[PPT] - Generalized Wavelets from a Representation Theory Viewpoint Vignon PowerPoint Presentation

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Generalized Wavelets from a Representation Theory Viewpoint

Vignon S. Oussa

Saint Louis University

October 2011

Vignon S. Oussa (Institute) Generalized Wavelets from a Representation Theory Viewpoint October 2011 1 / 25

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Content

1

Signals on sets, measures and group structures.

2

Representation theory.

3

Admissibility of representations and generalized wavelets.

4

Example of generalized wavelets on Zn and D2n.

5

Admissibility of representation on the circle and the real line.

6

Euclidean motion group.

7

Continuous wavelets on groups of matrices.

8

Conclusion.

Vignon S. Oussa (Institute) Generalized Wavelets from a Representation Theory Viewpoint October 2011 2 / 25

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Signals on a set

Important questions in signal analysis How do we represent data on some arbitrary set? How do we use the structure of the set to our advantage?

Vignon S. Oussa (Institute) Generalized Wavelets from a Representation Theory Viewpoint October 2011 3 / 25

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Sets will be groups endowed with some natural measure. Signals here will be square integrable functions. (Hilbert space) To represent signals: bases, discrete wavelets, continuous wavelets.

Examples

Sets Group structures measures Rn abelian Lebesgue measure T abelian Lebesgue measure Zn abelian counting measure D2n non abelian counting measure GL (n, R) non commutative Haar measure

Vignon S. Oussa (Institute) Generalized Wavelets from a Representation Theory Viewpoint October 2011 4 / 25

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What is a wavelet ? Classical de…nition. A wavelet is a function in f 2 L2 (R) such that the system n fj,k = 2j/2f

2jx k
: j, k 2 Z
forms an orthonormal basis in L2 (R) .

Vignon S. Oussa (Institute) Generalized Wavelets from a Representation Theory Viewpoint October 2011 5 / 25

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What is a wavelet ? Classical de…nition. A wavelet is a function in f 2 L2 (R) such that the system n fj,k = 2j/2f

2jx k
: j, k 2 Z
forms an orthonormal basis in L2 (R) .

Given g 2 L2 (R) , have unique representation. g (x) = ∑

j,k2Z

hg, fj,ki fj,k

Vignon S. Oussa (Institute) Generalized Wavelets from a Representation Theory Viewpoint October 2011 5 / 25

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What is a wavelet ? Classical de…nition. A wavelet is a function in f 2 L2 (R) such that the system n fj,k = 2j/2f

2jx k
: j, k 2 Z
forms an orthonormal basis in L2 (R) .

Given g 2 L2 (R) , have unique representation. g (x) = ∑

j,k2Z

hg, fj,ki fj,k

Example. The Haar wavelet

f (x) = 8 < : 1 if x 2 [0, 1/2) 1 if x 2 [1/2, 1) elsewhere

Vignon S. Oussa (Institute) Generalized Wavelets from a Representation Theory Viewpoint October 2011 5 / 25

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What is a wavelet ? Classical de…nition. A wavelet is a function in f 2 L2 (R) such that the system n fj,k = 2j/2f

2jx k
: j, k 2 Z
forms an orthonormal basis in L2 (R) .

Given g 2 L2 (R) , have unique representation. g (x) = ∑

j,k2Z

hg, fj,ki fj,k

Example. The Haar wavelet

f (x) = 8 < : 1 if x 2 [0, 1/2) 1 if x 2 [1/2, 1) elsewhere How do we generalize wavelets to other groups?

Vignon S. Oussa (Institute) Generalized Wavelets from a Representation Theory Viewpoint October 2011 5 / 25

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Representation Theory

A unitary representation is a strongly continuous homomorphism π from a group into a group of unitary operators on some Hilbert space H π : G ! U (H) A unitary representation π is reducible: existence non trivial closed subspace of H1 H s.t π (G) H1 H1. π is irreducible if all invariant subspaces of H are trivial. b G = unitary dual set of unitary irreducible rep. of G up to equivalence.

Fact

b G is needed for Fourier analysis on G.

Vignon S. Oussa (Institute) Generalized Wavelets from a Representation Theory Viewpoint October 2011 6 / 25

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Examples of unitary representations

Trivial representation π : C ! U (C) = T. π (z) = 1

Vignon S. Oussa (Institute) Generalized Wavelets from a Representation Theory Viewpoint October 2011 7 / 25

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Examples of unitary representations

Trivial representation π : C ! U (C) = T. π (z) = 1 Left regular representation on R. (L (x) F) (y) = F (y x) for any x 2 R.

