Math 357 - Topics in Applied Algebra
|
 |
Prerequisites: Math 250 Introduction to Linear Algebra and Math 251 Multivariable Calculus.
Introduction to Signal and Image Processing by
Discrete Fourier Transform and Wavelet Transforms
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), which you can download here as a zip file or as files and directories
Summer 2012 Schedule
This course is taught during the Spring term.
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
Revised 3/21/12
A. Jensen and A. la Cour-Harbo, Ripples in Mathematics: The Discrete Wavelet Transform (required)
Springer ISBN # 3-540--41662-5
Purchase at Rutgers book store or online
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
- Ripples in Mathematics text, homepage
- 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
S. Allen Broughton and Kurt Bryan, Discrete Fourier Analysis and Wavelets
(not required for course)
James S. Walker, A Primer on Wavelets and Their Scientific
Applications (Second Edition)
(not required for course)
Course Materials
Exams
- Midterm 1: Thursday, Feb. 23 (ARC 205)
- Review session: Wednesday, Feb. 22 at 7:30 pm (Hill 425)
- Midterm 2: Thursday, April 12 (ARC 205)
- Review session: Wednesday, April 11 at 7:30 pm (Hill 425)
- Final Exam: Thursday, May 3, 8-11 AM (ARC 205)
- Review session: Tuesday, May 1 at 5:00 pm (Hill 425)
MATLAB Assignments
- Project 1: Digital Signals and Computer Graphics
(Due February 6)
(pdf format) - Project 2: Convolution and Discrete Fourier Transform
(Due February 27)
(pdf format) - Project 3: Haar Wavelet Transform
(Due March 19)
(pdf format) - Project 4: Implementation of Wavelet Transforms
(Due April 2)
(pdf format) - Project 5: Image Analysis by Wavelet Transforms (Due April 23) (pdf format)
Course History
Taught by Professor Goodman 2005-2008, Professor Retakh 2009, Professor Goodman 2010-2012.
Disclaimer: Posted for informational purposes only
This material is posted by the faculty of the Mathematics Department at Rutgers New Brunswick for informational purposes. While we try to maintain it, information may not be current or may not apply to individual sections. The authority for content, textbook, syllabus, and grading policy lies with the current instructor.
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.