Teaching a Discrete Wavelets Transform Coursehelmut.knaust.info/presentations/2010/20100409_DWT_MAA.pdf · 2010-04-13 · Discrete Wavelet Transformations: An Elementary Approach

Setup The Problem Basic DSP Approximating Data JPEG2000 Learning Mathematica Final Projects

Teaching aDiscrete Wavelets Transform Course

Helmut KnaustDepartment of Mathematical Sciences

The University of Texas at El PasoEl Paso TX 79968-0514

[email protected]

Scottsdale AZ April 9, 2010


1 Course Setup

2 The Problem

3 Basics of Digital Signal Processing

4 Approximating Data

5 JPEG 2000 Algorithm

6 Learning Mathematica - Scaffolding during the Semester

7 Final Projects - Spring 2009


Course Objectives:

Students will make considerable progress in the followingareas:

1 Develop an understanding of the theoretical underpinningsof wavelet transforms and their applications.

2 Learn how to use a computer algebra system formathematical investigations, as a computational andvisualization aid, and for the implementation ofmathematical algorithms.

3 Get a flavor of the ideas and issues involved in applyingmathematics to a relevant engineering problem.

4 Be able to give and defend a mathematical presentation toa group of your peers.


Course Objectives:







Course Objectives:







Course Objectives:







Course Objectives:







Prerequisites:A thorough understanding of CalculusSome familiarity with matricesMathematical maturityWillingness to learn Mathematica


Course Structure:

Three semester credit hours (2 × 80 minutes per week)Four individual student projects and a final group projectTwo midterm exams, no final examHomework assigned regularly; students present solutionsin class.Classroom with LCD projector; computer laboratory is nextdoor.Night class at 6:30 p.m.


Course Structure:

Three semester credit hours (2 × 80 minutes per week)

Four individual student projects and a final group projectTwo midterm exams, no final examHomework assigned regularly; students present solutionsin class.Classroom with LCD projector; computer laboratory is nextdoor.Night class at 6:30 p.m.


Course Structure:

Three semester credit hours (2 × 80 minutes per week)Four individual student projects and a final group project

Two midterm exams, no final examHomework assigned regularly; students present solutionsin class.Classroom with LCD projector; computer laboratory is nextdoor.Night class at 6:30 p.m.


Course Structure:

Three semester credit hours (2 × 80 minutes per week)Four individual student projects and a final group projectTwo midterm exams, no final exam

Homework assigned regularly; students present solutionsin class.Classroom with LCD projector; computer laboratory is nextdoor.Night class at 6:30 p.m.


Course Structure:

Three semester credit hours (2 × 80 minutes per week)Four individual student projects and a final group projectTwo midterm exams, no final examHomework assigned regularly; students present solutionsin class.

Classroom with LCD projector; computer laboratory is nextdoor.Night class at 6:30 p.m.


Course Structure:

Three semester credit hours (2 × 80 minutes per week)Four individual student projects and a final group projectTwo midterm exams, no final examHomework assigned regularly; students present solutionsin class.Classroom with LCD projector; computer laboratory is nextdoor.

Night class at 6:30 p.m.


Course Structure:

Three semester credit hours (2 × 80 minutes per week)Four individual student projects and a final group projectTwo midterm exams, no final examHomework assigned regularly; students present solutionsin class.Classroom with LCD projector; computer laboratory is nextdoor.Night class at 6:30 p.m.


Time Spent:

40% Lecturing at the whiteboard40% Lecturing using Mathematica notebooks20% Students working on projects


Patrick van FleetDiscrete WaveletTransformations: AnElementary Approach withApplicationsWiley


An Alternative:

Arne Jensen and Anders LaCour-Harbo:Ripples in Mathematics: TheDiscrete Wavelet TransformSpringer


The Engineering Problem:Transmit digital information through a “narrow channel”.“Lossy compression”: The information received need notbe identical to the one sent, but the quality must be“acceptable”.

