Today’s Schedule Acknowledgements Workshop Structure The Course Review Material In the Classroom

Why Wavelets?

Patrick J. Van Fleet

Center for Applied MathematicsUniversity of St. Thomas

St. Paul, MN USA

PREP - Wavelet Workshop, 2006

Wednesday, 7 June, 2006 Lecture 1

Why Wavelets?

Today’s Schedule Acknowledgements Workshop Structure The Course Review Material In the Classroom

OutlineToday’s ScheduleAcknowledgementsWorkshop StructureThe Course

Intent of the CourseWhy Wavelets?Course Design

Review MaterialIn the Classroom

Teaching IdeasComputer UsageStudent Difficulties

Wednesday, 7 June, 2006 Lecture 1

Why Wavelets?

Today’s Schedule Acknowledgements Workshop Structure The Course Review Material In the Classroom

Today’s Schedule

9:00-10:15 ⇒Lecture One: Why Wavelets?10:15-10:30 Coffee Break (OSS 235)10:30-11:45 Lecture Two: Digital Images, Measures, and

Huffman Codes12:00-1:00 Lunch (Cafeteria)1:30-2:45 Lecture Three: Fourier Series, Convolution and

Filters2:45-3:00 Coffee Break (OSS 235)3:00-4:15 Lecture Four: 1D and 2D Haar Transforms5:30-6:30 Dinner (Cafeteria)

Wednesday, 7 June, 2006 Lecture 1

Why Wavelets?

Today’s Schedule Acknowledgements Workshop Structure The Course Review Material In the Classroom

Acknowledgements

I National Science FoundationI DUE-0442684

I Mathematical Association of AmericaI University of St. Thomas

Wednesday, 7 June, 2006 Lecture 1

Why Wavelets?

Today’s Schedule Acknowledgements Workshop Structure The Course Review Material In the Classroom

Workshop Structure

I LecturesI Mathematical ContentI Teaching IdeasI Use of ComputerI Student Difficulties

I Computer SessionsI Writing ModulesI Working Through LabsI Learn WaveletFunctions package

Wednesday, 7 June, 2006 Lecture 1

Why Wavelets?

Today’s Schedule Acknowledgements Workshop Structure The Course Review Material In the Classroom

Workshop Structure

I LecturesI Mathematical ContentI Teaching IdeasI Use of ComputerI Student Difficulties

I Computer SessionsI Writing ModulesI Working Through LabsI Learn WaveletFunctions package

Wednesday, 7 June, 2006 Lecture 1

Why Wavelets?

Today’s Schedule Acknowledgements Workshop Structure The Course Review Material In the Classroom

Workshop Structure

I LecturesI Mathematical ContentI Teaching IdeasI Use of ComputerI Student Difficulties

I Computer SessionsI Writing ModulesI Working Through LabsI Learn WaveletFunctions package

Wednesday, 7 June, 2006 Lecture 1

Why Wavelets?

Today’s Schedule Acknowledgements Workshop Structure The Course Review Material In the Classroom

Workshop Structure

I LecturesI Mathematical ContentI Teaching IdeasI Use of ComputerI Student Difficulties

I Computer SessionsI Writing ModulesI Working Through LabsI Learn WaveletFunctions package

Wednesday, 7 June, 2006 Lecture 1

Why Wavelets?

Today’s Schedule Acknowledgements Workshop Structure The Course Review Material In the Classroom

Workshop Structure

I LecturesI Mathematical ContentI Teaching IdeasI Use of ComputerI Student Difficulties

I Computer SessionsI Writing ModulesI Working Through LabsI Learn WaveletFunctions package

Wednesday, 7 June, 2006 Lecture 1

Why Wavelets?

Today’s Schedule Acknowledgements Workshop Structure The Course Review Material In the Classroom

Workshop Structure

I LecturesI Mathematical ContentI Teaching IdeasI Use of ComputerI Student Difficulties

I Computer SessionsI Writing ModulesI Working Through LabsI Learn WaveletFunctions package

Wednesday, 7 June, 2006 Lecture 1

Why Wavelets?

Today’s Schedule Acknowledgements Workshop Structure The Course Review Material In the Classroom

Workshop Structure

I LecturesI Mathematical ContentI Teaching IdeasI Use of ComputerI Student Difficulties

I Computer SessionsI Writing ModulesI Working Through LabsI Learn WaveletFunctions package

Wednesday, 7 June, 2006 Lecture 1

Why Wavelets?

Today’s Schedule Acknowledgements Workshop Structure The Course Review Material In the Classroom

Workshop Structure

I LecturesI Mathematical ContentI Teaching IdeasI Use of ComputerI Student Difficulties

I Computer SessionsI Writing ModulesI Working Through LabsI Learn WaveletFunctions package

Wednesday, 7 June, 2006 Lecture 1

Why Wavelets?

Today’s Schedule Acknowledgements Workshop Structure The Course Review Material In the Classroom

Workshop Structure

I LecturesI Mathematical ContentI Teaching IdeasI Use of ComputerI Student Difficulties

I Computer SessionsI Writing ModulesI Working Through LabsI Learn WaveletFunctions package

Wednesday, 7 June, 2006 Lecture 1

Why Wavelets?

Today’s Schedule Acknowledgements Workshop Structure The Course Review Material In the Classroom

Workshop Structure

I LecturesI Mathematical ContentI Teaching IdeasI Use of ComputerI Student Difficulties

I Computer SessionsI Writing ModulesI Working Through LabsI Learn WaveletFunctions package

Wednesday, 7 June, 2006 Lecture 1

Why Wavelets?

Today’s Schedule Acknowledgements Workshop Structure The Course Review Material In the Classroom

Intent of the Course

The CourseIntent of the Course

Many schools offer a Mathematical Modeling Course.I Students learn how to use math to solve real-world

problems.I Reinforces mathematics learned to date through modelling.I Invariably taken/offered after students have had an

analysis sequence.I Usually involves some computer work.

Wednesday, 7 June, 2006 Lecture 1

Why Wavelets?

Today’s Schedule Acknowledgements Workshop Structure The Course Review Material In the Classroom

Intent of the Course

The CourseIntent of the Course

Many schools offer a Mathematical Modeling Course.I Students learn how to use math to solve real-world

problems.I Reinforces mathematics learned to date through modelling.I Invariably taken/offered after students have had an

analysis sequence.I Usually involves some computer work.

Wednesday, 7 June, 2006 Lecture 1

Why Wavelets?

Today’s Schedule Acknowledgements Workshop Structure The Course Review Material In the Classroom

Intent of the Course

The CourseIntent of the Course

Many schools offer a Mathematical Modeling Course.I Students learn how to use math to solve real-world

problems.I Reinforces mathematics learned to date through modelling.I Invariably taken/offered after students have had an

analysis sequence.I Usually involves some computer work.

Wednesday, 7 June, 2006 Lecture 1

Why Wavelets?

Today’s Schedule Acknowledgements Workshop Structure The Course Review Material In the Classroom

Intent of the Course

The CourseIntent of the Course

Many schools offer a Mathematical Modeling Course.I Students learn how to use math to solve real-world

problems.I Reinforces mathematics learned to date through modelling.I Invariably taken/offered after students have had an

analysis sequence.I Usually involves some computer work.

Wednesday, 7 June, 2006 Lecture 1

Why Wavelets?

