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01:640:357 - Topics in Applied Algebra

Prerequisite: Math 250 Introduction to Linear Algebra and Math 251 Multivariable Calculus.

Introduction to Signal and Image Processing by Discrete Fourier and Wavelet Transforms
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).

Textbook

Chapters 1-4 of Discrete Fourier and Wavelet Transforms by Roe W. Goodman
Download from the Math 357 Sakai page (01:640:357:01 Sp 16)
(available only to students registered for the course)

Published version of the textbook available in Spring 2016 from  World Scientific Publishing

Other Resources

Other Recommended Books (not required for course)

A. Jensen and A. la Cour-Harbo, Ripples in Mathematics: The Discrete Wavelet Transform
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

Exams

  • Midterm 1: Thursday, Feb. 25 (ARC 205)
  • Midterm 2: Thursday, April 14 (ARC 205)
  • Final Exam: Thursday, May 5, 8-11 AM (ARC 205)

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 )

For Project 2 you will use the Finite Fourier transform graphic user interface.
Matlab m-file. Here is the link to download this m-file:  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)

Using Matlab

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 Matlab path to this directory.

Course History

Taught by Prof. R. Goodman 2005-2008 and 2010-2014, Prof. V. Retakh 2009 and 2016, Dr. M. Thibault 2015.


Sections Taught This Semester:

 

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Information posted prior to the beginning of the semester is frequently tentative, or based on previous semesters. Textbooks should not be purchased until confirmed with the instructor. For generally reliable textbook information—with the exception of sections with an alphabetic code like H1 or T1, and topics courses (197,395,495)—see the textbook list.

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