connection between zernike functions, corneal topography ...zernike coe cients computing the...

Table ofContents

Motivation –CORNEAproject

Zenikefunctions

Fritz Zernike

Orthogonality ofZernikefunctions

Representationof the corneasurface

Problems

Discreteorthogonality

The discreteZernikecoefficients

Computing thediscrete Zernikecoefficients

Zernikerepresentationof some testsurfaces

Connection tothe voicetransform

Unitaryrepresentation

Definition of thevoice transform

Special voicetransforms

The voicetransform of theBlaschke group

Properties

References

END

Numerical Harmonic Analysis Group

Connection between Zernike functions, cornealtopography and the voice transform

Margit Pap 1

[email protected], [email protected]

November 11, 2010

1University of Pécs, Hungary, NuHAGMargit Pap [email protected], [email protected] between Zernike functions, corneal topography and the voice transform

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Zenikefunctions

Fritz Zernike

Problems

Properties

References

END

Table of Contents

Motivation – CORNEA Project

Zernike functions

The Zernike representation used in ophtamology

Discrete orthogonality

Reconstruction of the corneal surface

Connection to the voice transform

Margit Pap http://nuhag.eu

Table ofContents

Zenikefunctions

Fritz Zernike

Problems

Properties

References

END

Motivation – CORNEA project

The corneal surface is frequently represented in terms ofthe Zernike functions.

The optical aberrations of human eyes (for ex. astigma,tilt) and optical systems are characterized with Zernikecoefficients.

Abberations are examined with Corneal topographer.

Measurements made by Shack – Hartmann wavefront -sensor.

Problem: Approximation of the Zernike coefficients andreconstruction of the corneal surface with minimal error.

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Zenikefunctions

Fritz Zernike

Problems

Properties

References

END

Fritz Zernike

Dutch physicist.

In 1934 he introduced the two variable orthogonal system– named later Zernike functions.

They are distinguished from the other orthogonal systemsby certain simple invariance properties which can beexplained from group theoretical considerations: for ex.they are invariant with respect to rotations of axes aboutorigin.

In 1953 winner of the Nobel prize for Physics.

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Zenikefunctions

Fritz Zernike

Problems

Properties

References

END

Zernike functions

Definition of Zernike functions

Z `n(ρ, θ) :=√

2n + |`|+ 1 R |`||`|+2n(ρ)ei`θ, ` ∈ Z , n ∈ N,

The radial terms R|`||`|+2n(ρ) are related to the Jacobi

polynomials in the following way:

R|`||`|+2n(ρ) = ρ

|`|P(0,|`|)n (2ρ

2 − 1).

Pictures

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Zenikefunctions

Fritz Zernike

Problems

Properties

References

END

Orthogonality of Zernike functions

1

π

∫ 2π0

∫ 10

Z `n(ρ, φ)Z`′n′(ρ, φ)ρdρdφ = δnn′δ``′ .

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Zenikefunctions

Fritz Zernike

Problems

Properties

References

END

Orthogonality of Zernike functions

1

π

∫ 2π0

∫ 10

Z `n(ρ, φ)Z`′n′(ρ, φ)ρdρdφ = δnn′δ``′ .

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Zenikefunctions

Fritz Zernike

Problems

Properties

References

END

Representation of the cornea surface

The corneal surface is described by a two variable functionover the unit disc.

g(x , y) or G (ρ, φ) = g(ρ cosφ, ρ sinφ)

The Zernike series expansion of G∑`,n

A`nZ`n(ρ, φ)

A`n =1

π

∫ 2π0

∫ 10

G (ρ, φ)Z `n(ρ, φ)ρdρdφ.

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Zenikefunctions

Fritz Zernike

Problems

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A`nZ`n(ρ, φ)

A`n =1

π

∫ 2π0

∫ 10

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Zenikefunctions

Fritz Zernike

Problems

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END

A`nZ`n(ρ, φ)

A`n =1

π

∫ 2π0

∫ 10

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Zenikefunctions

Fritz Zernike

Problems

Properties

References

END

A`nZ`n(ρ, φ)

A`n =1

π

∫ 2π0

∫ 10

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Zenikefunctions

Fritz Zernike

Problems

Properties

References

END

Problems

Open problems mentioned in: Wyant, J. C., Creath,K., Basic Wavefront Aberration Theory for OpticalMetrology, Applied Optics and Optical Engineering, VolXI, Academic Press (1992).

