640:357 Topics in Applied Algebra
Syllabus (Spring, 2009)
Texts for the course
1. Excerpts from:
Leon = Steven J. Leon, Linear Algebra with Applications (6th
edition)
(Purchase excerpts at Undergraduate Math Office -- Hill Center Room 303)
Don't buy the entire book!
2. Complete book:
Jensen = A. Jensen and A. la Cour-Harbo, Ripples in Mathematics: The Discrete Wavelet Transform
ISBN # 3-540--41662-5, Springer-Verlag, 2001
(Purchase entire book at Rutgers U. Store or order on-line.)
3. Lecture Notes:
Goodman = Roe Goodman, Discrete Fourier Transform and Wavelet Transforms
(Print from course web page.)
| Date | Lecture | Reading | Topics |
|---|---|---|---|
| 1/22 | 1 | Leon 3.1-3 Goodman 1.1-2 |
Vector Spaces and Subspaces Linear Independence; Function Spaces |
| 1/26 | 2 | Leon 3.4-5 | Basis, Dimension, Change of Basis |
| 1/29 | 3 | Leon 4.1-2 | Linear Transformations and Matrices |
| 2/02 | 4 | Leon 5.4 Goodman 1.3 |
Real and Complex Inner Product Spaces |
| 2/05 | 5 | Leon 5.5 Goodman 1.3 |
Orthonormal Bases; Approximation of Functions |
| Due: | 2/09 | Course web page | MATLAB Project #1: Visualizing Linear Transformations |
| 2/09 | 6 | Leon 5.5 Goodman 2.1 |
Fourier Series and Discrete Fourier Transform |
| 2/12 | 7 | Goodman 2.1 | Fourier Matrix |
| 2/16 | 8 | Goodman 2.2 | Shift-Invariant Linear Transformations and Circulant Matrices |
| 2/19 | 9 | Goodman 2.2 | Diagonalization of Circulant Matrices |
| 2/23 | 10 | Leon 5.5 Goodman 2.3 |
Downsampling; Fast Fourier Transform |
| 2/26 | 11 | Exam 1 | (closed book) |
| Due: | 3/02 | Course web page | MATLAB Project #2: Convolution and Discrete Fourier Transform |
| 3/02 | 12 | Jensen 2.1-2, 3.1-2 Goodman 3.1 |
Multiresolution Representation; Prediction and Update Transforms |
| 3/05 | 13 | Jensen 3.5, 5.1-2 Goodman 3.2 |
Multiple Scale Wavelet Transforms; Haar Transform |
| 3/09 | 14 | Jensen 3.3-4 Goodman 3.3 |
CDF(2,2) and Daubechies 4 Transforms |
| 3/12 | 15 | Jensen 5.3 Goodman 3.4 |
Wavelet Bases |
| Spring Break | |||
| Due: | 3/23 | Course web page | MATLAB Project #3: Haar Wavelet Transform |
| 3/23 | 16 | Jensen 6.1 Goodman 3.5 |
2D Wavelet Transforms (one scale) |
| 3/26 | 17 | Jensen 6.2, 7.1 Goodman 3.5, 4.1 |
2D Wavelet Transforms (multiple scales) Signals and z-transform |
| 3/30 | 18 | Jensen 6.2, 7.1 Goodman 4.1 |
DFT and z-transform; Convolution; Shift Operator |
| 4/02 | 19 | Jensen 7.2 Goodman 4.2 |
Downsampling, Upsampling, and Lazy Wavelet Transform |
| Due | 4/06 | Course web page | MATLAB Project #4: Implementation of Wavelet Transforms |
| 4/06 | 20 | Jensen 7.2 Goodman 4.2 |
Lifting and Polyphase Matrices |
| 4/09 | 21 | Jensen 7.3 Goodman 4.3 |
Filter Banks and Modulation Matrices Perfect Reconstruction (PR) Filter Banks |
| 4/13 | 22 | Jensen 7.3 Goodman 4.4 |
Constructing PR Filter Banks |
| 4/16 | 23 | Exam 2 | (closed book) |
| 4/20 | 24 | Jensen 12.4.1-2 Goodman 4.4 | Choosing PR Filters -- Bezout's Theorem |
| 4/23 | 25 | Jensen 7.3 Goodman 4.5 |
Comparison of Lifting and Filter Banks Factoring a polyphase matrix |
| Due: | 4/27 | Course web page | MATLAB Project #5: Image Analysis by Wavelet Transforms |
| 4/27 | 26 | Jensen 7.4-5 Goodman 4.6 |
Trend-Detail Decomposition for PR Filter Banks |
| 4/30 | 27 | Jensen 7.7 Goodman 4.7 |
Orthogonal Filter Banks |
| 5/04 | 28 | Jensen 7.8, 12.5 Goodman 4.7 |
Construction of Orthogonal Filter Banks |
| 5/13 | 12:00-3:00 pm | Final Exam | (closed book) |
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