausgeber: John Wiley & Sons
• Kategorie: Wissenschaft und neue Technologien
• Sprache: Englisch

Seitenzahl: 855

Veröffentlichungsjahr: 2019

Beschreibung

Updated and Expanded Textbook Offers Accessible and Applications-First Introduction to Wavelet Theory for Students and Professionals The new edition of Discrete Wavelet Transformations continues to guide readers through the abstract concepts of wavelet theory by using Dr. Van Fleet's highly practical, application-based approach, which reflects how mathematicians construct solutions to challenges outside the classroom. By introducing the Haar, orthogonal, and biorthogonal filters without the use of Fourier series, Van Fleet allows his audience to connect concepts directly to real-world applications at an earlier point than other publications in the field. Leveraging extensive graphical displays, this self-contained volume integrates concepts from calculus and linear algebra into the constructions of wavelet transformations and their applications, including data compression, edge detection in images and denoising of signals. Conceptual understanding is reinforced with over 500 detailed exercises and 24 computer labs. The second edition discusses new applications including image segmentation, pansharpening, and the FBI fingerprint compression specification. Other notable features include: * Two new chapters covering wavelet packets and the lifting method * A reorganization of the presentation so that basic filters can be constructed without the use of Fourier techniques * A new comprehensive chapter that explains filter derivation using Fourier techniques * Over 120 examples of which 91 are "live examples," which allow the reader to quickly reproduce these examples in Mathematica or MATLAB and deepen conceptual mastery * An overview of digital image basics, equipping readers with the tools they need to understand the image processing applications presented * A complete rewrite of the DiscreteWavelets package called WaveletWare for use with Mathematica and MATLAB * A website, www.stthomas.edu/wavelets, featuring material containing the WaveletWare package, live examples, and computer labs in addition to companion material for teaching a course using the book Comprehensive and grounded, this book and its online components provide an excellent foundation for developing undergraduate courses as well as a valuable resource for mathematicians, signal process engineers, and other professionals seeking to understand the practical applications of discrete wavelet transformations in solving real-world challenges.

Leseprobe

Discrete Wavelet Transformations

An Elementary Approach with Applications

Patrick J. Van Fleet

University of St. ThomasSt. Paul, Minnesota

Second Edition

Copyright

This edition first published 2019

© 2019 John Wiley & Sons, Inc.

Edition History

Wiley‐Interscience, (1e in English, 2008)

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Library of Congress Cataloging‐in‐Publication Data

Names: Van Fleet, Patrick J., 1962‐ author.

Title: Discrete wavelet transformations : an elementary approach with applications / Patrick J. Van Fleet (University of St. Thomas).

Description: 2nd edition. | Hoboken, NJ : John Wiley & Sons, Inc., [2019] | Includes bibliographical references and index. |

Identifiers: LCCN 2018046966 (print) | LCCN 2018055538 (ebook) | ISBN 9781118979327 (Adobe PDF) | ISBN 9781118979310 (ePub) | ISBN 9781118979273 (hardcover)

Subjects: LCSH: Wavelets (Mathematics) | Transformations (Mathematics) | Digital images‐Mathematics.

Classification: LCC QA403.3 (ebook) | LCC QA403.3 .V36 2019 (print) | DDC 515/.2433‐dc23

LC record available at https://lccn.loc.gov/2018046966

Cover design: Wiley

Cover image: Courtesy of Patrick J. Van Fleet

Dedication

For Andy

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 showcased, rigor is sometimes sacrificed for ease of presentation, and problem sets and labs sometimes elicit subjective answers. It is difficult as a mathematician to minimize the attention given to rigor and detail (convergence of Fourier series, for example) and it is often irresistible to omit ancillary topics (the FFT and its uses, other filtering techniques, wavelet packets) that almost beg to be included. It is important to remind yourself that you are preparing students for more rigorous mathematics and that any additional topics you introduce takes time away from a schedule that is already quite full.

