Math 4355: Mathematics of Signal Representations

Math 4355 -- Mathematics of Signal Representations
SPRING 2014

	Instructor: Demetrio Labate OFFICE: PGH 694 EMAIL ADDRESS: dlabate@math.uh.edu HOMEPAGE: http://www.math.uh.edu/~dlabate When and Where MEETING TIME: Mon, Wed, Fri 10-11, MEETING PLACE: AH 301 OFFICE HOURS: Mon, Wed 11-12 (or by appointment) Course Description: This course is a self-contained introduction to a very active and exciting area of applied mathematics which deals the representation of signals and images. It addresses fundamental and challenging questions like: how to efficiently and robustly store or transmit an image or a voice signal? how to remove unwanted noise and artifacts from data? how to identify features of interests in a signal? Students will learn the basic theory of Fourier series and wavelets which are omnipresent in a variety of emerging applications and technologies including image and video compression, electronic surveillance, remote sensing and data transmission. Some specific applications will also be discussed in the course. Textbook: A first course in wavelets with Fourier analysis by A. Boggess and F. Narcowich, Wiley, 2nd edition 2009. HOMEWORK: Homework 1 - Due Wed Jan 22 (latex file) - Solution Homework 2 - Due Mon Feb 3 - Solution Homework 3 - Due Mon Feb 10 - Solution Homework 4 - Due Wed Feb 19 - Solution Homework 5: Ex. 4, 18, 20, 23 (c,d), 26, p.83-86 - Due Wed Mar 5 - Solution Homework 6: Ex. 21, 24 (a,b,c,d), 33, p.85-89 - Due Wed Mar 19 - Solution Homework 7: Ex 2,4,6, p.128-129 - Due Fri Mar 28 - Solution Homework 8: Ex 5,10,12 (use Matlab to produce the graphs), p.128-129 - April 7 Solution Homework 9: Ex 7,8 (explore the use of Matlab commands fft and fftshift), p.156 - Due April 14 Solution Homework 10: Ex 1,2,9,10 p.186-188 - Due April 28 Solution	Fourier series approximation of square wave

Instructor: Demetrio Labate

dlabate@math.uh.edu

http://www.math.uh.edu/~dlabate

When and Where

Course Description:

This course is a self-contained introduction to a very active and exciting area of applied mathematics which deals the representation of signals and images. It addresses fundamental and challenging questions like: how to efficiently and robustly store or transmit an image or a voice signal? how to remove unwanted noise and artifacts from data? how to identify features of interests in a signal? Students will learn the basic theory of Fourier series and wavelets which are omnipresent in a variety of emerging applications and technologies including image and video compression, electronic surveillance, remote sensing and data transmission. Some specific applications will also be discussed in the course.

Textbook:

A first course in wavelets with Fourier analysis by A. Boggess and F. Narcowich, Wiley, 2nd edition 2009.

HOMEWORK:

Fourier series approximation of square wave

Useful Material:

Very brief linear algebra review [from http://alumni.media.mit.edu]
Another linear algebra review (with Matlab examples) [by G. Recktenwald Portland State University]

Test 1 with Solution - Test 2 with Solution - Test 3 with Solution

A list of old Tests #1,2,3 and their solutions: Tests+Solutions

Prerequisites:

MATH 2331 and one of the following: MATH 3333, MATH 3334, MATH 3330, MATH 3363. MATH 3321 can be used instead of MATH 2431. Students who wish to enroll without having one of the above junior-level courses are encouraged to discuss it with the instructor. While a prior knowledge of Matlab is not required, be aware that Matlab will be used for some homework. The use of the basic Matlab functions is very simple and it will be easy to acquire this basic-level knowledge during the course.

Course outline:

Inner product spaces [Ch.0, Sec.0.1-0.5]

Inner product spaces.
The spaces of square integrable functions and square summable series.
Schwarz and triangle inequalities.
Orthogonal projections and the least squares fit.

Fourier series and transform [Ch.1, Sec 1.1-1.3; Ch.2, Sec. 2.1-2.4]

Computation of Fourier series.
Convergence of Fourier series.
The Fourier transform.
Convolutions.
Linear filters.
The sampling theorem: Analog to digital and digital to analog conversions.
From analog to digital filters.
The Discrete Fourier transform (DFT), FFT, its use for the approximate computation of integral Fourier transforms [Ch.3,Sec.3.1].

Wavelets [Ch.4, Sec 4.1-4.3; Ch.5, Sec. 5.1-5.2]

The Haar wavelet.
Multiresolution analysis.
The scaling relation.
Properties of the scaling function.
Decomposition and reconstruction.
Wavelet design in the frequency domain.
The Daubechies wavelet.

Tests and Exam Dates:

The dates for the midterm exams are Fri Feb 21, Wed Mar 26 and We April 23. The final exam is scheduled on Mon May 5, 11-2 PM.

Grading:

Grades will be based on homework assignments counting 30% towards the final grade, on three midterm exams counting 40% towards the final grade (16% + 16% +8% for the worst one) and one final counting 30% towards the final grade.

The grade will be determined according to a set point scale: 90%-100%: A, 80%-89%: B, 70%-79%: C, 60-69% D; F is less than 60% (+ and - will also be used)..