Today’s Schedule Acknowledgements Workshop Structure The Course Review Material In the Classroom

Intent of the Course

The CourseWavelets and Applications Course

We want students to become immediately immersed inapplications:

I The applications should be current.I The students should actually be able to produce results -

keep the concocted examples to a minimum.I The math should be accessible to someone who has

completed Calculus II and Linear Algebra.I It is good if the modeling of the application is ad hoc and

draw from different areas of mathematics.

Wednesday, 7 June, 2006 Lecture 1

Why Wavelets?

Today’s Schedule Acknowledgements Workshop Structure The Course Review Material In the Classroom

Intent of the Course

The CourseWavelets and Applications Course

We want students to become immediately immersed inapplications:

I The applications should be current.I The students should actually be able to produce results -

keep the concocted examples to a minimum.I The math should be accessible to someone who has

completed Calculus II and Linear Algebra.I It is good if the modeling of the application is ad hoc and

draw from different areas of mathematics.

Wednesday, 7 June, 2006 Lecture 1

Why Wavelets?

Today’s Schedule Acknowledgements Workshop Structure The Course Review Material In the Classroom

Intent of the Course

The CourseWavelets and Applications Course

We want students to become immediately immersed inapplications:

I The applications should be current.I The students should actually be able to produce results -

keep the concocted examples to a minimum.I The math should be accessible to someone who has

completed Calculus II and Linear Algebra.I It is good if the modeling of the application is ad hoc and

draw from different areas of mathematics.

Wednesday, 7 June, 2006 Lecture 1

Why Wavelets?

Today’s Schedule Acknowledgements Workshop Structure The Course Review Material In the Classroom

Intent of the Course

The CourseWavelets and Applications Course

We want students to become immediately immersed inapplications:

I The applications should be current.I The students should actually be able to produce results -

keep the concocted examples to a minimum.I The math should be accessible to someone who has

completed Calculus II and Linear Algebra.I It is good if the modeling of the application is ad hoc and

draw from different areas of mathematics.

Wednesday, 7 June, 2006 Lecture 1

Why Wavelets?

Today’s Schedule Acknowledgements Workshop Structure The Course Review Material In the Classroom

Intent of the Course

The CourseWavelets and Applications Course

We want students to become immediately immersed inapplications:

I The applications should be current.I The students should actually be able to produce results -

keep the concocted examples to a minimum.I The math should be accessible to someone who has

completed Calculus II and Linear Algebra.I It is good if the modeling of the application is ad hoc and

draw from different areas of mathematics.

Wednesday, 7 June, 2006 Lecture 1

Why Wavelets?

Today’s Schedule Acknowledgements Workshop Structure The Course Review Material In the Classroom

Intent of the Course

The CourseWavelets and Applications Course

We want students to learn how to problem solve:I The new math should be by and large easy and much of

what is done uses Calculus and Linear Algebra.I They should be comfortable with things like looking up a

trig identity, exploiting symmetry, manipulating sums, etc.I Reverse engineering is a foreign concept to most

mathematics majors - we do a lot of that.I The ad hoc modeling should ultimately lead them to a brick

wall and require them to look at more abstractmathematics.

Wednesday, 7 June, 2006 Lecture 1

Why Wavelets?

Today’s Schedule Acknowledgements Workshop Structure The Course Review Material In the Classroom

Intent of the Course

The CourseWavelets and Applications Course

We want students to learn how to problem solve:I The new math should be by and large easy and much of

what is done uses Calculus and Linear Algebra.I They should be comfortable with things like looking up a

trig identity, exploiting symmetry, manipulating sums, etc.I Reverse engineering is a foreign concept to most

mathematics majors - we do a lot of that.I The ad hoc modeling should ultimately lead them to a brick

wall and require them to look at more abstractmathematics.

Wednesday, 7 June, 2006 Lecture 1

Why Wavelets?

Today’s Schedule Acknowledgements Workshop Structure The Course Review Material In the Classroom

Intent of the Course

The CourseWavelets and Applications Course

We want students to learn how to problem solve:I The new math should be by and large easy and much of

what is done uses Calculus and Linear Algebra.I They should be comfortable with things like looking up a

trig identity, exploiting symmetry, manipulating sums, etc.I Reverse engineering is a foreign concept to most

mathematics majors - we do a lot of that.I The ad hoc modeling should ultimately lead them to a brick

wall and require them to look at more abstractmathematics.

Wednesday, 7 June, 2006 Lecture 1

Why Wavelets?

Today’s Schedule Acknowledgements Workshop Structure The Course Review Material In the Classroom

Intent of the Course

The CourseWavelets and Applications Course

We want students to learn how to problem solve:I The new math should be by and large easy and much of

what is done uses Calculus and Linear Algebra.I They should be comfortable with things like looking up a

trig identity, exploiting symmetry, manipulating sums, etc.I Reverse engineering is a foreign concept to most

mathematics majors - we do a lot of that.I The ad hoc modeling should ultimately lead them to a brick

wall and require them to look at more abstractmathematics.

Wednesday, 7 June, 2006 Lecture 1

Why Wavelets?

Today’s Schedule Acknowledgements Workshop Structure The Course Review Material In the Classroom

Intent of the Course

The CourseWavelets and Applications Course

We want students to learn how to problem solve:I The new math should be by and large easy and much of

what is done uses Calculus and Linear Algebra.I They should be comfortable with things like looking up a

trig identity, exploiting symmetry, manipulating sums, etc.I Reverse engineering is a foreign concept to most

mathematics majors - we do a lot of that.I The ad hoc modeling should ultimately lead them to a brick

wall and require them to look at more abstractmathematics.

Wednesday, 7 June, 2006 Lecture 1

Why Wavelets?

Today’s Schedule Acknowledgements Workshop Structure The Course Review Material In the Classroom

Intent of the Course

The CourseWavelets and Applications Course

We want to establish the computer as a learning tool. Studentsshould:

I get to the point where they are comfortable testing ideason the computer.

I learn how to write modules.I shoot for efficient modules, but don’t give up total

readability.

Wednesday, 7 June, 2006 Lecture 1

Why Wavelets?

Today’s Schedule Acknowledgements Workshop Structure The Course Review Material In the Classroom

Intent of the Course

The CourseWavelets and Applications Course

We want to establish the computer as a learning tool. Studentsshould:

I get to the point where they are comfortable testing ideason the computer.

I learn how to write modules.I shoot for efficient modules, but don’t give up total

readability.

Wednesday, 7 June, 2006 Lecture 1

Why Wavelets?

Today’s Schedule Acknowledgements Workshop Structure The Course Review Material In the Classroom

Intent of the Course

The CourseWavelets and Applications Course

We want to establish the computer as a learning tool. Studentsshould:

I get to the point where they are comfortable testing ideason the computer.

I learn how to write modules.I shoot for efficient modules, but don’t give up total

readability.

Wednesday, 7 June, 2006 Lecture 1

Why Wavelets?

Today’s Schedule Acknowledgements Workshop Structure The Course Review Material In the Classroom

Intent of the Course

The CourseWavelets and Applications Course

We want to establish the computer as a learning tool. Studentsshould:

I get to the point where they are comfortable testing ideason the computer.

I learn how to write modules.I shoot for efficient modules, but don’t give up total

readability.

Wednesday, 7 June, 2006 Lecture 1

Why Wavelets?

Today’s Schedule Acknowledgements Workshop Structure The Course Review Material In the Classroom

Intent of the Course

The CourseWavelets and Applications Course

We want to establish the computer as a learning tool. Studentsshould:

I learn about the software package - explore help, try newfunctions, play with function parameters, etc.

I learn how to debug software they’ve written.I learn how to use the computer to learn mathematics

Wednesday, 7 June, 2006 Lecture 1

Why Wavelets?