Sender

Channel

Receiver


Applications:PhotosMP3 playersReal-time two-way audio (cellular telephones)

Streaming video (Netflix, Hulu)Real-time two-way audio and video (Skype)


Applications:PhotosMP3 playersReal-time two-way audio (cellular telephones)Streaming video (Netflix, Hulu)

Real-time two-way audio and video (Skype)


Applications:PhotosMP3 playersReal-time two-way audio (cellular telephones)Streaming video (Netflix, Hulu)Real-time two-way audio and video (Skype)


A color image consists of three color channels: Red, Green andBlue


Each pixel in a gray-scale image is represented by an integerbetween 0 and 255 = 28 − 1 (8 bit = 1 byte)

0=black, 255=white


“Raw” storage requirement:(512 × 768) bytes = 393.2 KB


“Naive compression” — Average of four neighboring pixels:Compression factor: 4


Mathematical Question: What ways are there to approximate afunction (“data”) other than just averaging?

Differentiation Techniques

Taylor series

Integration Techniques

Fourier seriesWavelets



Differentiation Techniques

Taylor seriesIntegration Techniques




Differentiation TechniquesTaylor series











Integration TechniquesFourier seriesWavelets


Fourier approximation

-Π -3 Π

4-

Π

2-

Π

4

Π

4

Π

2

3 Π

4Π

-2.0

-1.5

-1.0

-0.5

6

0.827958 sin(t)− 0.310564 sin(2t) + 0.191515 sin(3t)−0.139372 sin(4t) + 0.109891 sin(5t)− 0.0908419 sin(6t) +0.586021 cos(t)− 0.172359 cos(2t) + 0.079192 cos(3t)−0.0450785 cos(4t) + 0.0290109 cos(5t)− 0.0202076 cos(6t)−0.465052


The first Fourier functions:

-3 -2 -1 1 2 3

0.5

1.0

1.5

2.0

-3 -2 -1 1 2 3

-1.0

-0.5

0.5

1.0

-3 -2 -1 1 2 3

-1.0

-0.5

0.5

1.0

-3 -2 -1 1 2 3

-1.0

-0.5

0.5

1.0

-3 -2 -1 1 2 3

-1.0

-0.5

0.5

1.0

-3 -2 -1 1 2 3

-1.0

-0.5

0.5

1.0

-3 -2 -1 1 2 3

-1.0

-0.5

0.5

1.0

-3 -2 -1 1 2 3

-1.0

-0.5

0.5

1.0


The JPEG algorithm uses Fourier techniques - it employs the“Discrete Cosine Fourier Transform (DCT)”.

Here is a JPEG example with compression factor 6:


Wavelet pioneers: AlfredHaar (t-r, on the left),

Stephane Mallat (b-r), IngridDaubechies (t)


The first Haar functions:

1

-1

1

1

-1

1

1

-1

1

1

-1

1

1

-1

1

1

-1

1

1

-1

1

1

-1

1


Original image


1st color channel: Y


2nd color channel: Cr


3rd color channel: Cb


Applying the CDF97-wavelet transform to Y once


. . . and again. . .


. . . and one more time:


Quantizing the Y channel


The sender then encodes this “quantized image” andsends it through the narrow channel to the receiver.

The compression factor in this example is 10.2.

The receiver then decompresses the image to be able toview the approximation of the original image.


“Undoing” the transform (by receiver)


Original image, again. . .


Zooming in on the original image:


Zooming in on the received image:


Project 01

Due Tuesday, February 3, 2009

� Problem 1

Let B =

1 2 34 5 67 8 10

. Find matrices A and C such that

A.B.C =

7 8 104 5 61 2 3

. Can you choose A or C to be the identity matrix?