1. The discrete orthogonality of Zernike functions.

2. Addition formula for Zernike functions.

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Zenikefunctions

Fritz Zernike

Problems

Properties

References

END

Problems

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Zenikefunctions

Fritz Zernike

Problems

Properties

References

END

Problems

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Zenikefunctions

Fritz Zernike

Problems

Properties

References

END

Denote by λNj ∈ (−1, 1), j ∈ {1, ...,N} the roots ofLegendre polynomials PN of order N,

and for j = 1, ...,N, let

`Nj (x) :=(x − λN1 )...(x − λNj−1)(x − λNj+1)...(x − λNN)

(λNj − λN1 )...(λNj − λNj−1)(λNj − λNj+1)...(λNj − λNN),

be the corresponding fundamental polynomials ofLagrange interpolation.

ANj :=∫ 1−1`Nj (x)dx ,

the corresponding Cristoffel-numbers.

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Fritz Zernike

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ANj :=∫ 1−1`Nj (x)dx ,

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Zenikefunctions

Fritz Zernike

Problems

Properties

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END

ANj :=∫ 1−1`Nj (x)dx ,

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Fritz Zernike

Problems

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Let define the following numbers with the help of theroots of Legendre polynomials of order N

ρNk :=

√1 + λNk

2, k = 1, ...,N,

and the set of nodal points:

X := {zjk :=(ρNk ,

2πj

4N + 1

), k = 1, ...,N, j = 0, ..., 4N}

and let define

ν(zjk) :=ANk

2(4N + 1).

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Fritz Zernike

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ρNk :=

√1 + λNk

2, k = 1, ...,N,

X := {zjk :=(ρNk ,

2πj

4N + 1

), k = 1, ...,N, j = 0, ..., 4N}

and let define

ν(zjk) :=ANk

2(4N + 1).

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ρNk :=

√1 + λNk

2, k = 1, ...,N,

X := {zjk :=(ρNk ,

2πj

4N + 1

), k = 1, ...,N, j = 0, ..., 4N}

and let define

ν(zjk) :=ANk

2(4N + 1).

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Zenikefunctions

Fritz Zernike

Problems

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END

Let introduce the following discrete integral∫X

f (ρ, φ)dνN :=N∑

k=1

4N∑j=0

f (ρNk ,2πj

4N + 1)ANk

2(4N + 1).

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Zenikefunctions

Fritz Zernike

Problems

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Theorem Pap-Schipp 2005

If n + n′ + |m| ≤ 2N − 1,n + n′ + |m′| ≤ 2N − 1, n, n′ ∈ N,m,m′ ∈ Z, then∫

XZmn (ρ, φ)Z

m′n′ (ρ, φ)dνN = δnn′δmm′ .

For all f ∈ C (D)

limN→∞

∫X

fdνN =1

π

∫ 2π0

∫ 10

f (ρ, φ)ρdρdφ.

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Problems

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Theorem Pap-Schipp 2005

If n + n′ + |m| ≤ 2N − 1,n + n′ + |m′| ≤ 2N − 1, n, n′ ∈ N,m,m′ ∈ Z, then∫

XZmn (ρ, φ)Z

m′n′ (ρ, φ)dνN = δnn′δmm′ .

For all f ∈ C (D)

limN→∞

∫X

fdνN =1

π

∫ 2π0

∫ 10

f (ρ, φ)ρdρdφ.

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Zenikefunctions

Fritz Zernike

Problems

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END

The discrete Zernike coefficients

A′`n =

∫X

G (ρ, φ)Z `n(ρ, φ)dνN(ρ, φ) =

N∑k=1

4N∑j=0

G (ρNk ,2πj

4N + 1)Z `n(ρ

Nk ,

2πj

4N + 1)ANk

2(4N + 1)

The discrete Zernike coefficients of the function G fromC (D) tend to the corresponding continuous Zernikecoefficients if N → +∞.

