MATH 5483 - Wavelets, Section 001 - Spring 2008
TR 1:30-2:45 p.m., 1105 PHSC

Instructor: Nikola Petrov, 802 PHSC, (405)325-4316, npetrov AT math.ou.edu

Office Hours: Mon 2:30-3:30 p.m., Tue 2:45-3:30 p.m., or by appointment.

Prerequisites: 3113 (Introduction to ODEs) and 3333 (Linear algebra I).

Course catalog description: Fourier analysis on a finite cyclic group, the group of integers, and the real line. The matching pursuit algorithm. The Poisson summation formula and sampling. Multi-resolution analysis, various wavelet constructions (including those of Daubechies and Meyer) and filter banks. An introduction to the MATLAB wavelet toolbox.

Text: David F. Walnut, An Introduction to Wavelet Analysis, 1st edition, Birkhauser, 2002, ISBN-10: 0-8176-3962-4, ISBN-13: 978-0-8176-3962-4.

Homework: Library, 207 PHSC):

• Homework 1, due date Thu, Jan 24.
• Homework 2, due date Thu, Jan 31.
• Homework 3, due date Thu, Feb 7.
• Homework 4, due date Thu, Feb 14.
• Homework 5, due date Thu, Feb 21.
• Homework 6, due date Thu, Feb 28.
• Homework 7, due date Thu, Mar 6.
• Homework 8, due date Thu, Apr 3.
• Homework 9, due date Thu, Apr 10.

Content of the lectures:

• Lecture 1 (Tue, Jan 15): Introductory ideas about multiscale analysis. Review of linear algebra: linear spaces, norms, normed spaces, equivalence of norms in finite-dimensional spaces.
• Lecture 2 (Thu, Jan 17): Review of linear algebra (cont.): norms in infinite-dimensional spaces, spaces L^∞, L¹, L²; inner product spaces, Cauchy-Schwarz inequality, Minkowski inequality (pages 3-8 of Sec. 1.1).
• Lecture 3 (Tue, Jan 22): Dyadic step funcions, the Haar system: dyadic intervals I_j,k, properties, scale j dyadic step functions, Haar scaling functions p_j,k and Haar functions h_j,k, orthonormality of the Haar system {h_j,k}_j,k∈Z, the Splitting Lemma (Sec. 5.1, 5.2).
• Lecture 4 (Thu, Jan 24): Review of some concepts from analysis: convergence of sequences of numbers, convergence of a series of numbers, completeness property of R, Cauchy sequences, Weierstrass M-test, absolute convergence of series, rearrangements of series and relation with absolute convergence, doubly infinite sequences and series, Cⁿ, C^∞ and C^ω functions, uniform (L^∞) and mean-square (L²) convergence, interchange of limits and integrals (pages 9-16, 19-22 of Sec. 1.2).
  Generalized Fourier series: orthogonal systems, orthonormal systems, generalized Fourier coefficients (pages 47-49 of Sec. 2.3).
• Lecture 5 (Tue, Jan 29): Generalized Fourier series (cont.): remarks about functions in L^p as equivalence classes, Bessel's inequality, Lemma for optimal choice of coefficients to minimize the L² error in approximating with a finite linear combination of the functions of an orthonormal system; span of a system of functions {g_n}, mean-square (L²-) closure of span{g_n}, Theorem on a necessary and sufficient condition of an L² function to be in the L²-closure of span{g_n}, orthonormal basis, criteria for completeness (Plancherel formula, Parceval's equality) (pages 49-55 of Sec. 2.3).
  Haar bases on R: approximation operator P_j on L², approximation space V_j, the blurred version P_jf of f as the best approximaiton of f in V_j in L² sense (pages 133-134 of Sec. 5.5).
• Lecture 6 (Thu, Jan 31): Haar bases on R (cont.): properties of the approximation operators P_j (linearity, idempotency, ||P_jf||₂ is no greater than ||f||₂), approximation properties of P_j on compactly supported functions; detail operators Q_j on L²(R), wavelet space W_j (pages 134-136 of Sec. 5.5).

