Changed May 11, 2000
When it's appropriate, various links will appear here, along with news and announcements.
The homework part of the grade has two components: homework sets and Projects. Projects 1 and 2 will each be worth one homework set, and Project 3 will be worth one and one half homework sets. The gradelines for the sum of the "homework" scores (Project scores adjusted) will probably be lower than 80=A, 60=B, 40=C, 20=D (percents) but not much. You can calculate an estimate for your grade by following the steps described in the Syllabus (link below) under "Grading."
Your Project 3 papers look good!
Assuming for the moment that the 5.5 Homework sets averaged 40 points each, the Homework+Project Gradelines would be:
A, 280; B, 210; C, 140; D, 70.
Gradelines for Tests 1 and 2: A, 84; B, 67; C, 50; D, 37.
The Take-Home Final will be posted on the Web just before class on May 5. Those of you who want to take an in-class Final on Saturday May 13, from 4 - 6 pm should let me know this, in writing, by 3pm Monday May 8! Those of you who choose to take the Take-Home Final, please turn it in to me by Friday May 12 during building-open hours by sliding it under my office door (if I'm not there or not answering), or by giving it to Jan or Phong in Vincent 105 during their "business hours," or by bringing it to our classroom Friday May 12 between 4 and 6pm.
I received no notice by Monday at 3pm that anyone wanted to take an in-class final, so I assum I'll receive you Finals as described above. Also, see below for Questions and Answers!
The Take-Home Final is open-book and open-note. You must not consult with anyone in any way about it, except for clarification questions directed to me only, by email or in person. Answers to such questions will be sent to you and posted on the Web here.
Questions and Answers about the Take Home Final
Question(s):
I noticed for Question #16 (the last one), that somebody asked about proving (h5) to be true. You responded that this could be done with a computer program. But my question is: Do we actually have to show that (h5) is true?
Answer(s):
Yes.
Question(s):
In the Wavelet final, problem 4, when you say "the same" MRA, do you mean the same coefficients of h(n)'s?
Answer(s):
No, I mean the same collection of subspaces V_j.
Question(s):
Question 9) From what I know, a flow chart is a visual way of explaining an algorithm or a procedure. It is not clear to me what process is being asked to be explained in this question. Is it acceptable to draw a block-diagram that takes as input `f' and produces `P_N f' using the operators P_{N-M} and Q_{N-j}, j=1,...,M ?
Question 10) When you say `the next step', what parameter is to be incremented go to to the next step, M, or N? This question also suggests that the `flow chart' in question 9 should involve `steps' and the numbers h(n) and h_1(n). Is this true? If it is, why is it not mentioned in Question 9? If it is not, then how are we to do question 10?
Question 16 (the last question) Is it acceptable to utilize a computer program for this question to show that the condition h5 is satisfied? I can do that by finding the roots of h_0 + h_1 z + h_2 z^2 + h_3 z^3 + h_4 z^4 + h_5 z^5 and show that none of the 5 roots lie on the part of the unit circle so as to violate the h5 condition. Is this acceptable?
Answer(s): 9: Produce P_N f as the indicated sum, one step at a time: get P and Q for N-1, then go from there to N-2, and so on.
10: go from N to N-1, to N-2, and so on. I suppose it's j that is being incremented.
16: Yes, a computer program is fine: but you need to include the source code.
Question(s):
On Problem #7:
Do I assume the "first moment" of the scaling function is zero when I find the low-pass filter condition given when the "second moment" of the scaling function is zero? Or, am I supposed to treat these two moment equations independently and find the low-pass filter condition given by each?
Answer(s): You can proceed either way - assume 1 true and get 2, if that helps. Or, you can work them independently! This is a slightly deep question!
Question(s):
I'm stuck on #3 of the take-home final. I can not find a list of all Fourier facts. The "Fourier Transform Facts" PDF file on the web does not have the Fourier Inversion Formula or the Convolution Theorem. Are these properties in another document?
Also, do you want a functional form of the inverse transform of the statement in #3? That is, I assume it's not sufficient to write the inverse transform as a convolution.
Answer(s):
Yes, I want a formula in closed form for each transform.
The Fourier Inversion Formula is on page 6 of "From a low-pass filter candidate to a scaling function" on the Web. The Convolution Theorem:
The Fourier transform of the convolution is the product of the Fourier transforms. It's formula (8) in the "Fourier Transform facts: L^1" notes.
Convolution: \int f(t - s) g(s) ds =: f*g(t)
You're right: I never did get my list of L^2 Fourier Fact up on the Web! I'll work on that, but it may take a while. In the meantime, I hope you got what you need!
The Third Class project is due May 5.
Here is a statement of it:
You have discovered that a way to construct a scaling function and wavelets in two dimensions is
to form all possible f_1(x)f_2(y), where each f_j is either
\phi or \psi. In particular, \phi(x)\phi(y) gives the
two-dimensional scaling function.
