Math 357 - Topics in Applied Algebra

This course begins with some topics in linear algebra not covered in Math 250 (such as complex vector spaces, linear transformations, and Fourier series). It then develops the theory of the discrete Fourier transform and the new theory of discrete wavelet transforms. These mathematical tools can 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 some 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).

Fall 2019 Schedule

Textbook
  Discrete Fourier and Wavelet Transforms by Roe W. Goodman (World Scientific Publishing)
Available in hardcover, paperback, and e-book formats.

Other Resources

• Survey of course topics by Roe Goodman
  Discrete Fourier and Wavelet Transforms: Mathematical Microscopes for Signal Processing
  
• Fast Fourier Transform links
• Wikipedia page on Wavelets
• An article on Image Compression and the JPEG 2000 algorithm based on the CDF Wavelet transform (which is studied in this course).
• 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.
• The MIT Open CourseWare page of Gilbert Strang's course Wavelets and Filter Banks.
• Wavelet books and links

Other Recommended Books (not required for course)

S. Allen Broughton and Kurt Bryan, Discrete Fourier Analysis and Wavelets

James S. Walker, A Primer on Wavelets and Their Scientific Applications (Second Edition)

Course Materials

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

MATLAB Assignments

• Project 1: Digital Signals and Vector Graphics  
      (pdf format)
• Project 2: Convolution and Discrete Fourier Transform  
      (pdf format)
• Project 3: Haar Wavelet Transform  
      (pdf format)
• Project 4: Implementation of Wavelet Transforms  
      (pdf format)
• Project 5: Image Analysis by Wavelet Transforms  
    (pdf format)

    fftgui

For Projects 4 and 5 you will use the Uvi_Wave collection of Matlab m-files for wavelet transforms (developed at the University of Vigo, Spain). Here is the link to download these m-files:

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

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 fftgui.m file and the whole unzipped Uvi_Wave directory into a directory that your create on the X-apps server. Then set the Ma

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