I have prepared the book and software so that instructors have several options when offering the course. When I teach the course, I typically ask students to write modules (subroutines) for constructing wavelet filters or computing wavelet transformations and their inverses. Once the modules are working, students use them for work on inquiry‐type labs that are included in the text. For those faculty members who wish to offer a less programming‐intensive course and concentrate on the material in the labs, the complete software package is available for download.

In lieu of a final exam, I ask my students to work on final projects. I usually allow four or five class periods at the end of the semester for student work on the projects. Project topics are either theoretical in design, address a topic in the book we did not cover, or introduce a topic or application involving the wavelet transformation that is not included in the book. In many cases, the projects are totally experimental in nature. Projects are designed for students to work either on their own or in small groups. To make time for the projects, I usually have to omit some chapters from the book. Faculty members who do not require the programming described above and do not assign final projects can probably complete a large portion of the book in a single semester. Detailed course offerings follow in the preface to the second edition.

What You Won't Find in This Book

There are two points to be made in this section. The first laments the fact that several pet topics were omitted from the text and the second provides some rationale for the presentation of the material.

For those readers with a background in wavelets, it might pain you to know that except for a short discussion in Chapter 1 and asides in Sections 4.1 and 9.4, there is no mention of scaling functions, wavelet functions, dilation equations, or multiresolution analyses. In other words, this presentation does not follow the classical approach where wavelet filters are constructed in L2(ℝ). The approach here is entirely discrete – the point of the presentation is to draw as much as possible from courses early in a mathematics curriculum, augment it with the necessary amount of Fourier series, and then use applications to motivate more general filter constructions. For those students who might desire a more classical approach to the topic, I highly recommend the texts by Boggess and Narcowich [7], Frazier [42], Walnut [100] or even Ruch and Van Fleet [77].

Scientists who have used nonwavelet methods (Fourier or otherwise) to process signals and images will not find them discussed in this book. There are many methods for processing signals and images. Some work better than wavelet‐based methods, some do not work as well. We should view wavelets as simply another tool that we can use in these applications. Of course, it is important to compare other methods used in signal and image processing to those presented in this book. If those methods were presented here, it would not be possible to cover all the material in a single semester.

Some mathematicians might be disappointed at the lack of rigor in several parts of the text. This is the conundrum that comes with writing a book with modest prerequisites. If we get too bogged down in the details, we lose sight of the applications. On the other hand, if we concern ourselves only with applications, then we never develop a deep understanding of how the mathematics works. I have skewed my presentation to favor more applications and fewer technical details. Where appropriate, I have tried to point out arguments that are incomplete and provided suggestions (or references) that can make the argument rigorous.

Finally, despite the best efforts of friends, colleagues, and students, you will not find an error‐free presentation of the material in this book. For this, I am entirely to blame. Any corrections, suggestions, or criticisms will be greatly appreciated!

P. J. VAN FLEET

St. Paul, Minnesota USA

June 2007

PREFACE

What Has Changed?

Prior to the first edition appearing in January 2008, I had taught a version of this course eight times. Supported by National Science Foundation grants (DUE) I began offering workshops and mini‐courses about the wavelets course in 2006. Several of the attendees at these events taught a wavelets course at their home institutions and fortunately for me, provided valuable feedback about the course and text. A couple of popular points of discussion were the necessity of a sophomore linear algebra class as a prerequisite and possibility of using non‐Fourier methods for the ad hoc filter development. These two points were the primary motivating factors for the creation of the second edition of the text.

While I am not ready to claim that a sophomore linear algebra course is not needed as a prerequisite for a course in wavelets, I am comfortable saying that for a course constructed from particular topics in the book, the necessary background on vectors and matrices can be covered in Chapter 2. In such a case, a prerequisite linear algebra course serves to provide a student with a more mathematical appreciation for the material presented.