Lecture 7 (Tue, Feb 5): Haar bases on R (cont.): properties of the detail operators, formula for Q_jf as a linear combination of scale j Haar functions h_jk, the scale J Haar system on R, criterion for completeness of an orthonormal system {g_n} (every contionuous compactly supported function is in the L²-closure of span{g_n} (pages 136-138 of Sec. 5.5, pages 54-55 of Sec. 2.3).

• Lecture 8 (Thu, Feb 7): Haar bases on R (cont.): completeness of the scale J Haar system on R; the Haar system on R, completeness of the Haar system on R (pages 138-139 of Sec. 5.5).
  Regularity and rate of decay: Examples of the rate of decay of Fourier coefficients and Fourier transform of C^k functions, remarks about the rate of decay of Haar wavelet coefficients, remarks about the number of Haar wavelet coefficients affected by a singularity of the function (a discontinuity, a "corner", etc), (look at the theorems in Sec. 3.7, in particular Exercise 3.38 on page 79; look at pages 130-132 of Sec. 5.4).
  Orthonormal systems of translates: idea of multiresolution analysis, definition of orthonormal systems of translates, examples (page 163, page 164 of Sec. 7.1).
• Lecture 9 (Tue, Feb 12): Orthonormal systems of translates: derivation of the condition for a collection {T_n(g)} to be an orthonormal system of translates in terms of the periodized Fourier transform of g, examples, condition for a funciton to be in the L²-closure of the span of an ON system of translates (pages 164-166 of Sec. 165).
• Lecture 10 (Thu, Feb 14): Definition and properties of a multiresolution analysis (MRA): definition of a MRA, scaling function φ, approximation P_j:L²(R)→V_j and detail Q_j=P_j+1-P_j operators, {φ_jk}_k∈Z is an ON basis of V_j, approximation properties of P_j: ||P_jf-f||₂→0 and ||P_-jf||₂→0 as j→∞ for any continuous compactly supported function f on R, two-scale dilation equation and its Fourier transform, auxiliary function m₀(γ) (pages 169-173 of Sec. 7.2).
• Lecture 11 (Tue, Feb 19): The Haar MRA on R: definition and brief discussion.
  The piecewise lienar MRA on R: definition of V₀ and V_j, verifying conditions (a), (b), and (c) of the definition of an MRA (pages 174-176 of Sec. 7.3).
• Lecture 12 (Thu, Feb 21): The piecewise linear MRA on R (cont.): verifying conditions (e) of the definition of an MRA, Lemma 7.20, Lemma 7.21, Lemma 7.22, Lemma 7.23 (pages 175-178 of Sec. 7.3).
• Lecture 13 (Tue, Feb 26): The piecewise linear MRA on R (cont.): more on ON systems of translates: the orthonormality condition in terms of the Fourier transform (Lemma 7.4), condition for a funciton to be in the L²-span of an ON system of translates in terms of Fourier transforms (Lemma 7.5), construction of a function whose translates copmrise an ON system with the same span as a given system of translates (Lemma 7.7); finishing the verification of condition (e) from the definition of an MRA for the piecewise linear MRA on R (pages 164-168 of Sec. 7.2, page 178 of Sec. 7.3).
• Lecture 14 (Thu, Feb 28): Construction and examples of ON wavelet basis: main theorem (Theorem 7.35) on construction of a wavelet ON basis given an MRA, wavelet filter, wavelet, examples: the Haar wavelet, the piecewise linear wavelet (pages 185-188 of Sec. 7.4).
• Lecture 15 (Tue, Mar 4): Construction and examples of ON wavelet basis (cont.): bandlimited functions, Shannon's Sampling Theorem, construction of a bandlimited MRA on R (pages 81-85 of Sec. 3.9, page 179 of Sec. 7.3).
• Lecture 16 (Thu, Mar 6): Construction and examples of ON wavelet basis (cont.): checking all the conditions for the bandlimited MRA on R, bandlimited wavelet (pages 179-180 of Sec. 7.3, pages 188-189 of Sec. 7.4).
• Lecture 17 (Tue, Mar 11): Properties of the scaling function and the wavelet: |∫φ(x)dx|=1, ∫ψ(x)dx=0; Fourier transform of φ(x) is equal to 0 for integer nonzero argument and is equal to 1 by absolute value for argument 0; the periodized scaling function (i.e., the sum of φ(x+n) over all n∈Z) is 1 for any value of x (Sec. 7.6).
• Lecture 18 (Thu, Mar 13): From MRA to a Discrete Wavelet Transform (DWT): "wavelet crime", expressing the coefficients c_j of a function in the basis of V_-j in terms of the coefficients c_j+1 and d_j+1 in the bases in V_-j-1 and W_j-1 (pages 215-217 of Sec. 8.1).
• Lecture 19 (Tue, Mar 25): The Quadrature Mirror Filter conditions: motivation from MRA, shift operator, downsampling and upsampling operators, approximation H and detail G operators, adjoint O^* of an operator O, derivation of expressions of the adjoints H^* and G^* (pages 218-222 of Sec. 8.2).
• Lecture 20 (Thu, Mar 27): The Quadrature Mirror Filter conditions (cont.): frequency domain representations of τ_mc, ↑c, and ↓c, Hc, Gc, H^*c, G^*c; proof that HG^*=0 and GH^*=0 (pages 223-227 of Sec. 8.2).
• Lecture 21 (Tue, Apr 1): The Quadrature Mirror Filter conditions (cont.): theorem on the equivalence of the conditions on the function m₀(γ), the scaling filter h(k), and the identities H^*H+G^*G=I and HH^*=GG^*=I, geometric meaning of these identities, definition of a QMF filter, implications of the QMF conditions (pages 227-230 of Sec. 8.2).
• Lecture 22 (Thu, Apr 3): The discrete wavelet transform (DWT): definitions of the DWT corresponding to a given QMF, the DWT for finite-length signals - padding and periodization; orthogonal transformations, the DWT as an orthogonal transformation (Sec. 8.3).
• Lecture 23 (Tue, Apr 8): Scaling functions φ(x) from scaling filters h(k): the infinite product formula for Fourier transform of φ(x); infinite products - definition, convergence and absolute convergence, conditions for convergence of an infinite product, proof of convergence in the infinite product formula for Fourier transform of φ(x), problems with the L²(R) convergence in the the infinite product formula for Fourier transform of φ(x) and their solution (Theorem 8.35, without proof); the cascade algorithm - representing the 2-scale dilation equation as a fixed point of an operator, iterative method for solving operator equations, an example (pages 236-243 of Sec. 8.4).
  
