Course Proposal Introduction to Wavelets
Wavelet analysis is the unification of theories that were developed separately in many differing fields. These fields include, but are not limited to computer graphics, computer vision, medical imaging, signal processing, mathematics, geophysics, and engineering. Wavelets, in some sense, can be thought of as a generalization of the techniques of Fourier analysis. In a separate sense, however, wavelets are more of a complimentary tool to Fourier analysis rather than a generalization. Fourier analysis consists of analyzing which frequencies contribute and how much to a signal. The applications of this theory have extended far beyond what Fourier could have imagined. For example, this theory has allowed the telephone to become a practical form of communication. Using Fourier coefficients, signals can be temporarily shifted to different frequencies allowing a single phone line to transmit multiple signals simultaneously. A second example of the applications of Fourier analysis is in image compression. It allows one to extract the low frequency components of an image, the ones that contain the most visual content, while ignoring the high frequency components. It is this idea that lies at the heart of JPEG compression, one of the current benchmarks in lossy image compression. While Fourier analysis is a tremendous theoretical tool, its feasibility in applications did not become clear until the advent of the Fast Fourier Transform. This clever factorization was able to reduce running times from O(n2) to O(n log n). Such a reduction in the number of operations allows for the incorporation of the Fourier transform into applications where calculations must be done in real-time (e.g. telephone conversations). Where does Fourier analysis break down? Fourier analysis is a global scheme. While you are able to analyze the frequencies that make up a signal (and do so quickly), the local properties of the signal cannot be easily detected from the Fourier coefficients. For example, if your signal was a recording of a drum being struck once and silence otherwise, we could look at the Fourier coefficients to analyze the frequencies, but without significant efforts, we would not be able to tell when the beat of the drum happened. It is this phenomenon that is known as Heisenberg’s Uncertainty Principle. It says that a signal cannot be simultaneously localized in time and frequency. Wavelets, thus, are an attempt to overcome this shortcoming of Fourier analysis. They provide a way to do time-frequency analysis. The idea is that one chooses a “mother wavelet”, i.e. a function subject to some conditions like mean-value 0. Rather than using the pure harmonics eikx as in Fourier analysis, one uses shifted and dilated versions of the mother wavelet. By using a 2-variable base (one for the amount of shift and one for the amount of dilation), you are able to introduce enough redundancy to maintain the local properties of the original function.
While it may seem that such redundancy could only come at the cost of computational time, this is not entirely true. With the use of convolutions and filters, a Fast Wavelet Transform has also been developed which enables wavelets to be used in many of the same places where Fourier analysis was once used. In the proposed course, we would give an introductory overview of the topics discussed above. The course would begin with a review of linear algebra and an introduction to basic topics in complex analysis. We would move on to discuss the Fourier transform, the Fast Fourier Transform, and their applications. These topics should give enough background to appreciate the beauty of the wavelet transform. It is from here that we would explore wavelets and their applications. Such a course topic provides many opportunities that are not common among courses in the current mathematics curriculum at Johns Hopkins University. Rarely is there a topic that can be presented at a 300-level and is yet so applicable. In fact, unlike most fields in mathematics that are developed first in theory and then applied, the mathematical theory behind wavelets came after many of the applications. The diversity of the applications should appeal to students throughout the JHU community and should invite many students in other disciplines back to the mathematics department. The final project will give students the opportunity to and assistance with exploring applications to their particular fields of interest. Wavelets are also unique in their ability to introduce undergraduates to “new” math. Much of the material covered in today’s undergraduate mathematics courses was developed hundreds of years ago. While some of the results that are used with wavelets are nearly a hundred years old, the unification of the techniques and the development of the mathematical theory behind wavelets began only 16 years ago. Such recent results will hopefully not only captivate the interests of the students, but also give the undergraduate students a rare glimpse into mathematical research. The proposed course is an appropriate 300-level course. One of the most difficult things that I remember about being an undergraduate mathematics major is making the transition from computationally intensive courses, like calculus, to the more theoretical upper-level courses. A good 300-level course can greatly aid students in making this transition by providing a nice mix of computational and theoretical results. Throughout the semester, students will be introduced to the theoretical and be expected to begin proving results. By not losing sight of the applications, one can present increasingly abstract material in a way that is comprehensible to an undergraduate student.
October 10, 2000
Dear Committee Members: Please accept my application for the 2000 Dean’s Teaching Fellowship. I would like to propose a 300-level mathematics course titled “An Introduction to Wavelets”. Wavelets can be thought of as an extension of Fourier analysis that allows for localization in time. They are endlessly applicable and have been the focus of much research in recent history. I first became interested in wavelets as an undergraduate. They appealed to both my mathematics and computer science background. I was working as a research assistant in an industrial engineering lab where we were studying image compression schemes, among other things, and wavelet compression appeared to be giving a significant challenge to the long-standing benchmark, JPEG compression (which is based on Fourier analysis). A course on wavelets can fulfill many significant roles in a mathematics curriculum. Being as applicable as it is, wavelets should attract many students from diverse fields. I suspect that a number of students that would not have otherwise taken another math course at JHU may be interested in this course. By incorporating a final project, I hope to allow students to individualize a portion of the course and allow them to explore the applications that are most pertinent to their interests. Wavelets should also provide a course that will ease the transition from computationally intensive courses to the more theoretical upper-level courses for math majors. In the process, students will be introduced to some of the basic techniques that they will later encounter in a real analysis or complex analysis course. Wavelets, also, give undergraduates a rare glimpse at mathematical theory that has been developed largely within the last 20 years. Enclosed you will find my curriculum vitae and completed application form. Please do not hesitate to contact me at (410) 662-1185 or metcalfe@math.jhu.edu if I can provide any additional information. Thank you in advance for your time and consideration.
