Math 414-501 — Spring 2016
Test 3 Review
General Information
- Time and date. Test 3 will be given on Wednesday, 4/27/16,
at 10:20, in our usual classroom.
- Bluebooks. Please bring an 8½×11
bluebook.
- Office hours. I will have office hours on Monday
(4/25/16), 1:40-3:30, and on Tuesday (4/27/16), 1-3:30 pm.
-
Calculators. You may use calculators to do arithmetic, although
you will not need them. You may not use any calculator that
has the capability of doing linear algebra or storing programs or
other material.
-
Other devices. You may not use cell phones, computers, or any
other device capable of storing, sending, or receiving information.
-
Structure and coverage. There will be 4 to 6 questions, some
with multiple parts. The test will cover chapter 4 and sections 5.1,
5.2, 5.3.3, 5.3.4 and 6.2 in the text. The problems will be similar to
ones done
for
homework, and as examples in class and in the text. Here are links
to practice
tests:
2001
and 2009.
Be aware that these tests cover some material that will not be on the
test for this class. Note: this test will involve derivations
of many of the formulas we have discussed in class.
Topics Covered
Haar MRA. Know the Haar scaling function, wavelet,
approximation spaces (V's), and wavelet spaces (W's). Using the Haar
wavelet and scaling function, be able to carry out simple
decomposition and reconstruction algorithms. Know what the various
high pass and low pass filters associated with these algorithms are,
what down sampling and up sampling are, and finally be able to use
filter diagrams to describe the decomposition and reconstruction
algorithms.
Multiresolution analysis (MRA). You will be asked
to define Mallat's multiresolution analysis, including the
approximation spaces (V_j's). Also, know what the wavelet spaces
($W_j$'s) are and how they relate to the $V_j$'s. Be able to discuss
the details of the Haar MRA case the Shannon MRA (exercise 8 in §
5.4; be sure to know the Whittaker-Shannon Sampling Theorem, §2.4.)
Derivations and statements.
- Be able to derive the scaling relation, including the
$p_k$'s. Be able to state the formula for the wavelet.
- Be able to derive the decomposition formulas. Be able to state
the reconstruction formulas.
Filters. Know the high-pass and low-pass decomposition and
reconstruction filters, down sampling and up sampling. Know how to
implement both decomposition and reconstruction algorithms in terms of
filters. Be able to derive the filter version of the
decomposition algorithm.
Processing a signal. Be able to state the steps in processing a
signal. In the initialization step, Be able to show that, for the top
level $j$, $a^j_k \approx \,f(2^{-j}k)$, where we will use $m =
\int_{-\infty}^\infty \phi(x)dx=1$. (This is the "wavelet crime." See
Theorem 5.12, with $m=1$).
Fourier transform criteria for an MRA. Be able to
find the Fourier transformed (i.e., frequency space) form of the
scaling relation (Theorem 5.19) in terms of $P(z)$. Be able to state
Theorem 5.23.
Daubechies' wavelets and vanishing momnets. Know
how the Daubechies wavelets are classified using $N$, the largest
power of $z+1$ that divides $P(z)$, and also know how $N$ relates to
the number of vanishing moments of the Daubechies wavelet. For $N=2$,
be able to show that that linear polynomials are "reproduced" —
i.e., bjk=0 for them. Using this, be able to
explain how a wavelet analysis can be used in singularity detection
and data compression.
Updated 4/24/2016.