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.

  1. Be able to derive the scaling relation, including the $p_k$'s. Be able to state the formula for the wavelet.
  2. 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.