Introduction to Applied Harmonic Analysis

**Course: **M 393C / CSE 396

**Time/location: **Tuesday & Thursday 12:30 - 2:00 (RLM 10.176) and Wednesday 5:45-7 PM (RLM 5.104)

**Instructor:** Rachel Ward

**Office: **RLM 10.144

**Email: **__rward@math.utexas.edu__

**Office Hours: **Wednesday 3-5 PM, or by appointment

**Target Audience: **Graduate students in Math, ECE, ICES, CS, DSSC, and advanced undergraduate students

**Schedule: **Please note the additional time slot on Wednesday. This time slot will be used to make up for periods when class will not be held (see attached detailed schedule).

**Class webpage**: www.ma.utexas.edu/users/rward/courses/acha14/

**Course Objective**: This course should serve as an introduction to mathematical building blocks from time-frequency analysis (e.g. Fourier series, wavelets, sampling theorems) that can be used for signal and image processing, numerical analysis, and statistics. The course will emphasize the connection between the analog world and the discrete world, and focus on approximation and compression of functions and data. We will also discuss recent advances in sparse representations and compressive sensing.

**Prerequisites**: Linear algebra (M341 or equivalent), probability (M 362K or equivalent), real analysis (M365C or equivalent), functional analysis or second-semester real analysis, or consent of instructor.

**Topics:**

- Fourier Transforms, Sampling Theorems, Fourier Series
- Discrete Fourier Transform, Discrete Cosine Transform
- The Uncertainty Principle and Bandlimited Signals
- Frame Theory
- Tools using Time/Frequency Domain Partitioning
- Local Cosine/Sine Transform
- Fast Laplace/Poisson Solvers
- Haar Wavelets
- Multiresolution Analysis
- Discrete Wavelet Transform
- Image Approximation / Compression
- Principal Component Analysis
- Sparsity / Compressive Sensing

**Textbooks:**

** **The following textbooks are used as references, but are not required.

- S. Mallat: A Wavelet Tour of Signal Processing, 3rd Edition, Academic Press, 2009.
- I. Daubechies: Ten Lectures on Wavelets,1992.
- S. Foucart and H. Rauhut: A Mathematical Introduction to Compressive Sensing, Springer 2013.

**Grading Scheme: **50% Discussion/ Participation, 50% Final project

**Discussion sessions: **Relevant research papers will be handed out every two weeks or so, to be presented in class by students during discussion session. Each student must present at least once, but all students are expected to have read the papers prior to presentation. Performance during these sessions will determine the discussion/participation grade.

**Tentative Schedule**

*Many lectures will follow the **notes of Naoki Saito*.
Further reading on the topics of the lectures can be found * here*.

- Thursday, Aug. 28:
*Introduction/ Overview* - Tuesday, Sept. 2:
*Basics of Fourier Transforms* - Thursday, Sept. 4:
*Uncertainty Principles* - Tuesday, Sept. 9:
*Discretization via Sampling* - Wed., Sept. 10: Presentation: Uncertainty principles and signal recovery by Donoho, Stark
- Thursday, Sept. 11:
*Fourier Series on Intervals* - Tuesday, Sept. 23:
*Functions of Bounded Variation* - Wed., Sept. 24: Presentation: Approximating a bandlimited function using very coarsely quantized data by Daubechies, DeVore
- Thursday, Sept. 25:
*Discrete Fourier Transform* - Thursday, Oct. 2:
*Fast Fourier Transform* - Thursday, Oct. 9:
*Sturm-Liouville Thory* - Tuesday, Oct. 14:
*Principal Component Analysis* - Wed., Oct. 15: Presentation: Nonlinear total variation based noise removal algorithms by Rudin, Osher, Fatemi, and A non-local algorithm for image denoising by Buades, Coll, Morel
- Thursday, Oct. 16:
*Time-Frequency analysis* - Tuesday, Oct. 21:
*Introduction to Frame Theory* - Wed., Oct. 22: Presentation: A fast randomized algorithm for the approximation of matrices by Woolfe, Liberty, Rokhlin, Tygert
- Thursday, Oct. 23:
*Wavelet Transforms* - Tuesday, Nov 4:
*Wavelet Transforms II* - Wed., Nov 5: Presentation: The wavelet transform, time-frequency localization and signal analysis by Daubechies
- Thursday, Nov 6:
*Wavelet Bases, Fast Wavelet Transform* - Tuesday, Nov. 11:
*Sparsity and compressive sensing I* - Wed., Nov. 12: Presentation: Stable signal recovery from incomplete and inaccurate measurements by Candes, Romberg, Tao
- Thursday, Nov 13:
*Sparsity and compressive sensing II* - Tuesday, Nov 25:
*Sparsity and compressive sensing III* - Wed., Nov. 26:
*Presentation:* - Thursday, Dec. 4: Final Presentations I
- Tuesday, Dec 9: Final Presentations II