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Fritz Zernike

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A′`n =

∫X

N∑k=1

4N∑j=0

G (ρNk ,2πj

4N + 1)Z `n(ρ

Nk ,

2πj

4N + 1)ANk

2(4N + 1)

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Zenikefunctions

Fritz Zernike

Problems

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END

A′`n =

∫X

N∑k=1

4N∑j=0

G (ρNk ,2πj

4N + 1)Z `n(ρ

Nk ,

2πj

4N + 1)ANk

2(4N + 1)

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Zenikefunctions

Fritz Zernike

Problems

Properties

References

END

LetGN(ρ, φ) =

∑2n+|m|52N−1

AmnZmn (ρ, φ)

be an arbitrary linear combination of Zernike polynomialsof degree less than 2N.

The coefficients Amn can be expressed in the following twoways:

Amn =1

π

∫ 2π0

∫ 10

GN(ρ′, φ′)Zmn (ρ

′, φ′)ρ′dρ′dφ′,

Amn =

∫X

′, φ′)dνN(ρ′, φ′).

Measuring on X we can compute the exact values.

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Fritz Zernike

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LetGN(ρ, φ) =

∑2n+|m|52N−1

AmnZmn (ρ, φ)

Amn =1

π

∫ 2π0

∫ 10

Amn =

∫X

′, φ′)dνN(ρ′, φ′).

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Zenikefunctions

Fritz Zernike

Problems

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END

LetGN(ρ, φ) =

∑2n+|m|52N−1

AmnZmn (ρ, φ)

Amn =1

π

∫ 2π0

∫ 10

Amn =

∫X

′, φ′)dνN(ρ′, φ′).

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Zenikefunctions

Fritz Zernike

Problems

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END

LetGN(ρ, φ) =

∑2n+|m|52N−1

AmnZmn (ρ, φ)

Amn =1

π

∫ 2π0

∫ 10

Amn =

∫X

′, φ′)dνN(ρ′, φ′).

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Zenikefunctions

Fritz Zernike

Problems

Properties

References

END

Zernike representation of some test surfaces

A. Soumelidis, Z. Fazakas, M. Pap, F. Schipp: Discreteorthogonality of Zernike functions and its application tocorneal topography, Electronic Engineering and ComputingTechnology, Lecture notes in electrical engineering, 2010,Vol 60, 455-469, ISBN: 978-90-481-8775-1

Computer implementations, experimental results onartificial corneal-like surfaces.

Comparison of precision with the measurement methodsused by conventional topographers proved that theapproximations made using the discrete orthogonality arethe best.

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Zenikefunctions

Fritz Zernike

Problems

Properties

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END

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Zenikefunctions

Fritz Zernike

Problems

Properties

References

END

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Zenikefunctions

Fritz Zernike

Problems

Properties

References

END

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Zenikefunctions

Fritz Zernike

Problems

Properties

References

END

Classical methods

Approximation of Zernike coefficients is made on a set ofpoints which corresponds to the equidistant divisions alongthe Ox and Oy of the [−1, 1]× [−1, 1].Equidistant division along the radial line [0, 1] and theangular part [0, 2π].

The computation of discrete Zernike coefficients can bespeeded via FFT.

Future: measurements and reconstructions on real humancorneal surface and construction of new topographersbased on the measurements on X , applications in sight-correcting operations.

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Zenikefunctions

Fritz Zernike

Problems

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Classical methods

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Zenikefunctions

Fritz Zernike

Problems

Properties

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END

Classical methods

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Zenikefunctions

Fritz Zernike

Problems

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END

Classical methods

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Zenikefunctions

Fritz Zernike

Problems

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H. G. Feichtinger and K. H. Gröchenig unified the theoryof Gábor and wavelet transforms into a single theory. Thecommon generalization of these transforms is the so-calledvoice transform.

In the construction of the voice-transform the startingpoint will be a locally compact topological group (G , ·).

Let m be a left-invariant Haar measure of G :∫G

f (x) dm(x) =

∫G

f (a−1 · x) dm(x), (a ∈ G ).

