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 covered in Math 250 (such as complex vector spaces, linear transformations, and Fourier series). It then develops the theory of the finite Fourier transform and the new theory of discrete wavelet transforms. These transforms make it possible 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).
Fall 2014 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
Download the pdf file (revised 8/13/14)
A. Jensen and A. la Cour-Harbo, Ripples in Mathematics: The Discrete Wavelet Transform
Springer ISBN # 3-540--41662-5
Buy at Rutgers U. Store or use this link: Springer
Stanford FFT Laboratory
How to update JAVA and set Security Exception for FFT Lab
- 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.
- Survey talk by Roe Goodman
Discrete Fourier and Wavelet Transforms: Mathematical Microscopes for Signal Processing
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)
- Midterm 1: Thursday, Feb. 26 (ARC 205)
- Midterm 2: Thursday, April 16 (ARC 205)
- Final Exam: Thursday, May 7, 8-11 AM (ARC 205)
- Project 1: Digital Signals and Vector Graphics
(Due February 9)
- Project 2: Convolution and Discrete Fourier Transform
(Due March 2)
- Project 3: Haar Wavelet Transform
(Due March 23)
- Project 4: Implementation of Wavelet Transforms
(Due April 6)
- Project 5: Image Analysis by Wavelet Transforms (Due April 27) (pdf format)
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 Uvi_Wave files into a directory that your create on the X-apps server. Then set the Matlab path to this directory.
Taught by Prof. Goodman 2005-2008, Prof. Retakh 2009, Prof. Goodman 2010-2014.
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