CRN 23860

Discrete Wavelets and Image Processing

Syllabus

• Time and Place. MW 18:30-19:50 in UGLC 230

• Instructor. Helmut Knaust, Bell Hall 124, hknaust@utep.edu, 747-7002

• Office Hours. After class, or by appointment.

• 
  Textbook. Patrick Van Fleet. Discrete Wavelet Transformations: An Elementary Approach with Applications. Wiley-Interscience. The book has not appeared yet, but is due to come out in a week or two; in the meantime I will provide xerox copies of the introductory chapters.

• Prerequisites. The course has a very applied flavor. Knowledge of fundamental Calculus is required; some limited familiarity with matrices may be helpful. You will use Mathematica extensively, but prior knowledge is not expected. On the other hand, this is an advanced mathematics course, so you should have some mathematical maturity.

• Course Objectives. We will study a recent topic in mathematics (discrete wavelets), and how it is applied to the practical problem of image compression. While some of the underlying ideas go back to Joseph Fourier (1768-1830) and Alfred Haar (1885-1933), most of the material you will see is not older than 30 years. During the course you should expect (and I will expect) that you make considerable progress in the following areas:

1. Develop an understanding of the theoretical underpinnings of wavelet transforms and their applications.
2. Learn how to use a computer algebra system for mathematical investigations, as a computational and visualization aid, and for the implementation of mathematical algorithms.
3. Get a flavor of the ideas and issues involved in applying mathematics to a relevant engineering problem.
4. Be able to give and defend a mathematical presentation to a group of your peers.

• Class Participation and Activities. You are expected to always actively participate in class. There will also be some organized in-class activities including a project presentation. Students will present their final projects on May 7, 19:00-21:00. Your participation grade will contribute 20% to your grade.

• Homework. I will regularly assign homework. The homework will not be collected, but discussed in class.

• Tests. Exams will be given on the following dates: Monday, March 3, and Monday, April 28. Each exam counts 20% of your grade.

• Projects. You will complete several programming projects and a comprehensive final project. Some class time will be dedicated to work on these projects. The projects will contribute a combined total of 40% to your grade.

• Time Requirement. I expect that you spend an absolute minimum of six hours a week outside of class on reading the textbook, preparing for the next class, reviewing your class notes, and completing homework and project assignments. Not surprisingly, it has been my experience that there is a strong correlation between class grade and study time.

• Attendance. You are strongly encouraged to attend class. Students with five absences (excused or unexcused) will be dropped from the course with a grade of "F".

Mathematica Materials

Notebooks are written for Mathematica Version 6. "DW.m" last upload: 24-Apr-2008 11:23

• Mathematica Notebooks (directory)
• Images (download)

Projects

Final Projects

• Project 5A
• Project 5B
• Project 5C
• Project 5D

Projects should be emailed to hknaust@utep.edu on the due date before the beginning of class. Please make sure to delete the output in Mathematica notebooks you are attaching.

• Project 4 - due 4/10/08
• Project 3 - due 3/12/08
• Project 2 - due 2/18/08
• Project 1 - due 1/30/08

Homework

• 4/9/08: 5.2, 5.4, 5.11, 5.15. 5.25
• 3/17/08: Ex. 1-6 (handout)
• 3/12/08: 4.28, 4.32, 4.34bc
• 2/25/08: 3.21a, 3.23, 3.24a, 3.27, 3.40
• 2/11/08: 3.3, 3.7, 3.9, 3.10
• 1/16/08: 2.17, 2.22, 2.27
• 1/14/08: 2.10,2.13

Reference Material

• Rajendra Bhatia. Fourier Series.
• Albert Boggess & Francis J. Narcowich. A First Course in Wavelets with Fourier Analysis.
• Ingrid Daubechies. Ten Lectures on Wavelets.
• Michael W. Frazier. An Introduction to Wavelets Through Linear Algebra.
• Richard J. Gaylord, Samuel N. Kamin & Paul R. Wellin. Introduction to Programming with Mathematica.
• Rafael C. Gonzalez & Richard E. Woods. Digital Image Processing, 3rd ed.
• Arne Jensen & Anders la Cour-Harbo. Ripples in Mathematics: The Discrete Wavelet Transform.
• Thomas W. Körner. Fourier Analysis.
• Roman Maeder. Programming in Mathematica, 2nd ed.
• Patrick Van Fleet. Discrete Wavelet Transformations: An Elementary Approach with Applications.
• David F. Walnut. An Introduction to Wavelet Analysis.

Other Materials

• Wim Sweldens: The Lifting Scheme: A New Philosophy in Biorthogonal Wavelet Constructions
• C. M. Brislawn: Fingerprints Go Digital (Notices of the American Mathematical Society 42(11), 1995 , pp. 1278-83)
• Handout: Multi-Resolution Analysis
• C.E. Shannon: A Mathematical Theory of Communication (Bell System Technical Journal 27 (1948), pp. 379-423, pp. 623-656)
• Math Professor Wins Grammy
• S. Hughes, E. Brevdo, and I. Daubechies: Identifying Hidden Features: A Digital Characterization of Van Gogh’s Style (Proceedings of the First International Workshop on Image Processing for Artist Identification (2007), pp. 177-87)
• David Austin: What is JPEG? (Notices of the American Mathematical Society 55(2), 2008, pp. 226-9)
• Class Announcement
• Maxima

• Coefficients of Daubechies Wavelet Filters