Sincerely,
Jason Metcalfe
Course Description
INTRODUCTION TO WAVELETS An overview of wavelet analysis including basic Fourier analysis, the wavelet transform, Heisenberg uncertainty, Shannon’s Sampling Theorem, multiresolution, and applications (including signal processing, computer graphics, and vision).
Prerequisite: Calculus II and Linear Algebra, or permission of instructor Credits: 3
Syllabus
Introduction to Wavelets - Fall 2001
Instructor: Jason Metcalfe Krieger 219
metcalfe@math.jhu.edu
Required Text: • • An Introduction to Wavelets through Linear Algebra by Michael W. Frazier. Springer Undergraduate Texts in Mathematics Instructor Handouts
Recommended Text: • The World According to Wavelets, Second Edition by Barbara Burke Hubbard.
Grading: 20% 10% 20% 20% 30% - Weekly Homework Assignments - Final Project - In-class Midterm 1 - In-class Midterm 2 - Final Exam
Weekly Homework Assignments – Assignments will be distributed every Monday and collected the following Monday at the beginning of class. They will be a mix of computational problems and proving theoretical results (often filling in the steps of a long proof). Final Project – The final project will be a group project. The class will be divided into groups of 2-4 people (depending on the size of the class). Each group will pick a topic from the course to expand upon, a mathematical result in wavelet analysis that we have not covered, or an application of wavelets to present to the class. Some possible examples are Spline Wavelets, Wavelets in Image Querying, and Wavelets in Fractals. Each group will submit a 4-6 page paper on the topic of their choice and give a 25-minute presentation to the class during the last week of the semester.
Course Schedule Introduction to Wavelets - Fall 2001
Week 1: Review and Prerequisites
Linear Algebra Review, Infinite Series Review, Complex Numbers, Euler’s Formula Week 2-4: Basic Fourier Analysis
Discrete and Continuous Fourier Transform, Fast Fourier Transform, Convolutions, Plancherel’s Formula, Parseval’s Formula, Inversion Formulas, Windowed Fourier Transform Week 5: Two Classical Results
Heisenberg Uncertainty Principle and the Shannon Sampling Theorem Week 6: Our First Wavelet
Haar Wavelet, Haar Basis, Definition of a Wavelet Week 7-9: The Wavelet Transform
Definition, Examples, Inversion Formula, Plancherel’s Formula, Discrete Wavelet Transform, Continuous Wavelet Transform, Fast Wavelet Transform Week 10-11: Concluding Remarks Multiresolution analysis, wavelets in higher dimensions, orthonormal wavelets with compact support Week 11-12: Applications Focusing on applications in differential equations, signal processing, computer graphics and medical imaging, computer vision, and quantum mechanics. Week 13: Student Project Presentations
Curriculum Vitae
Personal Information: Name: Jason L. Metcalfe Address: 2813 St. Paul St.; Baltimore, MD 21218 Phone: (410) 662-1185 E-mail: metcalfe@math.jhu.edu
Education: Johns Hopkins University, Baltimore, MD: 1998 – present Department of Mathematics, pursuing Ph.D. Advisor: Dr. Christopher Sogge University of Washington, Seattle, WA: 1996 – 1998 B.S. in Mathematics, Magna Cum Laude B.S. in Computer Science, Magna Cum Laude University of Dayton, Dayton, OH: 1993 – 1996 Major: Mathematics and Computer Science
Honors: William Kelso Morrill Award of Excellence in the Teaching of Mathematics, Department of Mathematics, Johns Hopkins University, 2000 William Kelso Morrill Award of Excellence in the Teaching of Mathematics, Department of Mathematics, Johns Hopkins University, 1999 – Honorable Mention Phi Beta Kappa inductee, 1998
Grants and Scholarships: Full Tuition Fellowship and Stipend. Department of Mathematics, Johns Hopkins University, 1998 – present
Microsoft / UW Department of Computer Science and Engineering Scholarship, 1997 National Academy for Science, Space, and Technology Scholarship, 1993. Presidential Scholarship. University of Dayton, 1993 – 1996. Relevant Work and Teaching Experience: Graduate Teaching Assistant. Johns Hopkins University, Department of Mathematics. 1998 – present. Instructor. 110.109, Calculus II for Engineers. Summer 2000. Instructor. Johns Hopkins University, Institute for the Academic Advancement of Youth / CTY, Center for Distance Education. EPGY Grade 1-4. 1999-2000. Teaching Assistant. Johns Hopkins University, Institute for the Academic Advancement of Youth / CTY Summer Programs. “Investigations in Engineering”. Summer 1999. Research Assistant. University of Washington, Department of Industrial Engineering. Investigating quasi-Monte Carlo sequences in image compression, information theory (Kolmogorov complexity), and quaternion geometry in virtual reality. Advisor: Dr. Tony C. Woo. 1998.
Languages: Fluent in English; familiar with French