Vignon S. Oussa (Institute) Generalized Wavelets from a Representation Theory Viewpoint October 2011 7 / 25

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Examples of unitary representations

Trivial representation π : C ! U (C) = T. π (z) = 1 Left regular representation on R. (L (x) F) (y) = F (y x) for any x 2 R. Irreducible representation of R. χ (x) z = eixz for all x 2 R.

Vignon S. Oussa (Institute) Generalized Wavelets from a Representation Theory Viewpoint October 2011 7 / 25

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Examples of unitary representations

Trivial representation π : C ! U (C) = T. π (z) = 1 Left regular representation on R. (L (x) F) (y) = F (y x) for any x 2 R. Irreducible representation of R. χ (x) z = eixz for all x 2 R. b R = R, b T = Z and b Z = T. G = a x 1

: x 2 R, a > 0
,

b G = R [ f1, 1g

Vignon S. Oussa (Institute) Generalized Wavelets from a Representation Theory Viewpoint October 2011 7 / 25

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Examples of unitary representations

Trivial representation π : C ! U (C) = T. π (z) = 1 Left regular representation on R. (L (x) F) (y) = F (y x) for any x 2 R. Irreducible representation of R. χ (x) z = eixz for all x 2 R. b R = R, b T = Z and b Z = T. G = a x 1

: x 2 R, a > 0
,

b G = R [ f1, 1g G = 8 < : 2 4 1 x z 1 y 1 3 5 : x, y, z 2 R 9 = ; , b G = R2 [ R

Vignon S. Oussa (Institute) Generalized Wavelets from a Representation Theory Viewpoint October 2011 7 / 25

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Admissible representations and generalized wavelets

Admissibility A representation π of G acting in H is admissible if Wψ : H ! L2 (G) Wψφ (x) = hφ, π (x) ψi is an isometry

Wψφ
L2(G )

= kφkH ψ is a generalized wavelet or admissible vector.

Vignon S. Oussa (Institute) Generalized Wavelets from a Representation Theory Viewpoint October 2011 8 / 25

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Admissible representations and generalized wavelets

Admissibility A representation π of G acting in H is admissible if Wψ : H ! L2 (G) Wψφ (x) = hφ, π (x) ψi is an isometry

Wψφ
L2(G )

= kφkH ψ is a generalized wavelet or admissible vector. Wψ is called a wavelet transform.

Vignon S. Oussa (Institute) Generalized Wavelets from a Representation Theory Viewpoint October 2011 8 / 25

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Admissible representations and generalized wavelets

Admissibility A representation π of G acting in H is admissible if Wψ : H ! L2 (G) Wψφ (x) = hφ, π (x) ψi is an isometry

Wψφ
L2(G )

= kφkH ψ is a generalized wavelet or admissible vector. Wψ is called a wavelet transform. Reconstruction of functions. f (t) = R

G hf , π (x) ψi π (x) ψ (t) dx.

Vignon S. Oussa (Institute) Generalized Wavelets from a Representation Theory Viewpoint October 2011 8 / 25

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Admissibility of Left regular representation (sketch)

General idea Given L the left regular representation on a group G.

Vignon S. Oussa (Institute) Generalized Wavelets from a Representation Theory Viewpoint October 2011 9 / 25

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Admissibility of Left regular representation (sketch)

General idea Given L the left regular representation on a group G. Wψφ (x) = φ ψ (x1) (convolution). Put ψ (x) = ψ (x1).

Vignon S. Oussa (Institute) Generalized Wavelets from a Representation Theory Viewpoint October 2011 9 / 25

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Admissibility of Left regular representation (sketch)

General idea Given L the left regular representation on a group G. Wψφ (x) = φ ψ (x1) (convolution). Put ψ (x) = ψ (x1). Fourier transform. F : L2 (G) ! L2 b G

Ff (π) =

Z

G f (x) π

x1

dx

Vignon S. Oussa (Institute) Generalized Wavelets from a Representation Theory Viewpoint October 2011 9 / 25

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Admissibility of Left regular representation (sketch)

General idea Given L the left regular representation on a group G. Wψφ (x) = φ ψ (x1) (convolution). Put ψ (x) = ψ (x1). Fourier transform. F : L2 (G) ! L2 b G

Ff (π) =

Z

G f (x) π

x1

dx We want

Wψφ
= kφk for all φ 2 L2 (G) .

kφ ψk2 =

Z

b G

Fφ (π)

z }| { Fψ (π)

2

HS

dπ =

Z

b G kFφ (π)k2 HS dπ.

The way we choose Fψ (π) characterizes the construction of wavelets.

Vignon S. Oussa (Institute) Generalized Wavelets from a Representation Theory Viewpoint October 2011 9 / 25

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Discrete circle Z3

Problem

How do we obtain an orthonormal basis in l2 (Z3) ' C3 by simply shifting the components of a single vector u = (u1, u2, u3)?

