Yves Meyer

WaveletsLuiz Velho

IMPA

ContributionsYves Meyer was awarded the 2010 Gauss Prize for fundamental contributions to number theory, operator theory and harmonic analysis, and his pivotal role in the development of wavelets and

multiresolution analysis.

He also received the 2017 Abel Prize “for his pivotal role in the development of the

mathematical theory of wavelets.”

Stéphane Mallat calls him a “visionary” whose work cannot be labelled either

pure or applied mathematics, nor computer science either, but simply

“amazing”.

The ScientistFrom the mid-1980s, in what he called a

“second scientific life”, Meyer, together with Daubechies and Coifman, brought together

earlier work on wavelets into a unified picture.

In 1986 Meyer and Pierre Gilles Lemarié-Rieusset showed that wavelets may form mutually

independent sets of mathematical objects called orthogonal bases. Coifman, Daubechies and

Stéphane Mallat went on to develop applications to many problems in signal and image processing.

Wavelets

The Meyer Wavelet• First Non-Trivial Orthogonal Wavelet Basis

• Continuously Differentiable

• Non-Compact Support

• Defined in Frequency Domain

• Continuous Wavelet Transform

(1985)

Definition

The Function

• Space Domain

The Next Frontier

• “Big Questions”:

- How to Discretize the Wavelet Transform?

- How to Systematically Construct Wavelet Basis?

• The Answer

- Multiresolution Analysis

Multiresolution Analysis

• Yves Meyer and Stephane Mallat

• Foundations of Modern Signal Processing

• Many Applications in Science and Engineering

(1988)

Multiresolution Analysis and

Filter Banks

Outline• Scale Spaces

• Multiresolution Analysis

• Dilation Equations

• Fast Wavelet Transform • Two-Channel Filter Banks

• Matrix Implementation

• Examples

Scale

Natural Concept

• Physical

Measurements, Data Acquisition

• Perception

Focus, Features

• Applications

Units, Computation

Scale Spaces

Vs : Space of Scale s • Scaling Function

-1 1 2

)(||

1)(sx

sxs φφ =

1)( =∫ dxxsφ

{ })(| , kxsks −φφ

• Basis of Vs

Representation of a Function in VS

• Orthogonal Projection

• Representation Operator

• Reconstruction Operator

)( ,)(Proj ,, xff ksk

ksVs φφ∑=

φφ ∗== ffc kssk ,,

)()( , xcxfk

kssks ∑= φ

(continuous function)

(discrete sequence)

Resolution and Multiresolution

Scale ↔ Resolution

• Data Representation

Computation - Discrete Elements (Samples)

Resolution - Samples / Unit of Scale

• Need for Representation at Multiple Scales

Different Features - Different Scales

Efficient Computation - Coarse to Fine Scale

Sequence of Scale Spaces

1. Inclusion:

2. Scaling:

3. Density:

4. Maximality:

5. Scale Basis:

( Vj ) , j ∈ Z

Multiresolution Analysis

Vj ⊂ Vj-1

f(x) ∈ Vj ⇔ f(2 jx) ∈ V0

closure{∪j Vj } = L2(R)

∩j Vj = { 0 }

∃ φ (x) s.t. {φ (x-k) } basis of V0

Two Scale RelationAxioms 1. and 2. (Nested Spaces and Dyadic Scaling)

f(x) ∈ Vj ⇔ f(2x) ∈ Vj-1

f ∈ Vj ⇔ f ∈ Vj-1

Completeness

Axioms 3. and 4. (complete covering of function space)

closure{∪j Vj } = L2(R)

∩j Vj = { 0 }

Multiresolution Basis

φ j,k (x) = 2 - j/2 φ (2 - j x - k)

Vj+1

Vj

Vj-1

Dilations

Translations

Axioms 5. and 2. (basis family generated from a single φ)

From Multiresolution to Wavelets

Multiresolution Ladder • Coarse to Fine

• Fine to Coarse (information loss)

Wavelets: Complementary Subspaces Intuition: difference between two consecutive levels Wj : Details in Vj-1 that cannot be represented in Vj

f ∈ Vj ⇒ f ∈ Vj-1

f ∈ Vj-1 f ∉ Vj

Wavelet Spaces

Definition: Orthogonal Complement of Vj in V j-1

therefore: jjj WVV ⊕=−1

)(Proj)(Proj)(Proj wjVj1Vj- fff +=

Vj

f

( ) f

( ) f ProjVj

ProjWjWj

Vj-1

Wavelet BasisOrthogonality of Spaces • Decomposition of L2

• Basis of Wj

ψj,k (x) = 2 - j/2 ψ (2 - j x - k)

kj WW ⊥

jjWLZ

2 (R)∈⊕=

kj ≠

Multiresolution Decomposition• Scale and Wavelet Spaces • Function Decomposition

{0} … ⊂ V1 ⊂ V0 ⊂ V-1 ⊂ ... L2(R) | | | | | …

W1 W0 W-1 … …

f ∈ V0

f = ProjW0 ( f ) + ...+ ProjWn ( f ) + ProjVn ( f )