The major change in the second edition of the book is the clear delineation of filter development methods. The book starts largely unchanged through the first three chapters save for the inclusion of a section on convolution and filters in Chapter 2. The Haar wavelet transformation is introduced in Chapter 4 as a tool that efficiently concentrates the energy of a signal/image for applications such as compression. Daubechies filters and certain biorthogonal spline filter pairs are developed in Chapters 5 and 7 using conditions on the transformation matrix as well as requirements imposed on associated highpass filters. In particular, the concept of annihilating filters introduced in Section 2.4 takes the place of Fourier series in the derivation of wavelet filters. The ad hoc construction is limiting so complex numbers and Fourier series are introduced in Chapter 8 and these ideas are used in subsequent chapters to construct orthogonal filters and biorthogonal filter pairs in a more systematic way.

There have been chapter additions and subtractions as well. Gone is the chapter (previously Chapter 11) on algorithmic development of the biorthogonal wavelet transformation from the first edition although the material on symmetric biorthogonal filter pairs appears in Section 7.3 in this edition. Two new chapters have been added. Wavelet packets are introduced in Chapter 10. A main feature of this chapter is coverage of the FBI Wavelet Scalar Quantization Specification for compressing digital fingerprints. The LeGall wavelet transformation and lifting was discussed in Section 12.3 of the first edition of the text. This material now serves as an introduction for an entire chapter (Chapter 11) on the lifting method. This chapter makes heavy use of the Z‐transformation and serves as a more mathematically rigorous topic than what typically appears in the text.

Finally, due to strong reader feedback, the clown image has been removed from the new edition!

To the Instructor

For those instructors who have taught a course from the first edition of the book, you will notice a significant reordering of some topics in addition to a couple of new chapters. The book now naturally separates into two parts. After some background review in matrix algebra and digital images, the first part of the text can be viewed as ad hoc wavelet filter development devoid of Fourier methods. Old applications such as image edge detection, image compression, and de‐noising and still present. Applications of wavelet transformations to image segmentation and image pansharpening are new to this edition. When I last taught the course, I primarily covered Chapter 2–7. I also work through the labs (with a fair amount of in‐class time devoted to them) that emphasize the software development of transformations as well as applications. For a group of students with a good background in linear algebra, I would do a cursory review of Chapters 3 and 4 and then cover Chapters 8 – 11 with Chapter 12 optional. I would not assign much lab work instead emphasizing the live examples so that students can get an appreciation for applications. Detailed course outlines are given later in the preface.

If you do decide to have your students work through the transformation development labs (Labs 4.1–4.3, 5.1–5.3, 7.1, and 7.2), you should have your students work Problems 4.8, 4.10, 5.12, 5.13, 5.22, 5.23, 7.13, 7.14, 7.32, and 7.33. These problems are similar in nature but give some insight into developing efficient algorithms for computing wavelet transformations and their inverses. Problem 2.23 is useful for the construction of two‐dimensional transformations.

Problem Sets, Software Package, Labs, Live Examples

Problem sets and computers labs are given at the end of each section. There are 521 problems and 24 computer labs in the book. Some of the problems can be viewed as basic drill work or an opportunity to step through an algorithm “by hand.” There are basic problem‐solving exercises designed to increase student understanding of the material, problems where the student is asked to either provide or complete the proof of a result from the section, and problems that allow students to furnish details for ancillary topics that we do not completely cover in the text. To help students develop good problem‐solving skills, many problems contain hints, and more difficult problems are often broken into several steps. The results of problems marked with a ⋆ are used later in the text.

With regards to software, there are some new changes. The DiscreteWavelets package has been replaced by the package WaveletWare. The new package is a complete rewrite of DiscreteWavelets and among the many new improvements and enhancements are new visualization routines as well as routines to compute wavelet packet transformations.