Today’s Schedule Acknowledgements Workshop Structure The Course Review Material In the Classroom

Intent of the Course

The CourseWavelets and Applications Course

We want to establish the computer as a learning tool. Studentsshould:

I learn about the software package - explore help, try newfunctions, play with function parameters, etc.

I learn how to debug software they’ve written.I learn how to use the computer to learn mathematics

Wednesday, 7 June, 2006 Lecture 1

Why Wavelets?

Today’s Schedule Acknowledgements Workshop Structure The Course Review Material In the Classroom

Intent of the Course

The CourseWavelets and Applications Course

We want to establish the computer as a learning tool. Studentsshould:

I learn about the software package - explore help, try newfunctions, play with function parameters, etc.

I learn how to debug software they’ve written.I learn how to use the computer to learn mathematics

Wednesday, 7 June, 2006 Lecture 1

Why Wavelets?

Today’s Schedule Acknowledgements Workshop Structure The Course Review Material In the Classroom

Intent of the Course

The CourseWavelets and Applications Course

We want to establish the computer as a learning tool. Studentsshould:

I learn about the software package - explore help, try newfunctions, play with function parameters, etc.

I learn how to debug software they’ve written.I learn how to use the computer to learn mathematics

Wednesday, 7 June, 2006 Lecture 1

Why Wavelets?

Today’s Schedule Acknowledgements Workshop Structure The Course Review Material In the Classroom

Intent of the Course

The CourseWavelets and Applications Course

Students should leave the class with:I The confidence to at least try to solve any problem.I A motivation for why upper level courses such as Analysis

are important.I An appreciation of mathematics in the real world and an

understanding of where the current research is headed.I Confidence to use the computer as a learning tool in other

courses.I An understanding of the value of working in the transform

domain.

Wednesday, 7 June, 2006 Lecture 1

Why Wavelets?

Today’s Schedule Acknowledgements Workshop Structure The Course Review Material In the Classroom

Intent of the Course

The CourseWavelets and Applications Course

Students should leave the class with:I The confidence to at least try to solve any problem.I A motivation for why upper level courses such as Analysis

are important.I An appreciation of mathematics in the real world and an

understanding of where the current research is headed.I Confidence to use the computer as a learning tool in other

courses.I An understanding of the value of working in the transform

domain.

Wednesday, 7 June, 2006 Lecture 1

Why Wavelets?

Today’s Schedule Acknowledgements Workshop Structure The Course Review Material In the Classroom

Intent of the Course

The CourseWavelets and Applications Course

Students should leave the class with:I The confidence to at least try to solve any problem.I A motivation for why upper level courses such as Analysis

are important.I An appreciation of mathematics in the real world and an

understanding of where the current research is headed.I Confidence to use the computer as a learning tool in other

courses.I An understanding of the value of working in the transform

domain.

Wednesday, 7 June, 2006 Lecture 1

Why Wavelets?

Today’s Schedule Acknowledgements Workshop Structure The Course Review Material In the Classroom

Intent of the Course

The CourseWavelets and Applications Course

Students should leave the class with:I The confidence to at least try to solve any problem.I A motivation for why upper level courses such as Analysis

are important.I An appreciation of mathematics in the real world and an

understanding of where the current research is headed.I Confidence to use the computer as a learning tool in other

courses.I An understanding of the value of working in the transform

domain.

Wednesday, 7 June, 2006 Lecture 1

Why Wavelets?

Today’s Schedule Acknowledgements Workshop Structure The Course Review Material In the Classroom

Intent of the Course

The CourseWavelets and Applications Course

Students should leave the class with:I The confidence to at least try to solve any problem.I A motivation for why upper level courses such as Analysis

are important.I An appreciation of mathematics in the real world and an

understanding of where the current research is headed.I Confidence to use the computer as a learning tool in other

courses.I An understanding of the value of working in the transform

domain.

Wednesday, 7 June, 2006 Lecture 1

Why Wavelets?

Today’s Schedule Acknowledgements Workshop Structure The Course Review Material In the Classroom

Intent of the Course

The CourseWavelets and Applications Course

Students should leave the class with:I The confidence to at least try to solve any problem.I A motivation for why upper level courses such as Analysis

are important.I An appreciation of mathematics in the real world and an

understanding of where the current research is headed.I Confidence to use the computer as a learning tool in other

courses.I An understanding of the value of working in the transform

domain.

Wednesday, 7 June, 2006 Lecture 1

Why Wavelets?

Today’s Schedule Acknowledgements Workshop Structure The Course Review Material In the Classroom

Intent of the Course

The CourseWavelets and Applications Course

The instructor should:I learn to deal with subjective solutions to problems.I not worry about the technical details - at least in the

beginning.I be aware that students don’t really have that strong a grip

on concepts from previous mathematics courses and bepatient.

Wednesday, 7 June, 2006 Lecture 1

Why Wavelets?

Today’s Schedule Acknowledgements Workshop Structure The Course Review Material In the Classroom

Intent of the Course

The CourseWavelets and Applications Course

The instructor should:I learn to deal with subjective solutions to problems.I not worry about the technical details - at least in the

beginning.I be aware that students don’t really have that strong a grip

on concepts from previous mathematics courses and bepatient.

Wednesday, 7 June, 2006 Lecture 1

Why Wavelets?

Today’s Schedule Acknowledgements Workshop Structure The Course Review Material In the Classroom

Intent of the Course

The CourseWavelets and Applications Course

The instructor should:I learn to deal with subjective solutions to problems.I not worry about the technical details - at least in the

beginning.I be aware that students don’t really have that strong a grip

on concepts from previous mathematics courses and bepatient.

Wednesday, 7 June, 2006 Lecture 1

Why Wavelets?

Today’s Schedule Acknowledgements Workshop Structure The Course Review Material In the Classroom

Intent of the Course

The CourseWavelets and Applications Course

The instructor should:I learn to deal with subjective solutions to problems.I not worry about the technical details - at least in the

beginning.I be aware that students don’t really have that strong a grip

on concepts from previous mathematics courses and bepatient.

Wednesday, 7 June, 2006 Lecture 1

Why Wavelets?

Today’s Schedule Acknowledgements Workshop Structure The Course Review Material In the Classroom

Intent of the Course

The CourseWavelets and Applications Course

The instructor should:I realize that students cannot typically write code and debug

like we would hope.I be extra patient on things like working with Fourier series.I realize that on Day 30, you will still have students who

don’t immediately realize that eikπ = (−1)k !

Wednesday, 7 June, 2006 Lecture 1

Why Wavelets?

Today’s Schedule Acknowledgements Workshop Structure The Course Review Material In the Classroom

Intent of the Course

The CourseWavelets and Applications Course

The instructor should:I realize that students cannot typically write code and debug

like we would hope.I be extra patient on things like working with Fourier series.I realize that on Day 30, you will still have students who

don’t immediately realize that eikπ = (−1)k !

Wednesday, 7 June, 2006 Lecture 1

Why Wavelets?

Today’s Schedule Acknowledgements Workshop Structure The Course Review Material In the Classroom

Intent of the Course

The CourseWavelets and Applications Course

The instructor should:I realize that students cannot typically write code and debug

like we would hope.I be extra patient on things like working with Fourier series.I realize that on Day 30, you will still have students who

don’t immediately realize that eikπ = (−1)k !

Wednesday, 7 June, 2006 Lecture 1

Why Wavelets?

