Instructor: Willard Miller
Office: Vincent Hall 513
Phone: 612-624-7379
Classroom: Vincent Hall 6
Class times: 12:20-13:10 pm MWF
Prerequisites: [2243 or 2373 or 2573], [2283 or 2574 or 3283 or instr consent]; [[2263 or 2374], 4567] recommended;
Office Hours: M W 11:15-12:05, Th 12:20-13:10, or by appointment
Homepage: www.ima.umn.edu/~miller
Text: No text. Online class notes and supplementary materials.
Midterm Tests: M 29 March (including a take-home problem) and M 19 April
Final Exam: 10:30am-12:30pm Wednesday, May 12
Material covered in course:
Inner product spaces, Fourier series and transforms. Background theory/experience in wavelets. Multi-scale analysis, discrete wavelets, self-similarity. Computing techniques.
We will start at the beginning and cover the basics thoroughly. All of the later topics will be treated in some form. We will make use of the Wavelets Toolbox in MATLAB for class demonstrations and some of the homework. Filter banks from signal processing will be used to motivate the theory, and there will be applications to image processing. This is an interdisciplinary course, with a strong math core, meant for students in mathematics, science and engineering.
Assignments:
I will give out assignments in class and will announce due dates in class, as well.
Grading:
Homework 20%
First Midterm 20%
Second Midterm 20%
Final Exam 40%
Miscellaneous:
Grades will be deterrmined through an interaction of objective standards
and experience with the class. I don't preassign the number of students
who will receive a specific grade. On the other hand, neither will I preassign
the gradelines before seeing the distribution of grades. Gradelines will
be announced on the web, as soon as possible after
the quiz or exam.
Incompletes will be given only in cases where the student has completed all but a small fraction of the course with a grade of C or better and a severe unexpected event prevents completion of the course. In particular, if you get behind, you cannot ``bail out'' by taking an incomplete. The last day to cancel, without permission from your College office, is the last day of the sixth week.
Lecture Notes and Supplementary Notes for the Course (Postscript File)(PDF File)
A note on four types of convergence (Postscript File) (PDF FILE)
Introduction to MATLAB (courtesy of Professor Peter Olver) Postscript file PDF file
MATLAB and the Undergraduate Math Club
Maple plots
illustrating the Heisenberg principle for normal distributions
Maple plots
of Daubechies halfband distributions
Link to module on computing
the scaling function from the cascade algorithm
Homework Problem Set #1 (Postscript File) (PDF FILE)
Solutions (Postscript file) (PDF FILE)
Maple plots for Exercises 2 and 8, and other examples of uniform and non-uniform convergence
Homework Problem Set #2 (Postscript File) (PDF FILE)
Solutions (Postscript file) (PDF FILE)
Homework Problem Set #3 (Postscript File) (PDF
FILE)
Comments
on solution of problem 1 and code for problems 1 and 3. (HTML FILE)
Homework Problem Set #4 (Postscript File) (PDF
FILE)
Midterm 1 Takehome Problem (Postscript File)
(PDF FILE)
Homework Problem Set #5 (Postscript File) (PDF
FILE) Due April 30
Midterm 2 Takehome Problem (Postscript File)
(PDF FILE)
Solution to Midterm 2 Takehome Problem (Postscript File)
(PDF FILE)
Midterm 2 Solutions (Postscript File) (PDF FILE)
Sample Homework Problem Set #1 (Postscript File) (PDF FILE)
Solutions (Postscript file) (PDF FILE)
Sample Homework Problem Set #2 (Postscript File) (PDF FILE)
Solutions (Postscript file) (PDF FILE)
Sample Homework Problem Set #3 (Postscript File) (PDF FILE)
Solutions (Postscript file) (PDF FILE)
Sample Homework Problem Set #4 (Postscript File) (PDF FILE)
Sample Homework Problem Set #5 (Postscript File) (PDF FILE)
Solutions to Sample Midterm 2 (Postscript File) (PDF FILE)
Sample Homework Problem Set #6 (Postscript File) (PDF FILE)
Sample Homework Problem Set #7 (Postscript File) (PDF FILE)
Course Plan: Introduction to the Mathematics of WaveletsO. Motivation through signal processing, analysis and synthesis
I. Vector Spaces with Inner Product
Definitions
Bases
Schwarz inequality
Orthogonality, Orthonormal bases
Hilbert spaces.
L² and l². The Lebesgue integral.
Orthogonal projections
Gram-Schmidt orthogonalization
Linear operators and matrices
Least squares, applications
II. Fourier Series
Definitions
Real and complex Fourier series, Fourier series
on intervals of varying length, Fourier series for odd and even functions
Examples
Convergence results
Riemann-Lebesgue Lemma
Pointwise convergence, Gibbs phenomena, Cesàro
sums
Mean convergence, Bessel's inequality, Parseval's
equality,
Integration and differentiation of
Fourier series
III. The Fourier Transform
The transform as a limit of Fourier series
Convergence results
Riemann-Lebesgue Lemma
Pointwise convergence,
L¹ and L² convergence, Plancherel formula,
Properties of the Fourier transform
Examples
Relations between Fourier series and
Fourier integrals: sampling, aliasing,
The Fourier integral and the uncertainty
relation of quantum mechanics, Poisson Summation formula
IV. Discrete Fourier Transform
Definition and relation to Fourier series
Properties of the transform
Fast Fourier Transform (FFT)
Efficiency of the FFT algorithm
Approximation to the Fourier Transform
V. Linear Filters
Definition, Discrete and continuous filters
Time invariant filters and convolution
Causality
Filters in the time domain and in the frequency
domain
The Z-transform and Fourier series
Low pass and high pass filters
Analysis and synthesis of signals, downsampling
and upsampling
Filter banks, orthogonal filter banks and perfect
reconstruction of signals
Spectral factorization, Maxflat filters
VI. Multiresolution Analysis
Wavelets, multiple scales
Haar wavelets as motivation
Definitions
Scaling functions, The dilation equation, The
wavelet equation
Wavelets from filters
Lowpass iteration and the Cascade Algorithm
Daubechies wavelets
Scaling Function by recursion, Evaluation at
dyadic points
Infinite product formula for the scaling function
VII. Wavelet Theory
Accuracy of approximation, Convergence
Smoothness of scaling functions and wavelets
VIII. Other Topics (as time allows)
The Windowed Fourier transform and The Wavelet Transform
Bases and Frames, Windowed frames
Biorthogonal Filters and Wavelets
Multifilters and Multiwavelets
IX. Applications of Wavelets (as time allows)
Image compression
Digitized Fingerprints
Speech and Audio Compression
MATLAB exercises