Math 357 - Topics in Applied Algebra

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 audio signal (or two-dimensional 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 noise can be removed 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).

Summer 2013 Schedule

Texts for the course

1. Lecture notes:
Roe Goodman, Introduction to Signal and Image Processing by Discrete Fourier Transform and Wavelet Transforms (required)
  Download the   pdf file

3. Excerpts from:
Steven J. Leon, Linear Algebra with Applications (6th edition) (required)
Purchase a copy of selected parts of the book at the Undergraduate Math Office, Hill 303
Don't buy the whole book!

Web Resources

• Homepage for Ripples in Mathematics text
• Stanford FFT Laboratory
• Fast Fourier Transform links
• Wavelet Information
• Wikipedia 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 an article on Discrete Wavelet Transformations and Undergraduate Education by C. Beneteau and P. J. Van Fleet (from Notices of the American Mathematical Society, May 2011) that outlines all the mathematical topics covered in the course with many interesting examples of image processing.
• Here is the MIT Open CourseWare page of Gilbert Strang's course Wavelets and Filter Banks.

Recommended Books Emphasizing Applications

James S. Walker, A Primer on Wavelets and Their Scientific Applications (Second Edition)
  (not required for course)

Course Materials

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

Exams

• Midterm 1: Thursday, Feb. 28 (ARC 205)
  Review session: Wednesday, Feb. 27 at 7:30 pm (Hill 425)
• Midterm 2: Thursday, April 18 (ARC 205)
  Review session: Wednesday, April 17 at 7:30 pm (Hill 425)
• Final Exam: Thursday, May 9, 8-11 AM (ARC 205)
  Review session: Wednesday, May 8 at 4-6 PM (Hill 425)

MATLAB Assignments

• Project 1: Digital Signals and Computer Graphics   (Due February 11)
      (pdf format)

• Project 2: Convolution and Discrete Fourier Transform   (Due March 4)
      (pdf format)

• Project 3: Haar Wavelet Transform   (Due March 25)
      (pdf format)

• Project 4: Implementation of Wavelet Transforms   (Due April 8)
      (pdf format)

• Project 5: Image Analysis by Wavelet Transforms   (Due April 29)
    (pdf format)

    Uvi_Wave zip file (unzip the file to use the package)

    Uvi_Wave directories and files

Note: You can run Matlab on your own computer (without buying the program) by using the Rutgers X-application server.

• Click on this apps server link.
• Log in to the apps server using the connect button at the upper right-hand corner of the screen and your Rutgers NetID.
• From the Main Menu at the lower left corner of the apps server toolbar, click on Education and then on Matlab
• From the Main Menu click on Internet and then on Firefox Web Browser to access the Uvi_Wave files from the math 357 course web page.
• Copy the Uvi_Wave files into a directory that your create on the X-apps server. Then set the Matlab path to this directory.

Course History

Taught by Professor Goodman 2005-2008, Professor Retakh 2009, Professor Goodman 2010-2013.