Today’s Schedule Acknowledgements Workshop Structure The Course Review Material In the Classroom

Intent of the Course

The CourseWavelets and Applications Course

The instructor should:I realize that students cannot typically write code and debug

like we would hope.I be extra patient on things like working with Fourier series.I realize that on Day 30, you will still have students who

don’t immediately realize that eikπ = (−1)k !

Wednesday, 7 June, 2006 Lecture 1

Why Wavelets?

Today’s Schedule Acknowledgements Workshop Structure The Course Review Material In the Classroom

Why Wavelets?

The CourseWhy Wavelets?

What would a math lecture be without an exam? It’s time totake a test!

You are going to see three images. One is the original imageconsisting of 149604 bytes of information. A second image is awavelet-compressed version of the original using 12253 bytes(about 8% of the original size), and another image is awavelet-compressed version of the original using only 4452bytes (about 3% of the original size)!

Wednesday, 7 June, 2006 Lecture 1

Why Wavelets?

Today’s Schedule Acknowledgements Workshop Structure The Course Review Material In the Classroom

Why Wavelets?

The CourseWhy Wavelets?

Question: Which is which?

The following images come from a really neat site developed and maintainedby Osmar R. Zaïane at Simon Fraser University in Canada.

Wednesday, 7 June, 2006 Lecture 1

Why Wavelets?

Today’s Schedule Acknowledgements Workshop Structure The Course Review Material In the Classroom

Why Wavelets?

The CourseWhy Wavelets?

Image A

Wednesday, 7 June, 2006 Lecture 1

Why Wavelets?

Today’s Schedule Acknowledgements Workshop Structure The Course Review Material In the Classroom

Why Wavelets?

The CourseWhy Wavelets?

Image B

Wednesday, 7 June, 2006 Lecture 1

Why Wavelets?

Today’s Schedule Acknowledgements Workshop Structure The Course Review Material In the Classroom

Why Wavelets?

The CourseWhy Wavelets?

Image C

Wednesday, 7 June, 2006 Lecture 1

Why Wavelets?

Today’s Schedule Acknowledgements Workshop Structure The Course Review Material In the Classroom

Why Wavelets?

The CourseWhy Wavelets?

The Answer Key:

Image Number of Bytes Transfer Time*Image A 4452 0.15 secondsImage B 149604 5.00 secondsImage C 12253 0.40 seconds

*This rate assumes a standard 56K dial up modem thattypically connects at about 30K.

Wednesday, 7 June, 2006 Lecture 1

Why Wavelets?

Today’s Schedule Acknowledgements Workshop Structure The Course Review Material In the Classroom

Why Wavelets?

The CourseSo Why Wavelets?

I It is definitely new math - the subject took off in the late80’s/early 90’s.

I If you ask engineering or physics students to name someof the top accomplishments in their respective fields in thepast 100 years, they can.

I Mathematics students, when asked the same question,typically struggle to name one or two.

I The reason is most of the math we cover was developedhundreds of years ago and most of those famousmathematicians are ...

Wednesday, 7 June, 2006 Lecture 1

Why Wavelets?

Today’s Schedule Acknowledgements Workshop Structure The Course Review Material In the Classroom

Why Wavelets?

The CourseSo Why Wavelets?

I It is definitely new math - the subject took off in the late80’s/early 90’s.

I If you ask engineering or physics students to name someof the top accomplishments in their respective fields in thepast 100 years, they can.

I Mathematics students, when asked the same question,typically struggle to name one or two.

I The reason is most of the math we cover was developedhundreds of years ago and most of those famousmathematicians are ...

Wednesday, 7 June, 2006 Lecture 1

Why Wavelets?

Today’s Schedule Acknowledgements Workshop Structure The Course Review Material In the Classroom

Why Wavelets?

The CourseSo Why Wavelets?

I It is definitely new math - the subject took off in the late80’s/early 90’s.

I If you ask engineering or physics students to name someof the top accomplishments in their respective fields in thepast 100 years, they can.

I Mathematics students, when asked the same question,typically struggle to name one or two.

I The reason is most of the math we cover was developedhundreds of years ago and most of those famousmathematicians are ...

Wednesday, 7 June, 2006 Lecture 1

Why Wavelets?

Today’s Schedule Acknowledgements Workshop Structure The Course Review Material In the Classroom

Why Wavelets?

The CourseSo Why Wavelets?

I It is definitely new math - the subject took off in the late80’s/early 90’s.

I If you ask engineering or physics students to name someof the top accomplishments in their respective fields in thepast 100 years, they can.

I Mathematics students, when asked the same question,typically struggle to name one or two.

I The reason is most of the math we cover was developedhundreds of years ago and most of those famousmathematicians are ...

Wednesday, 7 June, 2006 Lecture 1

Why Wavelets?

Today’s Schedule Acknowledgements Workshop Structure The Course Review Material In the Classroom

Why Wavelets?

The CourseSo Why Wavelets?

Isaac Newton - Dead.

Wednesday, 7 June, 2006 Lecture 1

Why Wavelets?

Today’s Schedule Acknowledgements Workshop Structure The Course Review Material In the Classroom

Why Wavelets?

The CourseSo Why Wavelets?

Joseph Fourier - Dead.

Wednesday, 7 June, 2006 Lecture 1

Why Wavelets?

Today’s Schedule Acknowledgements Workshop Structure The Course Review Material In the Classroom

Why Wavelets?

The CourseSo Why Wavelets?

C.F. Gauss - Dead .... But Brain is Still Around.

Wednesday, 7 June, 2006 Lecture 1

Why Wavelets?

Today’s Schedule Acknowledgements Workshop Structure The Course Review Material In the Classroom

Why Wavelets?

The CourseSo Why Wavelets?

But wavelet theory is new math and the major players are...

Wednesday, 7 June, 2006 Lecture 1

Why Wavelets?

Today’s Schedule Acknowledgements Workshop Structure The Course Review Material In the Classroom

Why Wavelets?

The CourseSo Why Wavelets?

Ingrid Daubechies - Still Alive!

Wednesday, 7 June, 2006 Lecture 1

Why Wavelets?

Today’s Schedule Acknowledgements Workshop Structure The Course Review Material In the Classroom

Why Wavelets?

The CourseSo Why Wavelets?

David Donoho - Still Alive!

Wednesday, 7 June, 2006 Lecture 1

Why Wavelets?

Today’s Schedule Acknowledgements Workshop Structure The Course Review Material In the Classroom

Why Wavelets?

The CourseSo Why Wavelets?

Yves Meyer - Blurry, But Still Alive!

Wednesday, 7 June, 2006 Lecture 1

Why Wavelets?

Today’s Schedule Acknowledgements Workshop Structure The Course Review Material In the Classroom

Why Wavelets?

The CourseSo Why Wavelets?

Gil Strang, Still Alive!

Wednesday, 7 June, 2006 Lecture 1

Why Wavelets?

Today’s Schedule Acknowledgements Workshop Structure The Course Review Material In the Classroom

Why Wavelets?

The CourseSo Why Wavelets?

Applications involving wavelets are numerous. Here are two:I The FBI uses wavelets to digitally compress fingerprints.

The compression factor is between 40 and 100 times theoriginal size of the fingerprint!

I Starting in 2000, the Joint Photographic Experts Group,designers of the popular .jpg image format used on theweb, began implementing wavelet techniques in theirformat scheme.

Wednesday, 7 June, 2006 Lecture 1

Why Wavelets?

Today’s Schedule Acknowledgements Workshop Structure The Course Review Material In the Classroom

Why Wavelets?

The CourseSo Why Wavelets?

Applications involving wavelets are numerous. Here are two:I The FBI uses wavelets to digitally compress fingerprints.

