Math 5467, Spring '03
Last changed: January 4, 2003.
Old news:
NEWS Mar 4: Proposal IV for weights, added to the list below.
NEWS Mar 1: Proposals for weights, ideas for projects (from 2000).
So far there have been 3 proposals for weighting of
grading items for the course:
I: Three Tests, using the two with best GPA, 25% each;
Project and Special Problems lumped together, 26%;
Homework, 24%.
II: Three Tests, using the two with best GPA, 25% each;
Project, 30%;
Homework, 15%;
Special Problems, 5%;
(Discard 2 worst HW scores and worst Special Problem score).
III: Three Tests, using the two with best GPA, 20% each;
Project, 20%;
Homework, 20%.
Special Problems, 20%
(A challenge problem for 20% on 1 HW)
IV: Three tests, take two with best GPA, 26% each
Project - 28%
Homework - 15% (discard 2 worst HW)
Special Problems - 5%
Here are projects and ideas for projects from this course in 2000:
Project 3 in 2000:
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 from 2000:
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 in 2000:
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.
NEWS Feb 26: Assignment 4 is now posted.
NEWS Feb 25: Solutions and gradelines for Test 1 are here.
A link to the solutions is below.
Gradelines:
top: 110
p-top: 105
A 75
B 60
C 48
D 38
NEWS Feb 19: The test on Friday is closed-book closed-notes
(that's partly so I can ask you to state a Theorem or a Definition).
NEWS Feb 17: Assignment 1 has been returned. I'll be around today
a while if you want to pick your paper up. A corrected version
of Assignment 2 is now on the Web.
The midterm on Friday Feb. 21 will be based mostly on homework
and Lecture material thru Friday Feb. 14. Later material will
only be asked about in "define" questions. You should not expect to
finish the exam. I usually get a 50% average, which means you should
choose the problems you can do best on, and save the others until
last, if attempted at all. Partial credit is small unless significant
progress is made on a problem. This actually makes more work for me,
and, I hope, less pressure on you!
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