If the left invariant Haar measure of G is at the same timeright invariant then G is a unimodular group.

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In the construction of the voice-transform the startingpoint will be a locally compact topological group (G , ·).Let m be a left-invariant Haar measure of G :∫

Gf (x) dm(x) =

∫G

f (a−1 · x) dm(x), (a ∈ G ).

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Gf (x) dm(x) =

∫G

f (a−1 · x) dm(x), (a ∈ G ).

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Gf (x) dm(x) =

∫G

f (a−1 · x) dm(x), (a ∈ G ).

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Unitary representation

Unitary representation of the group (G , ·): Let usconsider a Hilbert-space (H, 〈·, ·〉).U denote the set of unitary bijections U : H → H.Namely, the elements of U are bounded linear operatorswhich satisfy 〈Uf ,Ug〉 = 〈f , g〉 (f , g ∈ H).

The set U with the composition operation(U ◦ V )f := U(Vf ) (f ∈ H) is a group.The homomorphism of the group (G , ·) on the group(U , ◦) satisfying

i) Ux ·y = Ux ◦ Uy (x , y ∈ G ),

ii) G 3 x → Ux f ∈ H is continuous for all f ∈ H

is called the unitary representation of (G , ·) on H.

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Unitary representation of the group (G , ·): Let usconsider a Hilbert-space (H, 〈·, ·〉).U denote the set of unitary bijections U : H → H.Namely, the elements of U are bounded linear operatorswhich satisfy 〈Uf ,Ug〉 = 〈f , g〉 (f , g ∈ H).The set U with the composition operation(U ◦ V )f := U(Vf ) (f ∈ H) is a group.

The homomorphism of the group (G , ·) on the group(U , ◦) satisfying

i) Ux ·y = Ux ◦ Uy (x , y ∈ G ),

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Zenikefunctions

Fritz Zernike

Problems

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END

Unitary representation of the group (G , ·): Let usconsider a Hilbert-space (H, 〈·, ·〉).U denote the set of unitary bijections U : H → H.Namely, the elements of U are bounded linear operatorswhich satisfy 〈Uf ,Ug〉 = 〈f , g〉 (f , g ∈ H).The set U with the composition operation(U ◦ V )f := U(Vf ) (f ∈ H) is a group.The homomorphism of the group (G , ·) on the group(U , ◦) satisfying

i) Ux ·y = Ux ◦ Uy (x , y ∈ G ),

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Zenikefunctions

Fritz Zernike

Problems

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Unitary representation of the group (G , ·): Let usconsider a Hilbert-space (H, 〈·, ·〉).U denote the set of unitary bijections U : H → H.Namely, the elements of U are bounded linear operatorswhich satisfy 〈Uf ,Ug〉 = 〈f , g〉 (f , g ∈ H).The set U with the composition operation(U ◦ V )f := U(Vf ) (f ∈ H) is a group.The homomorphism of the group (G , ·) on the group(U , ◦) satisfying

i) Ux ·y = Ux ◦ Uy (x , y ∈ G ),

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Definition of the voice transform

Definition

The voice transform of f ∈ H generated by therepresentation U and by the parameter ρ ∈ H is the(complex-valued) function on G defined by

(Vρf )(x) := 〈f ,Uxρ〉 (x ∈ G , f , ρ ∈ H).

Taking as starting point (not necessarily commutative)locally compact groups we can construct in this wayimportant transformations.

The affine wavelet transform is a voice transform of theaffine group.

The Gábor transform is a voice transform of theHeisenberg group.

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Definition

(Vρf )(x) := 〈f ,Uxρ〉 (x ∈ G , f , ρ ∈ H).

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Definition

(Vρf )(x) := 〈f ,Uxρ〉 (x ∈ G , f , ρ ∈ H).

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Definition

(Vρf )(x) := 〈f ,Uxρ〉 (x ∈ G , f , ρ ∈ H).

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Gábor transform

The Gábor transform is a voice transform generated bythe representation of the Weyl-Heisenberg group:

H := R× R× T

(a1, ω1, t1).(a2, ω2, t2) := (a1 + a2, ω1 + ω2, t1t2e2πiω1a2).

