Topics in Applied Algebra (Math 357)

This course begins with some topics in linear algebra not discussed in Math 250 (such as complex vector spaces, linear transformations, and change of basis). We then develop the theory of the finite Fourier transform and the new theory of discrete wavelet transforms. These transforms allow us to separate a digitized signal (or image) into low frequency components (coarse outline) and high frequency components (detailed features) in a computationally effective way. Then the signal or image can be compressed or enhanced (noise reduction) using these components.

The course will involve several MATLAB computer projects. Some prior knowledge of MATLAB is helpful but not necessary. A general familiarity with computers and basic programming skills are needed. Purchase of MATLAB software is not required, since you can use the MATLAB software in the ARC and other public computer labs at Rutgers. We will also use the public-domain wavelet software package Uvi_Wave (which runs under MATLAB), which you can download here as a zip file or as files and directories

Texts for the course

1. Excerpts from:
Steven J. Leon, Linear Algebra with Applications (6th edition) (required)
(Purchase course pack at Pequod Communications, 119 Somerset Street, New Brunswick -- don't buy the whole book)

2. Complete book:
A. Jensen and A. la Cour-Harbo, Ripples in Mathematics: The Discrete Wavelet Transform (required)
  ISBN # 3-540--41662-5, Springer-Verlag, 2001

3. Lecture notes:
Roe Goodman, Discrete Fourier Transform and Wavelet Transforms
(Download here as pdf file)
Last revised: 5/08/08

Web Resources

Here is the home page for the Ripples in Mathematics text.

Here is a page of Fast Fourier Transform links.

Here is a page of Wavelet Information.

Here is the Wikipedia article on Wavelets.

Here is an article on Image Compression and the JPEG 2000 algorithm based on the CDF Wavelet transform (which is studied in this course).

Here is the MIT Open CourseWare page of Gilbert Strang's course Wavelets and Filter Banks.

Course Materials

• General information and grading policy
• Course Syllabus
• Homework Problems

Exams

• Midterm 1: Thursday, Feb. 28 (ARC 203)
    Review session (optional): Wednesday, Feb. 27 at 7:30 pm (Hill 425)

• Midterm 2: Thursday, April 17 (ARC 203)
    Review session (optional): Monday, April 14 at 7:30 pm (Hill 425)

• Final Exam: Friday, May 9, 8-11 AM (ARC 203)
    Review session (optional): Wednesday, May 7 at 7:00 pm (Hill 425)
    Extra office hours:   3:00-5:00 Wednesday, May 7

MATLAB Assignments

• Project 1: Visualizing Linear Transformations   (Due February 11)
      (pdf format)     (postscript format)

• Project 2: Convolution and Discrete Fourier Transform   (Due March 3)
      (pdf format)     (postscript format)

• Project 3: Haar Wavelet Transform   (Due March 24)
      (pdf format)     (postscript format)

• Project 4: Implementation of Wavelet Transforms   (Due April 7)
      (pdf format)     (postscript format)

• Project 5: Image Analysis by Wavelet Transforms   (Due April 28)
      (pdf format)     (postscript format)