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.

References

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 for Fourier series, demonstrating the behavior of the kernel function D_k(t), Gibbs phenomena, the sinc function, and Cesàro sums

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)

   Solutions (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 Wavelets

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