Vignon S. Oussa (Institute) Generalized Wavelets from a Representation Theory ViewpointOctober 2011 10 / 25

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Wavelets on the discrete circle

Z3 Z3 = f0, 1, 2g with addition mod 3 and b Z3 = n e

2πik 3

2

k=0

Vignon S. Oussa (Institute) Generalized Wavelets from a Representation Theory ViewpointOctober 2011 11 / 25

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Wavelets on the discrete circle

Z3 Z3 = f0, 1, 2g with addition mod 3 and b Z3 = n e

2πik 3

2

k=0

Inner product on l2 (Z3) ' C3 is hf , gi =

2

∑

k=0

f (k) g (k)

Vignon S. Oussa (Institute) Generalized Wavelets from a Representation Theory ViewpointOctober 2011 11 / 25

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Wavelets on the discrete circle

Z3 Z3 = f0, 1, 2g with addition mod 3 and b Z3 = n e

2πik 3

2

k=0

Inner product on l2 (Z3) ' C3 is hf , gi =

2

∑

k=0

f (k) g (k) Fourier transform F on l2 (Z3) ' C3 [F] = 2 6 4 1 1 1 1 e

2πi 3

e

4πi 3

1 e

4πi 3

e

2πi 3

3 7 5

Vignon S. Oussa (Institute) Generalized Wavelets from a Representation Theory ViewpointOctober 2011 11 / 25

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Wavelets on the discrete circle

Z3 Z3 = f0, 1, 2g with addition mod 3 and b Z3 = n e

2πik 3

2

k=0

Inner product on l2 (Z3) ' C3 is hf , gi =

2

∑

k=0

3 7 5

Vignon S. Oussa (Institute) Generalized Wavelets from a Representation Theory ViewpointOctober 2011 11 / 25

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Wψ : C3 ! C3 with Wab (k) = b a (k) .

Vignon S. Oussa (Institute) Generalized Wavelets from a Representation Theory ViewpointOctober 2011 12 / 25

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Wψ : C3 ! C3 with Wab (k) = b a (k) . We want k(Wa) bk2 = kb ak2 = 1 3 kF (b a)k2 = 1 3

2

∑

j=0

jF (b) (j)j2

equal 1

z }| { jF (a) (j)j2 = kbk2 .

Vignon S. Oussa (Institute) Generalized Wavelets from a Representation Theory ViewpointOctober 2011 12 / 25

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Wψ : C3 ! C3 with Wab (k) = b a (k) . We want k(Wa) bk2 = kb ak2 = 1 3 kF (b a)k2 = 1 3

2

∑

j=0

3 7 5 2 4 eπi i e

πi 3

3 5 = 2 6 4

1 6i

p 3 1

6 + 1 3i 1 6

p 3 2

3 1 6i

p

3 6 + p 3 6 i

1

6 1 6i

3 7 5 Generalization to Zn. A generalized wavelet w 2 Cn.

Vignon S. Oussa (Institute) Generalized Wavelets from a Representation Theory ViewpointOctober 2011 13 / 25

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Wavelets on Dihedral group D8 D8 is the group of symmetries on the square: 4 rotations, and 4 re‡ections.

Vignon S. Oussa (Institute) Generalized Wavelets from a Representation Theory ViewpointOctober 2011 14 / 25

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Wavelets on Dihedral group D8 D8 is the group of symmetries on the square: 4 rotations, and 4 re‡ections. D8 =

1, r, r2, r3, s, sr, sr2, sr3

.

Vignon S. Oussa (Institute) Generalized Wavelets from a Representation Theory ViewpointOctober 2011 14 / 25

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Wavelets on Dihedral group D8 D8 is the group of symmetries on the square: 4 rotations, and 4 re‡ections. D8 =

1, r, r2, r3, s, sr, sr2, sr3

. b D8 = fχ0, χ1, χ2, χ3, σg , σ is a representation acting in C2.

Vignon S. Oussa (Institute) Generalized Wavelets from a Representation Theory ViewpointOctober 2011 14 / 25

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Wavelets on Dihedral group D8 D8 is the group of symmetries on the square: 4 rotations, and 4 re‡ections. D8 =

1, r, r2, r3, s, sr, sr2, sr3

. b D8 = fχ0, χ1, χ2, χ3, σg , σ is a representation acting in C2. Fourier transform Ff (π) = ∑x2D8 f (x) π (x) .