The representation of H on L2(R):

U(a,ω,t)f (x) := te2πiωx f (x − a) = tMωTaf (x)

the STFT- Gábor transform

Vϕf (a, ω) =

∫R

f (t)ϕ(t − a)e−2πiωtdt = 〈f ,MωTaϕ〉

= 〈f ,U(a,ω,1)ϕ〉Margit Pap http://nuhag.eu

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Affine wavelet transform

The affine group

G = {`(a,b)(x) = ax + b : R→ R : (a, b) ∈ R∗ × R}

`1◦`2(x) = a1a2x+a1b2+b1, (a1, b1)◦(a2, b2) = (a1a2, a1b2+b1)

The representation of G on L2(R)

U(a,b)f (x) = |a|−1/2f (a−1x − b)

The affine wavelet transform is a voice transformgenerated by this representation of affin group:

Wψf (a, b) = |a|−1/2∫R

f (t)ψ(a−1t − b)dt = 〈f ,TbDaψ〉

= 〈f ,U(a,b)ψ〉Margit Pap http://nuhag.eu

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The voice transform of the Blaschke group

The Blaschke group Let us denote by

Ba(z) := �z − b

1− b̄z(z ∈ C, a = (b, �) ∈ B := D×T, bz 6= 1)

the so called Blaschke functions,

D := {z ∈ C : |z | < 1}, T := {z ∈ C : |z | = 1}.

If a ∈ B, then Ba is an 1-1 map on T, D respectively.

The restrictions of the Blaschke functions on the set D oron T with the operation (Ba1 ◦ Ba2)(z) := Ba1(Ba2(z))form a group.

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Ba(z) := �z − b

1− b̄z(z ∈ C, a = (b, �) ∈ B := D×T, bz 6= 1)

D := {z ∈ C : |z | < 1}, T := {z ∈ C : |z | = 1}.

If a ∈ B, then Ba is an 1-1 map on T, D respectively.The restrictions of the Blaschke functions on the set D oron T with the operation (Ba1 ◦ Ba2)(z) := Ba1(Ba2(z))form a group.

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Ba(z) := �z − b

1− b̄z(z ∈ C, a = (b, �) ∈ B := D×T, bz 6= 1)

D := {z ∈ C : |z | < 1}, T := {z ∈ C : |z | = 1}.

If a ∈ B, then Ba is an 1-1 map on T, D respectively.The restrictions of the Blaschke functions on the set D oron T with the operation (Ba1 ◦ Ba2)(z) := Ba1(Ba2(z))form a group.

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In the set of the parameters B := D× T let us define theoperation induced by the function composition in thefollowing way Ba1 ◦ Ba2 = Ba1◦a2 .(B, ◦) will be the Blaschke group which is isomorphic withthe group ({Ba, a ∈ B}, ◦).

If we use the notations aj := (bj , �j), j ∈ {1, 2} anda := (b, �) =: a1 ◦ a2 then

b =b1�2 + b2

1 + b1b2�2, � = �1

�2 + b1b2

1 + �2b1b2.

The neutral element of the group (B, ◦) is e := (0, 1) ∈ Band the inverse element of a = (b, �) ∈ B isa−1 = (−b�, �).

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In the set of the parameters B := D× T let us define theoperation induced by the function composition in thefollowing way Ba1 ◦ Ba2 = Ba1◦a2 .(B, ◦) will be the Blaschke group which is isomorphic withthe group ({Ba, a ∈ B}, ◦).If we use the notations aj := (bj , �j), j ∈ {1, 2} anda := (b, �) =: a1 ◦ a2 then

b =b1�2 + b2

1 + b1b2�2, � = �1

�2 + b1b2

1 + �2b1b2.

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b =b1�2 + b2

1 + b1b2�2, � = �1

�2 + b1b2

1 + �2b1b2.

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b =b1�2 + b2

1 + b1b2�2, � = �1

�2 + b1b2

1 + �2b1b2.

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b =b1�2 + b2

1 + b1b2�2, � = �1

�2 + b1b2

1 + �2b1b2.