• Lecture 24 (Thu, Apr 10): Scaling functions φ(x) from scaling filters h(k)(cont.): the cascade algorithm for computing φ(x) as a fixed point of the operator A=∑_kh(k)D₂T_k; convergence of the iterates η_l=Aη_l-1 to the scaling function φ(x) and orthonormality of the system of translates {T_n&phi}_n∈Z as a consequence of the QMF conditions and the finite length of the scaling filter h(k) for η₀(x)=χ_[-1/2,1/2](x) (Theorem 8.36) (pages 243-245 of Sec. 8.4).
• Lecture 25 (Tue, Apr 15): Scaling functions φ(x) from scaling filters h(k)(cont.): relations between the support of the scaling function φ(x) and the length of the (finite) scaling filter h(k).
  Vanishing moments: moment of a function, vanishing moments; vanishing moments and smoothness of an ONS of translates {ψ_j,k}_j,k∈Z with ψ∈L¹(R) and the Fourier transform of ψ in L¹(R) (Theorem 9.1, idea of proof only); vanishing moments and approximation (Theorem 9.5, without proof) (pages 245-247 of Sec. 8.4, pages 249-254 of Sec. 9.1).
• Lecture 26 (Thu, Apr 17): Vanishing moments (cont.): vanishing moments and reproduction of polynomials (Theorem 9.7, without proof), equivalent conditions for vanishing moments (Theorem 9.11, with proof of (a), (b), and (c)) (pages 257, 258, 260-263 of Sec. 9.1).
• Lecture 27 (Tue, Apr 22): The Daubechies wavelets: using the QMF conditions and the conditions for vanishing moments to construct the frequency domain representation m₀(γ) of the scaling filter h(k), the Daubechies polynomials (pages 264-266 of Sec. 9.2).
• Lecture 28 (Thu, Apr 24): The Daubechies wavelets (cont.): derivation of the Daubechies wavelets in the cases N=1 and N=2 (pages 266-268 of Sec. 9.2).
• Lecture 29 (Tue, Apr 29): Projects: Qinghua Luo, "Approximating integral operators with wavelets"; Suyu Li, "B-spline wavelets".
• Lecture 30 (Tue, May 1): Projects: Bryan Burkholder, "Wavelets in studies of Kolmogorov scaling in turbulence"; Kesong Cheng, "Wavelets and regularity of functions".