Labs have become more streamlined. The primary focus of most labs are the development of algorithms either to construct wavelet filter (pairs) or wavelet transformations. Some labs do investigate new ideas or extensions of ideas introduced in the parent section. I often require my students to complete final projects and in these instances, I assign selected “programming” labs so that the students will have developed filter or transformation modules for use on the computational aspects of the projects.

Most of the computations are now addressed in the form of live examples. There are 123 examples in the text and 91 of these can be reproduced via live example software. Moreover, live example software often contains further investigation suggestions or “things to try” problems. Those examples that are live examples conclude with a link to relevant software.

Text Web Site

The software package, computer labs and live examples are available in MATLAB and Mathematica and are available on the course web site

stthomas.edu/wavelets.

Text Topics

The book begins with a short chapter (essay) entitled Why Wavelets? The purpose of this chapter is to give the reader information about the mathematical history of wavelets, how they are applied, and why we would want to learn about them. In Chapter Chapter 2, we review basic ideas about vectors and matrices and introduce the notions of convolution and filters. Vectors and matrices are central tools in our development. We review vector inner products and norms, matrix algebra, and block matrix arithmetic. The material in Sections 2.1 and 2.2 is included to make the text self‐contained and can be omitted if students have taken a sophomore linear algebra course. Students rarely see a detailed treatment of block matrices in lower level math courses, and because the text makes heavy use of block matrices, it is important to cover Section 2.3. The final section introduces the convolution product of bi‐infinite sequences and filters. After connecting filters to convolution, we discuss finite impulse response, lowpass and highpass filters. A new concept to the second edition is that of filters that annihilate polynomial data. The ad hoc construction of wavelet filters in Chapters 5 and 7 makes use of annihilating filters.

Since most of our applications deal with digital images, we cover basic ideas about digital images in Chapter 3. We show how to write images as matrices, discuss some elementary digital processing tools, and define popular color spaces. The application of image compression is visited often in the text. The final step in an image compression algorithm is the encoding of data. There are many sophisticated coding methods but for the ease of presentation, we utilize Huffman coding. This method is presented in Section 3.3. Quantization is a typical step in image compression or audio denoising. Quantization functions specific to particular applications as needed in the text, but for generic quantization needs, we use cumulative energy. This function and its role in compression is presented in Section 3.4. Entropy and peak signal‐to‐noise ratio are tools used to measure the effectiveness of image compression methods. They are also introduced in Section 3.4.

We learn in Section 3.3 that encoding image data without preprocessing it leads to an inefficient image compression routine. The Haar wavelet transformation is introduced in Chapter 4 as one possible way to process data ahead of encoding it. The first three sections discuss the one‐dimensional transformation, the iterated transformation, and the two‐dimensional transformation. The chapter concludes with a section on applications of the transformation to image compression and image edge detection.

Chapters 5 and 7 consider ad hoc construction of filters needed to create wavelet transformation matrices. In Chapter 5, we construct filters from Daubechies' family of orthogonal filters [32]. The motivation for this construction is the inability for the Haar filter to correlate data – longer filter are better in this regard and can be constructed so that the wavelet transformation matrix is orthogonal. We learn in Chapter 7 that symmetric filters are best equipped to handle values at “beginning” or “end” of a data set. Unfortunately, Daubechies filters, other than the Haar filter, are not symmetric, so we continue our ad hoc construction by deriving biorthogonal spline filter pairs. Each filter is symmetric but the matrices they generate are not. However, the wavelet matrices are related in that the inverse of one is the transpose of the other. Thus the construction preserves the basic structure of the wavelet transformation matrix. After deriving pairs of lengths 3 and 5 (Section 7.1) and 4 and 8 (Section 7.2), we discuss ways to exploit the symmetry of the filters in order to further improve the wavelet transformation's ability to process data ahead of encoding in image compression. Chapter 7 concludes with a section on applications of biorthogonal wavelet transformations to image compression and image panharpening. Sandwiched between Chapters 5 and 7 is a chapter that considers wavelet‐based methods for denoising data. In particular, we introduce the VISUShrink [36] and SUREShrink [37] methods and illustrate how the latter can be employed in the application of image segmentation.