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The integral of the function f : B→ C, with respect tothis left invariant Haar measure m of the group (B, ◦), isgiven by∫

Bf (a) dm(a) =

1

2π

∫I

∫D

f (b, e it)

(1− |b|2)2db1db2dt,

where a = (b, e it) = (b1 + ib2, eit) ∈ D× T.

It can be shown that this integral is invariant under theinverse transformation a→ a−1, so this group isunimodular.

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The integral of the function f : B→ C, with respect tothis left invariant Haar measure m of the group (B, ◦), isgiven by∫

Bf (a) dm(a) =

1

2π

∫I

∫D

f (b, e it)

(1− |b|2)2db1db2dt,

where a = (b, e it) = (b1 + ib2, eit) ∈ D× T.

It can be shown that this integral is invariant under theinverse transformation a→ a−1, so this group isunimodular.

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The representation of the Blaschke group on H2(T)

Denote by �n(t) = eint (t ∈ I = [0, 2π], n ∈ N), let

consider the Hilbert space H = H2(T), the closure inL2(T)-norm of the set

span{�n, n ∈ N}.

The inner product is given by

〈f , g〉 := 12π

∫I

f (e it)g(e it) dt (f , g ∈ H).

The representation of the Blaschke group on H2(T): for(z = e it ∈ T, a = (b, e iθ) ∈ B

), f ∈ H2(T):

(Ua−1f )(z) :=

√e iθ(1− |b|2)(1− bz)

f(e iθ(z − b)

1− bz)

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〈f , g〉 := 12π

∫I

), f ∈ H2(T):

(Ua−1f )(z) :=

√e iθ(1− |b|2)(1− bz)

f(e iθ(z − b)

1− bz)

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〈f , g〉 := 12π

∫I

), f ∈ H2(T):

(Ua−1f )(z) :=

√e iθ(1− |b|2)(1− bz)

f(e iθ(z − b)

1− bz)

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The voice transform generated by Ua (a ∈ B) is given bythe following formula

(Vρf )(a−1) := 〈f ,Ua−1ρ〉 (f , ρ ∈ H2(T)).

Pap M., Schipp F., The voice transform on the Blaschkegroup I., PU.M.A., Vol 17, (2006), No 3-4, pp. 387-395.

Pap M., Schipp F., The voice transform on the Blaschkegroup II., Annales Univ. Sci. (Budapest), Sect. Comput.,29 (2008) 157-173.

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(Vρf )(a−1) := 〈f ,Ua−1ρ〉 (f , ρ ∈ H2(T)).

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(Vρf )(a−1) := 〈f ,Ua−1ρ〉 (f , ρ ∈ H2(T)).

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The matrix elements of representations of theBlaschke group

The matrix elements can be expressed by Zernikefunctions.

The matrix elements vmn(a−1) := 〈�n,Ua−1�m〉 of

representation U with respect to the basis {�n : n ∈ N}and a = (re iϕ, e iψ) are given by

vmn(a−1) =

√1− r2√

m + n + 1e−i(m+1/2)ψ(−1)mZ |m−n|min{n,m}(r , ϕ).

An important consequence of this connection is theaddition formula for Zernike functions.

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vmn(a−1) =

√1− r2√

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vmn(a−1) =

√1− r2√

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vmn(a−1) =

√1− r2√

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It is known that in general the matrix elements of therepresentations satisfy the following so called additionformula:

vmn(a1 ◦ a2) =∑k

vmk(a1)vkn(a2) (a1, a2 ∈ B).

From this relation we obtain the following addition formulafor Zernike functions: if aj := (rje

iϕj , e iψj ), j ∈ {1, 2} anda := (re iϕ, e iψ) = a1 ◦ a2 then

√1− r2√

(n + m + 1)(1− r21 )(1− r22 )e−i(m+1/2)ψZ

|n−m|min{m,n}(r , ϕ) =

∑k

(−1)ke−i(m+1/2)ψ1e−i(k+1/2)ψ2√(m + k + 1)(n + k + 1)

Z|k−m|min{m,k}(r1, ϕ1)Z

|n−k|min{k,n}(r2, ϕ2),

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It is known that in general the matrix elements of therepresentations satisfy the following so called additionformula:

vmn(a1 ◦ a2) =∑k

vmk(a1)vkn(a2) (a1, a2 ∈ B).