(I) Your assignment for this project is to do something "real" with two-dimensional wavelets. I hope you'll work with something relevant to your fields of interest. Possibilities include, and are by no means limited to:
Working with images approximated by various kinds of 2D wavelets (Haar, Daubechies 4, etc.),
to see which work best for a given image, for compression, etc.;
Analyze an image into its "components" at different scales and selectively reassemble the image
to examine the effect of omitting various components;
Observe "edge-effects" in image reconstruction. Experiment with ways to minimize it,
such as reflection across an edge, and so on;
Try "enhancing" an image by modifying wavelet coefficients;
Try out using different wavelets in different parts of an image, or at different scales:
"adaptive" use of wavelets;
(II) You should use wavelet resources on the Web, for references, data, etc.;
(III) You should use computers to do the "number-crunching," but you need to include the "code" you used to make the computer do the work, and describe, for each block of code, what "job" you are having the computer do;
(IV) The Project should be well-written in a narrative style that you can quickly and effectively use and understand six months from now.
The Second Class project is tentatively due April 17. Here is a statement of it:
Define MRA for the two-dimensional space of square-integrable functions, L^2(\real^2).
Work out the example of such an MRA determined by the two-dimensional box function
B(s,t):= 1 when 0<= s < 1 and 0<= t < 1, and B(s,t):= 0 otherwise.
Also, find the wavelets that "go with" B(s,t). There will be more than one of them!
In other words, W_0 will have to be expressed as an orthogonal direct sum of
smaller spaces, each one being generated by the same sort of translates as V_0.
There are only a few.
The first Class project is now due March 17. Here is a statement of it:
Find all sequences h(n) that have only two non-zero terms, and that
"generate" a scaling function whose integer translates form an orthonormal set.
Hint: First step: by (h1)-(h4) you can find out what values the h's must have, but not
where the h's are located, except that one has to be at an odd integer and
the other at an even one.
Second step: Then you calculate m_o(\xi) and simplify so it resembles
the Fourier transform of the Box function.
Third step: You actually construct the (Fourier transform of the) scaling function using
the Cascade formula! The infinite product can be calculated the same way we did
the one for the Box function. You should arrive at: an exponential with imaginary exponent
(so it corresponds to a translation before taking the Fourier transform), times the
Fourier transform of a Box function centered at 0.
Fourth step: You figure out what the scaling function is, and check whether its
integer translates form an orthogonal set. If they do, you have one of the solutions.
If not, you discard it.
The files below are all in PDF format -- you'll need Acrobat Reader to view them. Some are from a previous version of the course.
Math 5467: Syllabus.
Math 5467: Assignments.
From a scaling function to a wavelet 3/19/00
From a low-pass filter candidate to a scaling function 3/13/00
From a MRA to a scaling function, via a low-pass filter 3/9/00
The Lebesgue facts 3/5/00
Fourier Trasform Facts: L^1 3/2/00
Fourier Trasform Facts: L^2 5/9/00
A review list before Test 1 on Friday Feb 25
Partial solutions: HW 1, 3iii & 4
Partial solutions: HW 2, 2(b) & 3
Partial solutions: HW 3, 1 & 2
Partial solutions: HW 4, 2 & 4
Partial solutions: HW 5: 3; Remarks on 1, 2, 4
Existence of L^2 Fourier Transform
A Brief Introduction to Inner Product Spaces and Hilbert Spaces. Feb 24, 2000.
There will be plenty of exercises, hopefully both fun and intriguing.
The reference text four years ago was a book by Ingrid Daubechies, "Ten
lectures on Wavelets." it's great, but advanced.
Our official text in '97 was
A first course on wavelets , by Eugenio Hernández and
Guido Weiss, CRC Press, 1996. It's too big, too advanced, too expensive.
Here are some other sources:
An Introduction to Wavelets, by Charles K. Chui, Academic Press, 1992.
A survey article in the Bulletin of the Belgian Math Society:
"Learning to swim in a sea of wavelets," by Adhemar Bultheel,
Bull. Belg. Math. Soc. 2 (1995) 1-44.
I almost made this the "core" text.
Wavelets : Algorithms and applications
by Yves Meyer, translated and revised by Robert D. Ryan
Soc. for Industrial and Applied Math., 1993.
This is quite readable, especially for overview, by one of the developers
of the subject.
The world according to wavelets,
by Barbara Burke Hubbard,
A K Peters 1996,
is fun to read, and gives some mathematics too. The book's subtitle describes
it well:
"The Story of a Mathematical Technique in the Making."
If you have questions, please call Prof. Max Jodeit at 625-3855, send e-mail to jodeit@math.umn.edu. The course may be cross-listed; we can find out for you.
You may also check my Web page, "http://www.math.umn.edu/~jodeit/"
or Gerry Naughton's, "http://www.math.umn.edu/~naughton/" for
possible further information.
Brief overview, examples and history of wavelets and wavelet transforms,
description of background needed:
2 lectures
Review of, or introduction to, Inner Product spaces and Hilbert spaces
(of countable dimension); orthonormal bases, Bessel's inequality and Parseval's
Theorem, examples:
4 lectures
Orthogonal and Unitary operators, examples, the Fourier transform:
2 lectures
Overview, Haar wavelets and the fast Haar transform, applications:
4 lectures
Wavelet based Multi-resolution Analyses, with examples:
6-8 lectures
Daubechies wavelets and the Fast Wavelet Transform:
4 lectures
Wavelets and image compression; Wavelets and Numerical Analysis:
4-2 lectures
Review and Outlook:
2-0 lectures