**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 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**