Vignon S. Oussa (Institute) Generalized Wavelets from a Representation Theory ViewpointOctober 2011 14 / 25

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Wavelets on Dihedral group D8 D8 is the group of symmetries on the square: 4 rotations, and 4 re‡ections. D8 =

1, r, r2, r3, s, sr, sr2, sr3

. b D8 = fχ0, χ1, χ2, χ3, σg , σ is a representation acting in C2. Fourier transform Ff (π) = ∑x2D8 f (x) π (x) . Fourier transform inverse. F 1a (x) = 1

8 ∑π2 b D8 dπTr

π
x1

a (π)

.

Vignon S. Oussa (Institute) Generalized Wavelets from a Representation Theory ViewpointOctober 2011 14 / 25

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Wavelets on Dihedral group D8 D8 is the group of symmetries on the square: 4 rotations, and 4 re‡ections. D8 =

1, r, r2, r3, s, sr, sr2, sr3

. b D8 = fχ0, χ1, χ2, χ3, σg , σ is a representation acting in C2. Fourier transform Ff (π) = ∑x2D8 f (x) π (x) . Fourier transform inverse. F 1a (x) = 1

8 ∑π2 b D8 dπTr

π
x1

a (π)

.

Construction of wavelets. Pick g such that Fg (χk) = eiθχ, Fg (σ) = M22 isometry on C2

Vignon S. Oussa (Institute) Generalized Wavelets from a Representation Theory ViewpointOctober 2011 14 / 25

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Wavelets on Dihedral group D8 D8 is the group of symmetries on the square: 4 rotations, and 4 re‡ections. D8 =

1, r, r2, r3, s, sr, sr2, sr3

. b D8 = fχ0, χ1, χ2, χ3, σg , σ is a representation acting in C2. Fourier transform Ff (π) = ∑x2D8 f (x) π (x) . Fourier transform inverse. F 1a (x) = 1

8 ∑π2 b D8 dπTr

π
x1

a (π)

.

Construction of wavelets. Pick g such that Fg (χk) = eiθχ, Fg (σ) = M22 isometry on C2 g(x) = 1

8 ∑π2 b D8 dπ Tr

π
x1 Fg (π)
generates ONB by shifts.

Vignon S. Oussa (Institute) Generalized Wavelets from a Representation Theory ViewpointOctober 2011 14 / 25

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Generalized wavelets on Z b Z = unit circle T.

Vignon S. Oussa (Institute) Generalized Wavelets from a Representation Theory ViewpointOctober 2011 15 / 25

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Generalized wavelets on Z b Z = unit circle T. Fourier transform. (Ff ) (x) = ∑k2Z f (k) eikx.

Vignon S. Oussa (Institute) Generalized Wavelets from a Representation Theory ViewpointOctober 2011 15 / 25

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Generalized wavelets on Z b Z = unit circle T. Fourier transform. (Ff ) (x) = ∑k2Z f (k) eikx. Fourier inverse transform.

F 1g

(k) =

1 2π

R 2π g (x) eixkdx.

Vignon S. Oussa (Institute) Generalized Wavelets from a Representation Theory ViewpointOctober 2011 15 / 25

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Generalized wavelets on Z b Z = unit circle T. Fourier transform. (Ff ) (x) = ∑k2Z f (k) eikx. Fourier inverse transform.

F 1g

(k) =

1 2π

R 2π g (x) eixkdx. What kind of orthonormal basis (generalized wavelets) are generated by translations of integers in l2 (Z) .

Vignon S. Oussa (Institute) Generalized Wavelets from a Representation Theory ViewpointOctober 2011 15 / 25

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Generalized wavelets on Z b Z = unit circle T. Fourier transform. (Ff ) (x) = ∑k2Z f (k) eikx. Fourier inverse transform.

F 1g

(k) =

1 2π

R 2π g (x) eixkdx. What kind of orthonormal basis (generalized wavelets) are generated by translations of integers in l2 (Z) . Wa : l2 (Z) ! l2 (Z) with Wab (k) = b a (k) .

Vignon S. Oussa (Institute) Generalized Wavelets from a Representation Theory ViewpointOctober 2011 15 / 25

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Generalized wavelets on Z b Z = unit circle T. Fourier transform. (Ff ) (x) = ∑k2Z f (k) eikx. Fourier inverse transform.

F 1g

(k) =

1 2π

R 2π g (x) eixkdx. What kind of orthonormal basis (generalized wavelets) are generated by translations of integers in l2 (Z) . Wa : l2 (Z) ! l2 (Z) with Wab (k) = b a (k) . Sequences fakgk2Z such that for …xed t 2 R, ak = (

e2πi(tk)1 2πi(kt)

if k 6= t 1 if k = t .

Vignon S. Oussa (Institute) Generalized Wavelets from a Representation Theory ViewpointOctober 2011 15 / 25

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