# EE 596 Wavelets

## Contents

## EE 596, Wavelets, Spring 2016

**Course Description:** The theory and application of wavelet decomposition of signals. Includes subband coding, image compression, multiresolution signal processing, filter banks, and time-frequency tilings.

**Prerequisites:** *EE 483, Introduction to Digital Signal Processing* and *EE 441, Applied Linear Algebra for Engineering*, or equivalent courses. Please note that the course will assume some knowledge of standard DSP concepts as well as of some basic linear algebra. If you took these two courses some time ago it would be a good idea to review some of the key material early in the semester.

## Instructor

*Signal and Image Processing Institute*

*Department of Electrical Engineering*

*University of Southern California*

*3740 McClintock Ave., EEB 436*

*Los Angeles, CA 90089-2564*

*Tel: (213) 740-2320*

*Fax: (213) 740-4651*

*Email: antonio DOT ortega AT sipi DOT usc DOT edu*

## Schedule

**Lectures**Tuesday and Thursday, 11-12:20pm, KAP 158**Office hours**Tuesday and Thursday, 1:30-3:00pm, EEB 436, and by appointment.**Midterm 1**TBD, in class**Midterm 2**TBD, in class**Final**TBD

## Grading

Each midterm will account for 30% of the grade. The remaining 40% will be based on a project. The final project report will be due on Dec 14, 2012. Project presentations will be on Dec 10, 2012.

## Lectures

- Lecture 1 (8/28/12)
- Introduction, goals, historical perspective

- Lecture 2 (8/30/12)
- Uncertainty principle
- Practical time frequency localization example
- Signal spaces

- Lecture 3 (9/4/12)
- Piecewise constant signals and Haar Wavelets
- Bases
- Norms

- Lecture 4 (9/6/12)
- View lectures 1-4 from 2010

- Lecture 5 (9/11/12)
- View lecture 4-5 from 2010
- Spaces, subspaces, orthogonal complements, successive approximation

- Lecture 6 (9/13/12)
- Haar Wavelet construction, discrete time Haar construction example

- Lecture 7 (9/18/12)
- View lecture 6-7 from 2010
- Bi-orthogonal bases, overcomplete representations

- Lecture 8 (9/20/12)
- View Lecture 8-9 from 2010

- No Lecture on 9/25/12
- Lecture 9 (9/27/12)
- View Lecture 10 from 2010
- Criteria to select a representation in an overcomplete set
- Why is sparsity useful?
- Least squares solution, brute force search

- Lecture 10 (10/2/12)
- Matching pursuits and Orthogonal Matching Pursuits
- Why does l1 promote sparsity?
- Basis pursuit

- Lecture 11 (10/4/12)
- Compressed sensing
- Discussion of compressed sensing requirements and applications

- Lecture 12 (10/9/12)
- View Lectures 11-12 from 2010
- Multirate signal processing
- Modulation domain representation of filterbanks

- Lecture 13 (10/11/12)
- Time domain representation, polyphase domain representation

- Lecture 14 (10/16/12)
- Polyphase domain representation, QMF solutions

- Lecture 15 (10/18/12)
- Review session -- Problems

- Midterm #1 (10/23/12)
- Lecture 16 (10/25/12)
- View lectures 13-15, 2010
- Orthogonal filterbank solutions

- Lecture 17 (10/30/12)
- View Lecture 16-17, 2010
- Adaptive bases
- Wavelet packets
- Examples

- Lecture 18 (11/1/12)
- Bi-orthogonal conditions and solutions

- Lecture 19 (11/6/12)
- View lectures 18-20, 2010
- Lifting

- Lecture 20 (11/8/12)
- Multichannel transforms
- Multidimensional transforms

## Texbooks

**Required:**

- Martin Vetterli and Jelena Kovacevic, Wavelets and Subband Coding, Prentice Hall, 1995. This textbook is now available electronically at http://www.waveletsandsubbandcoding.org
- Matlab Wavelet Toolbox, This toolbox is available on the student computer accounts.

**Recommended:**

- Gilbert Strang and Truong Q. Nguyen, Wavelets and Filter Banks, Wellesley-Cambridge Press, 1995
- Stephane Mallat, A Wavelet Tour of Signal Processing: The Sparse Way, 3rd Ed., Academic Press - Elsevier, 2009
- P. P. Vaidyanathan, Multirate Systems and Filter Banks , Prentice Hall, 1993

## Material Covered (Subject to Change)

**Weeks 1 and 2**Introduction and Motivation. Signal representation using bases. Hilbert spaces. Orthogonal, bi-orthogonal basis and overcomplete expansions. Example: representing finite energy continuous signals using Haar basis. Example of construction of Haar basis**Week 3**Bases for discrete signals. Finite and infinite dimensional spaces.**Week 4**Overcomplete expansions. Searching for the best representation. Matching pursuits and variations. Compressed sensing.**Weeks 5 and 6**Multirate signal processing. Filterbanks and discrete wavelet transforms. Time domain, frequency domain and polyphase domain representations.**Week 7 and 8**2-Channel orthogonal filterbanks. Iterated filterbanks. Bi-orthogonal filterbanks. Lifting factorizations. Multichannel filterbanks. Modulated filterbanks.**Weeks 9 and 10**Multidimensional wavelets. Edgelets, bandlets, ridgelets and other extensions. Lifting for video representation.**Week 11**Continuous time wavelets. Series expansions of continuous signals. Haar, Sinc, Meyer, Daubechies and Spline wavelets. Mallat algorithm.**Weeks 12, 13, 14 and 15**Applications. Compression. Classification. Graphics. Class Projects.