The compression factor is between 40 and 100 times theoriginal size of the fingerprint!

I Starting in 2000, the Joint Photographic Experts Group,designers of the popular .jpg image format used on theweb, began implementing wavelet techniques in theirformat scheme.

Wednesday, 7 June, 2006 Lecture 1

Why Wavelets?

Today’s Schedule Acknowledgements Workshop Structure The Course Review Material In the Classroom

Why Wavelets?

The CourseSo Why Wavelets?

Applications involving wavelets are numerous. Here are two:I The FBI uses wavelets to digitally compress fingerprints.

The compression factor is between 40 and 100 times theoriginal size of the fingerprint!

I Starting in 2000, the Joint Photographic Experts Group,designers of the popular .jpg image format used on theweb, began implementing wavelet techniques in theirformat scheme.

Wednesday, 7 June, 2006 Lecture 1

Why Wavelets?

Today’s Schedule Acknowledgements Workshop Structure The Course Review Material In the Classroom

Why Wavelets?

The CourseSo Why Wavelets?

And two more:I Kodak Polychrome Graphics, a Twin Cities’ based

company, produces extremely high resolution digitalimages for advertisements that appear in publications likeTime Magazine. They use wavelets to allow their clients toquickly analyze parts of the image.

I Wavelets have been heavily utilized in the modeling ofdistant galaxies. Wavelets have helped astronomers locatea subcluster of galaxies in the Coma supercluster of 1,400galaxies! (Quantum Vol. 15 No. 3)

Wednesday, 7 June, 2006 Lecture 1

Why Wavelets?

Today’s Schedule Acknowledgements Workshop Structure The Course Review Material In the Classroom

Why Wavelets?

The CourseSo Why Wavelets?

And two more:I Kodak Polychrome Graphics, a Twin Cities’ based

company, produces extremely high resolution digitalimages for advertisements that appear in publications likeTime Magazine. They use wavelets to allow their clients toquickly analyze parts of the image.

I Wavelets have been heavily utilized in the modeling ofdistant galaxies. Wavelets have helped astronomers locatea subcluster of galaxies in the Coma supercluster of 1,400galaxies! (Quantum Vol. 15 No. 3)

Wednesday, 7 June, 2006 Lecture 1

Why Wavelets?

Today’s Schedule Acknowledgements Workshop Structure The Course Review Material In the Classroom

Why Wavelets?

The CourseSo Why Wavelets?

And two more:I Kodak Polychrome Graphics, a Twin Cities’ based

company, produces extremely high resolution digitalimages for advertisements that appear in publications likeTime Magazine. They use wavelets to allow their clients toquickly analyze parts of the image.

I Wavelets have been heavily utilized in the modeling ofdistant galaxies. Wavelets have helped astronomers locatea subcluster of galaxies in the Coma supercluster of 1,400galaxies! (Quantum Vol. 15 No. 3)

Wednesday, 7 June, 2006 Lecture 1

Why Wavelets?

Today’s Schedule Acknowledgements Workshop Structure The Course Review Material In the Classroom

Why Wavelets?

The CourseSo Why Wavelets?

And one more:I Musicologists restore an 1889 recording of Brahms playing

his Hungarian Dance Number 1 from a wax cylinderrecording in such bad shape that many listeners failed torecognize that a piano was even being played. (Quantum Vol.15 No. 3)

Wednesday, 7 June, 2006 Lecture 1

Why Wavelets?

Today’s Schedule Acknowledgements Workshop Structure The Course Review Material In the Classroom

Why Wavelets?

The CourseSo Why Wavelets?

Wavelets also fill my wish list in other ways:I If you stay away from the classical approach, the derivation

of the Discrete Wavelet Transform involves a lot of linearalgebra, some trigonometry, a little calculus, and anintroduction to convolution and Fourier series.

I Students can dive right into applications.I The derivation is largely ad hoc . . . until we need better

wavelet filters!

Wednesday, 7 June, 2006 Lecture 1

Why Wavelets?

Today’s Schedule Acknowledgements Workshop Structure The Course Review Material In the Classroom

Why Wavelets?

The CourseSo Why Wavelets?

Wavelets also fill my wish list in other ways:I If you stay away from the classical approach, the derivation

of the Discrete Wavelet Transform involves a lot of linearalgebra, some trigonometry, a little calculus, and anintroduction to convolution and Fourier series.

I Students can dive right into applications.I The derivation is largely ad hoc . . . until we need better

wavelet filters!

Wednesday, 7 June, 2006 Lecture 1

Why Wavelets?

Today’s Schedule Acknowledgements Workshop Structure The Course Review Material In the Classroom

Why Wavelets?

The CourseSo Why Wavelets?

Wavelets also fill my wish list in other ways:I If you stay away from the classical approach, the derivation

of the Discrete Wavelet Transform involves a lot of linearalgebra, some trigonometry, a little calculus, and anintroduction to convolution and Fourier series.

I Students can dive right into applications.I The derivation is largely ad hoc . . . until we need better

wavelet filters!

Wednesday, 7 June, 2006 Lecture 1

Why Wavelets?

Today’s Schedule Acknowledgements Workshop Structure The Course Review Material In the Classroom

Why Wavelets?

The CourseSo Why Wavelets?

Wavelets also fill my wish list in other ways:I If you stay away from the classical approach, the derivation

of the Discrete Wavelet Transform involves a lot of linearalgebra, some trigonometry, a little calculus, and anintroduction to convolution and Fourier series.

I Students can dive right into applications.I The derivation is largely ad hoc . . . until we need better

wavelet filters!

Wednesday, 7 June, 2006 Lecture 1

Why Wavelets?

Today’s Schedule Acknowledgements Workshop Structure The Course Review Material In the Classroom

Course Design

The CourseCourse Design

I Start with a quick review of vectors, inner products, andnorms, and also a couple of days on matrix analysis(multiplication, orthogonal, transpose, and block matrices).

I Learn about digital images, Fourier series (more in anengineering sense), convolution, and filters.

I Ad hoc development of the Haar Wavelet Transform(averages and differences).

I Applications - compression and edge detection.

Wednesday, 7 June, 2006 Lecture 1

Why Wavelets?

Today’s Schedule Acknowledgements Workshop Structure The Course Review Material In the Classroom

Course Design

The CourseCourse Design

I Start with a quick review of vectors, inner products, andnorms, and also a couple of days on matrix analysis(multiplication, orthogonal, transpose, and block matrices).

I Learn about digital images, Fourier series (more in anengineering sense), convolution, and filters.

I Ad hoc development of the Haar Wavelet Transform(averages and differences).

I Applications - compression and edge detection.

Wednesday, 7 June, 2006 Lecture 1

Why Wavelets?

Today’s Schedule Acknowledgements Workshop Structure The Course Review Material In the Classroom

Course Design

The CourseCourse Design

I Start with a quick review of vectors, inner products, andnorms, and also a couple of days on matrix analysis(multiplication, orthogonal, transpose, and block matrices).

I Learn about digital images, Fourier series (more in anengineering sense), convolution, and filters.

I Ad hoc development of the Haar Wavelet Transform(averages and differences).

I Applications - compression and edge detection.

Wednesday, 7 June, 2006 Lecture 1

Why Wavelets?

Today’s Schedule Acknowledgements Workshop Structure The Course Review Material In the Classroom

Course Design

The CourseCourse Design

I Start with a quick review of vectors, inner products, andnorms, and also a couple of days on matrix analysis(multiplication, orthogonal, transpose, and block matrices).