Attendance: You are required to attend class on those days when an examination is being given; attendance during other class periods is also strongly encouraged. You are fully responsible for the material covered in each class, whether or not you attend. Make-ups for missed exams will be given only if there is a compelling reason for the absence, which I know about beforehand and can document independently of your testimony (for example, via a note or a phone call from a doctor or a parent).

Homework: It is absolutely essential to solve the assigned homework problems! Homework assignments will be given regularly throughout the semester and will be posted on this web-site. Usually the homeworks will be due at the start of class on Thursday. Each homework will consist of several problems, of which some pseudo-randomly chosen problems will be graded. Your lowest homework grade will be dropped. All homework should be written on a 8.5"×11" paper with your name clearly written, and should be stapled. No late homework will be accepted!

You are encouraged to discuss the homework problems with other students. However, you have to write your solutions clearly and in your own words - this is the only way to achieve real understanding! It is advisable that you first write a draft of the solutions and then copy them neatly. Please write the problems in the same order in which they are given in the assignment.

Project: Each student must write a project for the class, and give a 20-minute talk about it during the last lectures. The project topics should be determined by February 15; but you are encouraged to discuss the topics with me as soon as possible.

Exams: There will be one in-class midterm and a comprehensive final. All tests must be taken at the scheduled times, except in extraordinary circumstances.

Grading: Your grade will be determined by your performance on the following coursework:

Academic calendar for Spring 2008.

Policy on W/I Grades : Through February 22, you can withdraw from the course with an automatic W. In addition, it is my policy to give any student a W grade, regardless of his/her performance in the course, through the extended drop period that ends on May 2. However, after March 31, you can only drop via petition to the Dean of your college. Such petitions are not often granted. Furthermore, even if the petition is granted, I will give you a grade of "Withdrawn Failing" if you are indeed failing at the time of your petition.

The grade of I (Incomplete) is not intended to serve as a benign substitute for the grade of F. I only give the I grade if a student has completed the majority of the work in the course (for example everything except the final exam), the coursework cannot be completed because of compelling and verifiable problems beyond the student's control, and the student expresses a clear intention of making up the missed work as soon as possible.

Academic Misconduct: All cases of suspected academic misconduct will be referred to the Dean of the College of Arts and Sciences for prosecution under the University's Academic Misconduct Code. The penalties can be quite severe. Don't do it! For more details on the University's policies concerning academic misconduct see http://www.ou.edu/provost/integrity/. See also the Academic Misconduct Code, which is a part of the Student Code and can be found at http://www.ou.edu/studentcode/.

Students With Disabilities: The University of Oklahoma is committed to providing reasonable accommodation for all students with disabilities. Students with disabilities who require accommodations in this course are requested to speak with the instructor as early in the semester as possible. Students with disabilities must be registered with the Office of Disability Services prior to receiving accommodations in this course. The Office of Disability Services is located in Goddard Health Center, Suite 166: phone 405-325-3852 or TDD only 405-325-4173.

Good to know:

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