## Projects

- Project requirements:
- Projects should be done individually.
- Each project must involve using the wavelet transform as a tool. A signal is analyzed/classified, etc by computing its wavelet transform and then the required task (e.g. denoising/classification) is performed in the transform domain.
- The Matlab toolbox or C libraries can be used for the project. C libraries are available at Dartmouth and Rutgers.
- Whichever method is used, the source code will have to be made available along with the project report (only for the routines that you write, which could call those available in matlab or C.)

- Reporting requirements: a final report and a class presentation.
- Project descriptions and references
- Test data for the projects
- Software packages

Demos on the web

- Jelena Kovacevic's webpage contains numerous pointers to books, projects, demos, applets, etc.
- Wavelet Library Demo at South Carolina
- Bell Labs: Wim Sweldens' Wavelet Cascade Applet
- Biomedical Group at EPFL - Fractional Splines Demo
- SIMPLIcity Content Based Image Retrieval - Search
- Wavelet Resources

## Sample Project Topics - Organized by Areas

- Coding
- Implementation of a Pyramidal Image Coder
- Compression of finite-length discrete-time signals using flexible adaptive wavelet packets<
- Wavelet Descriptors for Planar Curves
- Sinusoidal Modeling of Audio Signals Using Frame-Based Perceptually Weighted Matching Pursuits
- Low Complexity Motion Estimation Algorithm for Long-term Memory Motion Compensation Using Hierarchical Motion Estimation
- Global/Local Motion Compensation for 3D Video Coding Based on Lifting Techniques

- Classification/Recognition
- Shift Invariant Texture Classification by Using Wavelet Frame
- Texture Feature Extraction with Non-Separable Wavelet Transforms
- Comparison of Two Wavelet-Based Image Watermarking Techniques
- Application of Wavelet Transform in Analysis of Fractal Signals
- Human-Face Detection and Location in Color Images Using Wavelet Decomposition
- Music/Speech Classifier using Wavelets
- Wavelet Decomposition for the Analysis of Heart Rate Variability
- Wavelet-based fMRI dynamic activation detection
- Wavelet analysis of evoked potentials
- Detection of Microcalcifications in Mammograms using Wavelet Transforms
- Wavelet-based Tone Classification for Thai

- Denoising
- Comparison of Denoising via Block Wiener Filtering in Wavelet Domain with Existing Ad-hoc Linear and Non-linear Denoising Techniques
- Wavelet-domain filtering of data with Poisson noise
- Contrast Enhancement and De-noising using Wavelets
- Wavelet Denoising Applied to Time Delay Estimation
- Comparison of image denoising using Wavelet Shrinkage vs. MMSE using an exponential decay autocorrelation model
- Threshold Denoising Effects on Covariance Matrices
- Comparing Performance of Different Wavelet De-noising algorithms with Basic Noise Removal Techniques
- Information Driven Denosing of MEG data in the Wavelets Domain
- Two Methods for Image Enhancement

- Watermarking/Halftoning
- Introduction of IWT to wavelet-based watermarking and its effect on performance
- Inverse Halftoning using Wavelets

- Communications
- Wavelets Based MC-CDMA System
- MMSE Estimation Multi-user detection for CDMA System based on Wavelet Transform

## Statement for Students with Disabilities

Any student requesting academic accommodations based on a disability is required to register with Disability Services and Programs (DSP) each semester. A letter of verification for approved accommodations can be obtained from DSP. Please be sure the letter is delivered to me (or to TA) as early in the semester as possible. DSP is located in STU 301 and is open 8:30 a.m.--5:00 p.m., Monday through Friday. The phone number for DSP is (213) 740-0776.

## Statement on Academic Integrity

USC seeks to maintain an optimal learning environment. General principles of academic honesty include the concept of respect for the intellectual property of others, the expectation that individual work will be submitted unless otherwise allowed by an instructor, and the obligations both to protect oneÃs own academic work from misuse by others as well as to avoid using anotherÃs work as oneÃs own. All students are expected to understand and abide by these principles. Scampus, the Student Guidebook, contains the Student Conduct Code in Section 11.00, while the recommended sanctions are located in Appendix A http://www.usc.edu/dept/publications/SCAMPUS/gov/

Students will be referred to the Office of Student Judicial Affairs and Community Standards for further review, should there be any suspicion of academic dishonesty. The Review process can be found at http://www.usc.edu/student-affairs/SJACS/.