I Learn about digital images, Fourier series (more in anengineering sense), convolution, and filters.

I Ad hoc development of the Haar Wavelet Transform(averages and differences).

I Applications - compression and edge detection.

Wednesday, 7 June, 2006 Lecture 1

Why Wavelets?

Today’s Schedule Acknowledgements Workshop Structure The Course Review Material In the Classroom

Course Design

The CourseCourse Design

I Start with a quick review of vectors, inner products, andnorms, and also a couple of days on matrix analysis(multiplication, orthogonal, transpose, and block matrices).

I Learn about digital images, Fourier series (more in anengineering sense), convolution, and filters.

I Ad hoc development of the Haar Wavelet Transform(averages and differences).

I Applications - compression and edge detection.

Wednesday, 7 June, 2006 Lecture 1

Why Wavelets?

Today’s Schedule Acknowledgements Workshop Structure The Course Review Material In the Classroom

Course Design

The CourseCourse Design

I More ad hoc development of the Daubechies waveletfilters.

I Application - Audio and image de-noising.I At this point, we hit a wall - we need a little more formal

theory before proceeding. So we have basically spent 8weeks motivating why someone would want to learn aboutthese theoretical results.

Wednesday, 7 June, 2006 Lecture 1

Why Wavelets?

Today’s Schedule Acknowledgements Workshop Structure The Course Review Material In the Classroom

Course Design

The CourseCourse Design

I More ad hoc development of the Daubechies waveletfilters.

I Application - Audio and image de-noising.I At this point, we hit a wall - we need a little more formal

theory before proceeding. So we have basically spent 8weeks motivating why someone would want to learn aboutthese theoretical results.

Wednesday, 7 June, 2006 Lecture 1

Why Wavelets?

Today’s Schedule Acknowledgements Workshop Structure The Course Review Material In the Classroom

Course Design

The CourseCourse Design

I More ad hoc development of the Daubechies waveletfilters.

I Application - Audio and image de-noising.I At this point, we hit a wall - we need a little more formal

theory before proceeding. So we have basically spent 8weeks motivating why someone would want to learn aboutthese theoretical results.

Wednesday, 7 June, 2006 Lecture 1

Why Wavelets?

Today’s Schedule Acknowledgements Workshop Structure The Course Review Material In the Classroom

Course Design

The CourseCourse Design

I More ad hoc development of the Daubechies waveletfilters.

I Application - Audio and image de-noising.I At this point, we hit a wall - we need a little more formal

theory before proceeding. So we have basically spent 8weeks motivating why someone would want to learn aboutthese theoretical results.

Wednesday, 7 June, 2006 Lecture 1

Why Wavelets?

Today’s Schedule Acknowledgements Workshop Structure The Course Review Material In the Classroom

Course Design

The CourseCourse Design

I There is computer involvement from the outset - theinstructor has the option of having the students write theirown modules and plug them into the waveletfunctionspackage or just work through the labs using a completewaveletfunctions package.

I Instead of introducing the theory first and then showingsome examples and applications, we work in reverse order.

Wednesday, 7 June, 2006 Lecture 1

Why Wavelets?

Today’s Schedule Acknowledgements Workshop Structure The Course Review Material In the Classroom

Course Design

The CourseCourse Design

I There is computer involvement from the outset - theinstructor has the option of having the students write theirown modules and plug them into the waveletfunctionspackage or just work through the labs using a completewaveletfunctions package.

I Instead of introducing the theory first and then showingsome examples and applications, we work in reverse order.

Wednesday, 7 June, 2006 Lecture 1

Why Wavelets?

Today’s Schedule Acknowledgements Workshop Structure The Course Review Material In the Classroom

Course Design

The CourseCourse Design

I There is computer involvement from the outset - theinstructor has the option of having the students write theirown modules and plug them into the waveletfunctionspackage or just work through the labs using a completewaveletfunctions package.

I Instead of introducing the theory first and then showingsome examples and applications, we work in reverse order.

Wednesday, 7 June, 2006 Lecture 1

Why Wavelets?

Today’s Schedule Acknowledgements Workshop Structure The Course Review Material In the Classroom

Course Design

The CourseCourse Design

I Students are highly motivated to learn the theoreticalresults - at this point, they have working transform modulesand understand the potential of the wavelet transform.

I Develop results in the Fourier domain and then use themto design biorthogonal filter pairs.

I Applications - JPG2000 Compression Standard and FBIFingerprint Compression Scheme.

I End with class projects and presentations.

Wednesday, 7 June, 2006 Lecture 1

Why Wavelets?

Today’s Schedule Acknowledgements Workshop Structure The Course Review Material In the Classroom

Course Design

The CourseCourse Design

I Students are highly motivated to learn the theoreticalresults - at this point, they have working transform modulesand understand the potential of the wavelet transform.

I Develop results in the Fourier domain and then use themto design biorthogonal filter pairs.

I Applications - JPG2000 Compression Standard and FBIFingerprint Compression Scheme.

I End with class projects and presentations.

Wednesday, 7 June, 2006 Lecture 1

Why Wavelets?

Today’s Schedule Acknowledgements Workshop Structure The Course Review Material In the Classroom

Course Design

The CourseCourse Design

I Students are highly motivated to learn the theoreticalresults - at this point, they have working transform modulesand understand the potential of the wavelet transform.

I Develop results in the Fourier domain and then use themto design biorthogonal filter pairs.

I Applications - JPG2000 Compression Standard and FBIFingerprint Compression Scheme.

I End with class projects and presentations.

Wednesday, 7 June, 2006 Lecture 1

Why Wavelets?

Today’s Schedule Acknowledgements Workshop Structure The Course Review Material In the Classroom

Course Design

The CourseCourse Design

I Students are highly motivated to learn the theoreticalresults - at this point, they have working transform modulesand understand the potential of the wavelet transform.

I Develop results in the Fourier domain and then use themto design biorthogonal filter pairs.

I Applications - JPG2000 Compression Standard and FBIFingerprint Compression Scheme.

I End with class projects and presentations.

Wednesday, 7 June, 2006 Lecture 1

Why Wavelets?

Today’s Schedule Acknowledgements Workshop Structure The Course Review Material In the Classroom

Course Design

The CourseCourse Design

I Students are highly motivated to learn the theoreticalresults - at this point, they have working transform modulesand understand the potential of the wavelet transform.

I Develop results in the Fourier domain and then use themto design biorthogonal filter pairs.

I Applications - JPG2000 Compression Standard and FBIFingerprint Compression Scheme.

I End with class projects and presentations.

Wednesday, 7 June, 2006 Lecture 1

Why Wavelets?

Today’s Schedule Acknowledgements Workshop Structure The Course Review Material In the Classroom

Linear Algebra Review

I We start with a basic review of vectors, norms, and innerproducts - spend some time on orthogonal vectors.

I In particular, constructions like g = (d ,−c, b,−a) fromh = (a, b, c, d) produce orthogonal vectors.

I Basic matrix arithmetic, but from an imaging perspective:

Wednesday, 7 June, 2006 Lecture 1

Why Wavelets?

Today’s Schedule Acknowledgements Workshop Structure The Course Review Material In the Classroom

Linear Algebra Review

I We start with a basic review of vectors, norms, and innerproducts - spend some time on orthogonal vectors.

I In particular, constructions like g = (d ,−c, b,−a) fromh = (a, b, c, d) produce orthogonal vectors.

I Basic matrix arithmetic, but from an imaging perspective:

Wednesday, 7 June, 2006 Lecture 1

Why Wavelets?