The ad hoc construction methods employed in Chapters 5 and 7 are limiting. In particular, we are unable to use these ideas to construct the entire family of biorthogonal spline filter pairs or an orthogonal family of filters called Coiflets. In order to perform more systematic construction of wavelet filters, we need ideas from complex analysis and Fourier series. These concepts are introduced in Chapter 8. The chapter concludes with a section that connects Fourier series to filter construction. The background material in Chapter 8 is implemented in Chapter 9 where Fourier methods are used to characterize filter construction (both lowpass and highpass) in the Fourier domain. The general results are given in Section 9.1 and Daubechies filters, Coiflets, biorthogonal spline filter pairs and the CDF97 biorthogonal filter pair are constructed in subsequent sections.

Chapters 10 and 11 are new to the second edition of the book. The wavelet packet transformation (Chapter 10) is a generalization of the wavelet transformation and allows facilitates a more application‐dependent decomposition of data. In Section 10.1, the wavelet packet transformation is introduced for both one‐ and two‐dimensional data and the best basis algorithm is outlined in Section 10.2 as a way of determining the most efficient way to transform the data, relative to a given cost function. Probably the most well‐known application of the wavelet packet transformation is the FBI Wavelet Scalar Quantization Specification for compressing digital fingerprints and this method is presented in Section 10.3. The first edition of the book contained a section on the LeGall filter and its implementation via lifting. The immediate application of this filter and lifting is in the lossless compression component of the JPEG2000 image compression standard. In the lossless compression case, it is imperative that integer‐valued data are mapped to integers so that the quantization step can be skipped. Computation of the wavelet transformation via lifting facilitates this step. Chapter 11 begins with lifting and the LeGall filter pair and then in Section 11.2 introduces the Z‐transform and Laurent polynomials. These are the necessary tools to perform lifting for any given orthogonal filter or biorthogonal filter pair. The general lifting method is given in Section 11.3 with three example constructions give in the final section of the chapter.

The book concludes with a presentation (Chapter 12 of the JPEG2000 image compression standard. This standard makes use of lifting for lossless compression and the CDF97 filter pair in conjunction with the biorthogonal wavelet transformation for lossy compression.

Course Outlines

The book contains more material than can be covered in a one‐semester course and this allows some flexibility when teaching the course.

I taught a version of the course using this manuscript during the Fall 2016 semester. The course was comprised of 40 class periods meeting three days a week for 65 minutes per meeting. The students were best suited for an applications‐driven version of the course so the design included time for in‐class lab work as well as work on final projects that occurred during the last seven meetings of the class. Here is a breakdown of the material covered.

1 meeting: Outline the ideas in Chapter 1.

2 meetings: Exams.

4 meetings: Chapter 2 with a main focus on the material in Section 2.4.

7 meetings: Chapter 3. We did not cover conversion to and from HSI space and spent roughly two days on in‐class labs.

10 meetings: Probably the most important material for this group of students is in Chapter 4. Along with the chapter material, we worked on three in‐class labs. The labs consisted of writing code to implement the one‐ and two‐dimensional Haar wavelet transformation and then labs each on image compression and image edge detection.

4 meetings: Sections 5.1 and 5.2 with one in‐class lab on the implementation of the wavelet transformation.

3 meetings: Sections 7.1 and 7.3 with some time allotted for an in‐class lab on image compression.

2 meetings: Section 11.1 and an in‐class lab on the associated lifting method

7 meetings: Final projects. Topic material was taken from Sections 5.3, 7.4, Chapter 10 and ideas in [1] on CAPTCHAs.

For a more mathematically mature audience, I might use the following outline.

1 meeting: Outline the ideas in Chapter 1.

3 meetings: Exams.

2 meetings: Section 2.4.

3 meetings: Chapter 3