Two-Scale Operators

• Transform Representation Sequences

Discrete Operators

Move Between Consecutive Levels j and j+1

• Act on Multiresolution Hierarchy

Fine ↔ Coarse

Fine ↔ Detail

Key to Efficient Computation

Dilation Equations• Scaling Basis:

100 −⊂∈ VVφ

)(,1)(,1 ,100 , xkk

kxkk

k h−−− ∑=∑= φφφφφ

100 −⊂∈ VWψ

)(,1)(,1 ,100 , xkk

kxkk

k g−−− ∑=∑= φφφψψ

• Wavelet Basis:

Expression of φ and ψ

• Reflexive Definition

• ψ defined in terms of φ

)2(2)( kxk

kx h −= ∑ φφ

)2(2)( kxk

kx g −= ∑ φψ

Shape of φ and ψ

)(xφ

11 −−φh

)(xψ

00φh 11φh 11 −−φg 00φg 11φg

Fine to Coarse: Operator H• Recursive Computation of Inner Products with φ

kjn

knkj hff ,12, ,, −−∑= φφ

∑= −−n

kjkn fh ,12 ,φ

( ) 1 2 −↓= ∗ jn

jn cHc

• Discrete Convolution and Decimation

• Change of Representation

1−⎯⎯← jn

Hjn cc

1−⊂ jj VV

Fine to Detail: Operator G• Recursive Computation of Inner Products with ψ

kjn

knkj gff ,12, ,, −−∑= φψ

∑= −−n

kjkn fg ,12 ,φ

( ) 1 2 −↓= ∗ jn

jn cGd

• Discrete Convolution and Decimation

• Change of Representation

1−⎯⎯← jn

Gjn cd

1−⊂ jj VW

Wavelet Decomposition

• Assume:

!

!

21

21

++

+⎯→⎯+⎯→⎯

jn

jn

GG

jn

Hjn

Hjn

dd

ccc

kjjn fc ,,φ=• Start with:

• Decomposition Scheme

jj Vf ∈

Coarse and Detail to Fine• Recover c j-1 from c j and d j

njjj

n fc ,111 , −−− = φ

njkjjkkj

jk dc ,1,, ,

−∑+∑= φψφ

( ) ( ) jn

jn

jn dGcHc 221 ↑+↑=− ∗∗

• Reconstruction Operator ( Expansion and Convolution )

∑+∑= −− njkjjknjkj

jk dc ,1,,1, ,, φψφφ

∑+∑= −−jkkn

jkkn dgch 22

Wavelet Reconstruction

1

1

++

⎯→⎯+⎯→⎯+

jn

mjn

GG

jn

Hjn

Hmjn

dd

ccc

!

!

( )1,,, +++ jmjmj ddc …• Start with:

• Decomposition Scheme

)

Wavelets and Filters

A Look in Frequency Domain

• Operators H and G

Discrete Filters

• Fast Wavelet Transform

Sub-band Filtering

• Signal Processing Framework

Filter Design Techniques

Implementation Scheme

Low-Pass FilterDiscrete Operator H

• Impulse Response

• Transfer Function

( ) ( )…… ,,,, 101 hhhhk −=

∑=−

k

ikkehH ωω)(

Periodic-2 is πH

High-Pass Filter

Discrete Operator G • Impulse Response

• Transfer Function

( ) ( )…… ,,,, 101 ggggk −=

∑=−

k

ikkegG ωω)(

Periodic-2 is πG

Multiresolution in Frequency Domain

],[ˆjjjjj aafVf −∈∈ ⇒

],[ˆ111 +++

−∈ jjj aaf

],[],[ˆ 111 jjjjj aaaao+++

∪−−∈

ja− ja

1+− ja 1+ja

ja− 1+− ja 1+ja ja

Two Channel Filter Banks

• One-Level Transform: Sub-band Filtering Perfect Reconstruction

c cj j

j+1H H

G G↓2 ↑2

↑2↓2 c

j+1d

Analysis Bank

Synthesis Bank

Tree-Structured Filter Bank

Wavelet Transform

• Recursive Filtering • Same Filters (invariance: dyadic scale, integer shift)