Today’s Schedule Acknowledgements Workshop Structure The Course Review Material In the Classroom

Linear Algebra Review

I We start with a basic review of vectors, norms, and innerproducts - spend some time on orthogonal vectors.

I In particular, constructions like g = (d ,−c, b,−a) fromh = (a, b, c, d) produce orthogonal vectors.

I Basic matrix arithmetic, but from an imaging perspective:

Wednesday, 7 June, 2006 Lecture 1

Why Wavelets?

Today’s Schedule Acknowledgements Workshop Structure The Course Review Material In the Classroom

Linear Algebra Review

I We start with a basic review of vectors, norms, and innerproducts - spend some time on orthogonal vectors.

I In particular, constructions like g = (d ,−c, b,−a) fromh = (a, b, c, d) produce orthogonal vectors.

I Basic matrix arithmetic, but from an imaging perspective:

Wednesday, 7 June, 2006 Lecture 1

Why Wavelets?

Today’s Schedule Acknowledgements Workshop Structure The Course Review Material In the Classroom

Linear Algebra Review

Let A be the image below, c = .5, and T be a matrix whoseentries are 255. Then

A c · A

and

Wednesday, 7 June, 2006 Lecture 1

Why Wavelets?

Today’s Schedule Acknowledgements Workshop Structure The Course Review Material In the Classroom

Linear Algebra Review

Let A be the image below, c = .5, and T be a matrix whoseentries are 255. Then

A T − A

Wednesday, 7 June, 2006 Lecture 1

Why Wavelets?

Today’s Schedule Acknowledgements Workshop Structure The Course Review Material In the Classroom

Linear Algebra ReviewI We review matrix multiplication and try to emphasize that

A · B should be viewed as

A · B = [A · b1, . . . A · bN ]

A operating on the columns of B.I We discuss transposes and orthogonal matrices.I We look at A · x = b from a reverse engineering standpoint.

That is, for a given function f(x) and x, find matrix A so that

Ax = f(x)

Wednesday, 7 June, 2006 Lecture 1

Why Wavelets?

Today’s Schedule Acknowledgements Workshop Structure The Course Review Material In the Classroom

Linear Algebra ReviewI We review matrix multiplication and try to emphasize that

A · B should be viewed as

A · B = [A · b1, . . . A · bN ]

A operating on the columns of B.I We discuss transposes and orthogonal matrices.I We look at A · x = b from a reverse engineering standpoint.

That is, for a given function f(x) and x, find matrix A so that

Ax = f(x)

Wednesday, 7 June, 2006 Lecture 1

Why Wavelets?

Today’s Schedule Acknowledgements Workshop Structure The Course Review Material In the Classroom

Linear Algebra ReviewI We review matrix multiplication and try to emphasize that

A · B should be viewed as

A · B = [A · b1, . . . A · bN ]

A operating on the columns of B.I We discuss transposes and orthogonal matrices.I We look at A · x = b from a reverse engineering standpoint.

That is, for a given function f(x) and x, find matrix A so that

Ax = f(x)

Wednesday, 7 June, 2006 Lecture 1

Why Wavelets?

Today’s Schedule Acknowledgements Workshop Structure The Course Review Material In the Classroom

Linear Algebra ReviewI We review matrix multiplication and try to emphasize that

A · B should be viewed as

A · B = [A · b1, . . . A · bN ]

A operating on the columns of B.I We discuss transposes and orthogonal matrices.I We look at A · x = b from a reverse engineering standpoint.

That is, for a given function f(x) and x, find matrix A so that

Ax = f(x)

Wednesday, 7 June, 2006 Lecture 1

Why Wavelets?

Today’s Schedule Acknowledgements Workshop Structure The Course Review Material In the Classroom

Linear Algebra Review

I We spend a fair amount of time on block matrix arithmetic -in particular, block matrix multiplication and transpose.

I A typical problem might be to find block matrix W so that

W

H

G

=

G

H

Wednesday, 7 June, 2006 Lecture 1

Why Wavelets?

Today’s Schedule Acknowledgements Workshop Structure The Course Review Material In the Classroom

Linear Algebra Review

I We spend a fair amount of time on block matrix arithmetic -in particular, block matrix multiplication and transpose.

I A typical problem might be to find block matrix W so that

W

H

G

=

G

H

Wednesday, 7 June, 2006 Lecture 1

Why Wavelets?

Today’s Schedule Acknowledgements Workshop Structure The Course Review Material In the Classroom

Linear Algebra Review

I We spend a fair amount of time on block matrix arithmetic -in particular, block matrix multiplication and transpose.

I A typical problem might be to find block matrix W so that

W

H

G

=

G

H

Wednesday, 7 June, 2006 Lecture 1

Why Wavelets?

Today’s Schedule Acknowledgements Workshop Structure The Course Review Material In the Classroom

Linear Algebra Review

Find a block matrices V and W so that V ·A ·W , where A is theimage on the left, produces the image on the right.

A V · A ·W

Wednesday, 7 June, 2006 Lecture 1

Why Wavelets?

Today’s Schedule Acknowledgements Workshop Structure The Course Review Material In the Classroom

Teaching Ideas

In The ClassroomTeaching Ideas

I I spend a lot of time with inner products - this is primarilyfor later use.

I For example 3 + 2x + 5x2 − x3 can be viewed as(3, 2, 5,−1) dotted with (1, x , x2, x3). This is really helpfulfor generating Fourier series in Mathematica.

I We do a lot of visual matrix multiplication - i.e., if we arecomputing C = AB and the first row of A consists entirelyof 1’s, can you describe the top row of C?

Wednesday, 7 June, 2006 Lecture 1

Why Wavelets?

Today’s Schedule Acknowledgements Workshop Structure The Course Review Material In the Classroom

Teaching Ideas

In The ClassroomTeaching Ideas

I I spend a lot of time with inner products - this is primarilyfor later use.

I For example 3 + 2x + 5x2 − x3 can be viewed as(3, 2, 5,−1) dotted with (1, x , x2, x3). This is really helpfulfor generating Fourier series in Mathematica.

I We do a lot of visual matrix multiplication - i.e., if we arecomputing C = AB and the first row of A consists entirelyof 1’s, can you describe the top row of C?

Wednesday, 7 June, 2006 Lecture 1

Why Wavelets?

Today’s Schedule Acknowledgements Workshop Structure The Course Review Material In the Classroom

Teaching Ideas

In The ClassroomTeaching Ideas

I I spend a lot of time with inner products - this is primarilyfor later use.

I For example 3 + 2x + 5x2 − x3 can be viewed as(3, 2, 5,−1) dotted with (1, x , x2, x3). This is really helpfulfor generating Fourier series in Mathematica.

I We do a lot of visual matrix multiplication - i.e., if we arecomputing C = AB and the first row of A consists entirelyof 1’s, can you describe the top row of C?

Wednesday, 7 June, 2006 Lecture 1

Why Wavelets?

Today’s Schedule Acknowledgements Workshop Structure The Course Review Material In the Classroom

Teaching Ideas

In The ClassroomTeaching Ideas

I I spend a lot of time with inner products - this is primarilyfor later use.

I For example 3 + 2x + 5x2 − x3 can be viewed as(3, 2, 5,−1) dotted with (1, x , x2, x3). This is really helpfulfor generating Fourier series in Mathematica.

I We do a lot of visual matrix multiplication - i.e., if we arecomputing C = AB and the first row of A consists entirelyof 1’s, can you describe the top row of C?

Wednesday, 7 June, 2006 Lecture 1

Why Wavelets?