From this relation we obtain the following addition formulafor Zernike functions: if aj := (rje

iϕj , e iψj ), j ∈ {1, 2} anda := (re iϕ, e iψ) = a1 ◦ a2 then

√1− r2√

(n + m + 1)(1− r21 )(1− r22 )e−i(m+1/2)ψZ

|n−m|min{m,n}(r , ϕ) =

∑k

(−1)ke−i(m+1/2)ψ1e−i(k+1/2)ψ2√(m + k + 1)(n + k + 1)

Z|k−m|min{m,k}(r1, ϕ1)Z

|n−k|min{k,n}(r2, ϕ2),

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Properties

Discrete Laguerre functions

Let ϕ = 1 be the mother wavelet, the shift operator:

(Sϕ)(z) = zϕ(z) (ϕ ∈ H2(D), z ∈ D ∪ T).

Then the discrete Laguerre functions

ϕa,m(z) := (Ua−1Smϕ)(z) =

√�(1− |b|2)(1− bz)

(�(z − b)1− bz

)m.

Let V�m f (a−1) = 〈f ,Ua−1�m〉 and let define the following

projection operator

Pf (a, z) :=∞∑

m=0

(V�m f )(a−1)ϕa,m(z),

where the infinite series is absolute convergent for z ∈ D.Margit Pap http://nuhag.eu

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Properties

Reconstruction formula

Theorem For every f ∈ H2(T), for every z = r1e it ∈ D andfor every a ∈ B

limr1→1

Pf (a, z) = f (e it)

a.e. t ∈ I and in H2 norm. If f ∈ C (T), then the convergenceis uniform.

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Properties

Infinite series representation of voice – transform

Inner product generated by the weight w = (1− r2) on B

〈〈F ,G 〉

〉:=

1

2π

∫ 10

∫I

rF (re iϕ)G (re iϕ)

1− r2dϕdr . (1)

Theorem Let us consider ρ ∈ H2(T), let us denote bybn := 〈ρ, �n〉 and suppose that

∑∞n=0 |bn|

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Properties

Coefficients

The coefficients c`n can be expressed using the trigonometricFourier coefficients of f and ρ :

c`n := 〈f , �n〉〈ρ, �n+`〉 (` ≥ 0), c`n := 〈f , �n−`〉〈ρ, �n〉 (` < 0),

furthermore

〈〈Vρf ,Vρf 〉

〉=

1

2

∞∑m=0

∞∑n=0

|〈f , �n〉|2|〈ρ, �m〉|2

n + m + 1.

c`n =2n + `+ 1

2π

∫ 10

∫I(Vρf )(re

iϕ)r |`|e i`ϕP`n(r2)

r√1− r2

dϕdr (` ∈ Z, n ∈ N).

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Properties

Representation of the voice – transform as a differentialoperator

Let fix a polynomial

κ(z) := c0 + c1z + · · ·+ cNzN (z ∈ C)

and a complex number b ∈ C and let denote by A the set ofanalytic functions on D. Denote by αb(z) := 1− bz (z ∈ C).For every f ∈ A let be

Lbκf :=N∑

n=0

cnn!

(αnbf )(n).

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Properties

For an arbitrary function

f (e it) =∞∑

n=−∞ane

int (t ∈ I)

let denote by

f ∗(z) :=∞∑n=0

anzn, f∗(z) =

∞∑n=0

a−n−1zn (z ∈ D).

Then f ∗, f∗ ∈ H2(D) and

f (e it) = f ∗(e it) + e−it f∗(e−it) (for almost every t ∈ I).

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Properties

Theorem. For every function f ∈ L2(T) and for everytrigonometric polynomial ρ ∈ L2(T) the voice transform Vρf off can be represented as

Vρf (a−1) =

√1− |b|2[(Lbρ∗f ∗)(b)+(Lbρ∗f∗)(b)] (a = (b, 1) ∈ B).

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Properties

Admissible functions

Th

connection between zernike functions, corneal topography ...zernike coe cients computing the...

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