c cj j

j+2

H

H

H

H

G

G

G

G

↓2

↓2

↑2

↑2↑2

↑2

↓2

↓2 c

j+1

j+2

d

d

Frequency Splitting• Recursive Filtering

Dyadic Bands Energy Preservation

0G1G

π

2H 2G

2π4π

Matrix Implementation of FWT

• Filtering with Subsampling

• Two Channel Filter Bank

• Tree-Structured Filter Banks

c j

j+2

H

H

G

G

↓2

↓2

↓2

↓2 c

j+1

j+2

d

d

Filtering with Subsampling

• Convolution and Decimation: y = (↓2)F x

• Filter Matrix : y = ↓F x

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛

⎟⎟⎠

⎞⎜⎜⎝

⎛=⎟⎟⎠

⎞⎜⎜⎝

⎛

3

2

1

0

01

01

01

10

1

0

01000001

xxxx

ffff

ffff

yy

!!

!!

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛

⎟⎟⎠

⎞⎜⎜⎝

⎛=⎟⎟⎠

⎞⎜⎜⎝

⎛

3

2

1

0

01

10

1

0

xxxx

ffff

yy

!!

Two-Channel Filter Bank

• Analysis Bank

• Analysis Matrix

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛

=

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛

03

02

01

00

01

10

01

10

11

10

11

10

cccc

gggg

hhhh

ddcc

!!

!!

⎟⎟⎠

⎞⎜⎜⎝

⎛↓↓

=⎟⎟⎠

⎞⎜⎜⎝

⎛↓↓

=GH

GHM)2()2(

• Synthesis BankT1 MMS == −

* Critical Sampling

Tree-Structured Filter Bank

• Decomposition and Reconstruction Matrices for Level j

• Example of Decomposition Sequence

( )⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛

⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛=

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛

07

06

05

04

03

02

01

00

13

12

11

10

21

20

30

30

84

4

6

2

c

c

c

c

c

c

c

c

d

d

d

d

d

d

d

c

MI

MI

M

⎟⎟⎠

⎞⎜⎜⎝

⎛=

−kn

kj I

MD

00

⎟⎟

⎠

⎞

⎜⎜

⎝

⎛==

−

−

kn

kjj I

MDR0

0T1

jNk 2with =

FWT Matrix Computation

• Direct Transform (Analysis Bank)

• Inverse Transform (Synthesis Bank)

• Data Vector( ))())(( mjjjj ddcv −= !

jj

j vDv =+1

110

0 += mmvRRRc !

1+= jj

j vRv

001

1 cDDDv mm !=+

)

Example

Haar: Simplest Case

• Scaling Function and Wavelet

• Transform Scheme

• Matrix Computation • Function Decomposition

• Change of Basis

Haar Wavelets

• Scaling Function and Wavelet

• Low / High Pass Filters

φ ψ

)1 ,1( 2

1=H )1- ,1( 2

1=G

Haar Transform

• Input:

c

c

c

c

c

c c0

0

0

1

1

2

0

1

0 00

0

0

11

2

1

1

2 3

d

d

d

• output: ),,,( 11

10

20

20 dddcf =

),,,( 03

02

01

00 ccccf =

Haar Decomposition

Projection

),,,( 03

02

01

00

0 ccccf =

),( 11

10

1 ddo =

),( 11

10

1 ccf =

)( 20

2 do =)( 20

2 cf =

Representation

Haar Matrix Computation

• Normalization:

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛

=

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛

−

−⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛

−

1423

0219

11

rrrr

rrrr

rrrr

21=r

First LevelSecond Level

• Wavelet Matrix: (H and G)

Change of Basis Interpretation

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛

−

+

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛

−+

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛

−

−+

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛

=

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛

=

1100

1

0011

4

1111

2

1111

3

0219

f

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛

+

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛

+

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛

+

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛

=

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛

=

1000

0

0100

2

0010

1

0001

9

0219

f

• Scale Representation

• Multiresolution Representation

Representation in the Two Bases

Scale Basis

0V2V

2W

1W

Multiresolution Basis

Bi-Orthogonal Wavelets

• Represent with one Basis, Reconstruct with the Other

Bi-Orthonormal Basis• Dual Basis

• Computationally Similar to Orthonormal Basis

• More Degrees of Freedom to Construct Basis

Meyer Bi-orthogonal Wavelets

Discrete Filter Pairs Conclusions

Computational Scheme for Wavelets

• Multiresolution Analysis • Non-Redundant Wavelets

• Two-Channel Filter Banks

• Function Decompostion

• Wavelet Transform as a Basis Change

• Bi-Orthogonal Wavelets