Today’s Schedule Acknowledgements Workshop Structure The Course Review Material In the Classroom

Teaching Ideas

In The ClassroomTeaching Ideas

I I talk about left multiplication by a circulant matrix - inparticular, a circulant matrix that contains only 2 to 6nonzero elements per row. This work pays off in laterchapters.

I We spend a significant amount of time manipulating blockmatrices and doing block matrix arithmetic and operations.This again is very important in Chapters 6 and 7 of the text.

Wednesday, 7 June, 2006 Lecture 1

Why Wavelets?

Today’s Schedule Acknowledgements Workshop Structure The Course Review Material In the Classroom

Teaching Ideas

In The ClassroomTeaching Ideas

I I talk about left multiplication by a circulant matrix - inparticular, a circulant matrix that contains only 2 to 6nonzero elements per row. This work pays off in laterchapters.

I We spend a significant amount of time manipulating blockmatrices and doing block matrix arithmetic and operations.This again is very important in Chapters 6 and 7 of the text.

Wednesday, 7 June, 2006 Lecture 1

Why Wavelets?

Today’s Schedule Acknowledgements Workshop Structure The Course Review Material In the Classroom

Teaching Ideas

In The ClassroomTeaching Ideas

I I talk about left multiplication by a circulant matrix - inparticular, a circulant matrix that contains only 2 to 6nonzero elements per row. This work pays off in laterchapters.

I We spend a significant amount of time manipulating blockmatrices and doing block matrix arithmetic and operations.This again is very important in Chapters 6 and 7 of the text.

Wednesday, 7 June, 2006 Lecture 1

Why Wavelets?

Today’s Schedule Acknowledgements Workshop Structure The Course Review Material In the Classroom

Computer Usage

In The ClassroomComputer Usage

I This is a good chapter to get the students used to playingwith the computer.

I You can introduce syntax for vectors and matrices and thevarious operations here.

I I usually give them short labs on Inner Products andNorms, Matrix Arithmetic, and Block MatrixArithmetic.

I These labs are available athttp://cam.mathlab.stthomas.edu/pvf/Math316

Wednesday, 7 June, 2006 Lecture 1

Why Wavelets?

Today’s Schedule Acknowledgements Workshop Structure The Course Review Material In the Classroom

Computer Usage

In The ClassroomComputer Usage

I This is a good chapter to get the students used to playingwith the computer.

I You can introduce syntax for vectors and matrices and thevarious operations here.

I I usually give them short labs on Inner Products andNorms, Matrix Arithmetic, and Block MatrixArithmetic.

I These labs are available athttp://cam.mathlab.stthomas.edu/pvf/Math316

Wednesday, 7 June, 2006 Lecture 1

Why Wavelets?

Today’s Schedule Acknowledgements Workshop Structure The Course Review Material In the Classroom

Computer Usage

In The ClassroomComputer Usage

I This is a good chapter to get the students used to playingwith the computer.

I You can introduce syntax for vectors and matrices and thevarious operations here.

I I usually give them short labs on Inner Products andNorms, Matrix Arithmetic, and Block MatrixArithmetic.

I These labs are available athttp://cam.mathlab.stthomas.edu/pvf/Math316

Wednesday, 7 June, 2006 Lecture 1

Why Wavelets?

Today’s Schedule Acknowledgements Workshop Structure The Course Review Material In the Classroom

Computer Usage

In The ClassroomComputer Usage

I This is a good chapter to get the students used to playingwith the computer.

I You can introduce syntax for vectors and matrices and thevarious operations here.

I I usually give them short labs on Inner Products andNorms, Matrix Arithmetic, and Block MatrixArithmetic.

I These labs are available athttp://cam.mathlab.stthomas.edu/pvf/Math316

Wednesday, 7 June, 2006 Lecture 1

Why Wavelets?

Today’s Schedule Acknowledgements Workshop Structure The Course Review Material In the Classroom

Student Difficulties

In The ClassroomStudent Difficulties

I Students grasp the block matrix arithmetic quickly - in fact,they seem to like the fact that working with blockssimplifies many matrix computations.

I The idea of AB being viewed as [Ab1, . . . , AbN ] bothersthem.

I Here is the first chance the students have to becomeinfuriated with the software package. Bugs are rampantduring the first few labs. I make sure they work in pairs sothey can correct errors - most are simply typos.

Wednesday, 7 June, 2006 Lecture 1

Why Wavelets?

Today’s Schedule Acknowledgements Workshop Structure The Course Review Material In the Classroom

Student Difficulties

In The ClassroomStudent Difficulties

I Students grasp the block matrix arithmetic quickly - in fact,they seem to like the fact that working with blockssimplifies many matrix computations.

I The idea of AB being viewed as [Ab1, . . . , AbN ] bothersthem.

I Here is the first chance the students have to becomeinfuriated with the software package. Bugs are rampantduring the first few labs. I make sure they work in pairs sothey can correct errors - most are simply typos.

Wednesday, 7 June, 2006 Lecture 1

Why Wavelets?

Today’s Schedule Acknowledgements Workshop Structure The Course Review Material In the Classroom

Student Difficulties

In The ClassroomStudent Difficulties

I Students grasp the block matrix arithmetic quickly - in fact,they seem to like the fact that working with blockssimplifies many matrix computations.

I The idea of AB being viewed as [Ab1, . . . , AbN ] bothersthem.

I Here is the first chance the students have to becomeinfuriated with the software package. Bugs are rampantduring the first few labs. I make sure they work in pairs sothey can correct errors - most are simply typos.

Wednesday, 7 June, 2006 Lecture 1

Why Wavelets?

Today’s Schedule Acknowledgements Workshop Structure The Course Review Material In the Classroom

Student Difficulties

In The ClassroomStudent Difficulties

I Students grasp the block matrix arithmetic quickly - in fact,they seem to like the fact that working with blockssimplifies many matrix computations.

I The idea of AB being viewed as [Ab1, . . . , AbN ] bothersthem.

I Here is the first chance the students have to becomeinfuriated with the software package. Bugs are rampantduring the first few labs. I make sure they work in pairs sothey can correct errors - most are simply typos.

Wednesday, 7 June, 2006 Lecture 1

Why Wavelets?

Today’s Schedule Acknowledgements Workshop Structure The Course Review Material In the Classroom

Student Difficulties

In The ClassroomStudent Difficulties

I The mathematics majors are a bit uncomfortable with thereverse engineering problems while the physics andengineering majors embrace them.

Wednesday, 7 June, 2006 Lecture 1

Why Wavelets?

Today’s Schedule Acknowledgements Workshop Structure The Course Review Material In the Classroom

Student Difficulties

In The ClassroomStudent Difficulties

I The mathematics majors are a bit uncomfortable with thereverse engineering problems while the physics andengineering majors embrace them.

Wednesday, 7 June, 2006 Lecture 1

Why Wavelets?

Today’s Schedule Acknowledgements Workshop Structure The Course Review Material In the Classroom

Today’s Schedule

9:00-10:15 Lecture One: Why Wavelets?10:15-10:30 ⇒Coffee Break (OSS 235)10:30-11:45 Lecture Two: Digital Images, Measures, and

Huffman Codes12:00-1:00 Lunch (Cafeteria)1:30-2:45 Lecture Three: Fourier Series, Convolution and

Filters2:45-3:00 Coffee Break (OSS 235)3:00-4:15 Lecture Four: 1D and 2D Haar Transforms5:30-6:30 Dinner (Cafeteria)

Wednesday, 7 June, 2006 Lecture 1

Why Wavelets?