Due February 17, 2009

� Problem 2

Pick a small grayscale image (your choice) and use matrix transposition and multiplication with suitable matrices to

produce the following six images :


Project 04

Due April 9, 2009

� Problem 1 (15 points)

Write a module ' InverseHaarWLTransform[pic_, n_]' to apply the inverse Haar Transform 'n' times to the matrix ' pic'.

Check (using PSNR) that if you apply the Haar transform three times and then your module (with n=3), you obtain the

original image back (modulo loss due to round-off error).

� Problem 2 (15 points)

Write a procedure (preferably a module) to graph a function and its approximation in the vector spaceV4 defined in the handout "Multi-resolution Analysis".

Here is the definition of the father wavelet and the plot of one of the basis elements in V4 :

phiAx_E = Piecewise@881, 0 < x < 1<<, 0D


0.2 0.4 0.6 0.8 1.0

0.8

1.0

1.2


Final Projects - Setup:

Three student groups with three studentsEach group prepares a 20-minute oral presentation using aMathematica notebook.Project topics:

1 Matrix Completion

2 Huffman Tree and Coding3 Zooming in on an Image



Three student groups with three students

Each group prepares a 20-minute oral presentation using aMathematica notebook.Project topics:

1 Matrix Completion




Three student groups with three studentsEach group prepares a 20-minute oral presentation using aMathematica notebook.

Project topics:1 Matrix Completion





1 Matrix Completion





1 Matrix Completion2 Huffman Tree and Coding

3 Zooming in on an Image




1 Matrix Completion2 Huffman Tree and Coding3 Zooming in on an Image


Project A: Matrix Completion∗

Objective: Replace the standard D4-filter by a modified filterthat avoids coefficient wrapping of the last row/column.

Here is the shape of the modified D4-filter:

W8 =

a2 a1 a0 0 0 0 0 00 h3 h2 h1 h0 0 0 00 0 0 h3 h2 h1 h0 00 0 0 0 0 b2 b1 b0

c2 c1 c0 0 0 0 0 00 −h0 h1 −h2 h3 0 0 00 0 0 −h0 h1 −h2 h3 00 0 0 0 0 d2 d1 d0

∗Based on Problem 7.19 of the textbook


Project A: Matrix Completion∗

Objective: Replace the standard D4-filter by a modified filterthat avoids coefficient wrapping of the last row/column.Here is the shape of the modified D4-filter:

W8 =

a2 a1 a0 0 0 0 0 00 h3 h2 h1 h0 0 0 00 0 0 h3 h2 h1 h0 00 0 0 0 0 b2 b1 b0

c2 c1 c0 0 0 0 0 00 −h0 h1 −h2 h3 0 0 00 0 0 −h0 h1 −h2 h3 00 0 0 0 0 d2 d1 d0

∗Based on Problem 7.19 of the textbook


Matrix Completion

Original D4


Matrix Completion

Modified D4


Project B: Huffman Tree and Coding

Objective: Write Mathematica commandsto Huffman encode,to Huffman decode, andto produce a Huffman tree.

In[6]:= CodeWord@"Mississippi", 88i, "0"<, 8M, "100"<, 8p, "101"<, 8s, "11"<<DThe coded bit stream for Mississippi is 100011110111101011010 .

In[10]:= Uncodestring@"100011110111101011010",88i, "0"<, 8M, "100"<, 8p, "101"<, 8s, "11"<<D

The word for the bit stream 100011110111101011010 is Mississippi.














Huffman Tree and Coding

In[59]:= HuffTree@"Supercalifragilisticexpialidocious"D

0 1

0 1 0 1

0 1 0 10 1

0 1 0 10 1 0 1 0 1 0 1

0 1 0 1 0 1

i

a cu le p s ro

f S t gd x


Project C: Zooming In

bB

vV

hH

dD


Any Questions?

Contact: [email protected]


Documents

Teaching a Discrete Wavelets Transform Coursehelmut.knaust.info/presentations/2010/20100409_DWT_MAA.pdf · 2010-04-13 · Discrete Wavelet Transformations: An Elementary Approach