connection between zernike functions, corneal topography and the voice transform

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

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 Pecs, Hungary, NuHAGMargit Pap [email protected], [email protected] between Zernike functions, corneal topography and the voice transform

Table ofContents

Zenikefunctions

Fritz Zernike

Problems

Properties

References

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

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

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

Zernike functions

Definition of Zernike functions

Z `n(ρ, θ) :=

√2n + |`|+ 1 R

|`||`|+2n(ρ)e i`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 (2ρ2 − 1).

Pictures

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Zenikefunctions

Fritz Zernike

Problems

Properties

References

Orthogonality of Zernike functions

∫ 2π

0Z `n(ρ, φ)Z `′

n′(ρ, φ)ρdρdφ = δnn′δ``′ .

Table ofContents

Zenikefunctions

Fritz Zernike

Problems

Properties

References

Orthogonality of Zernike functions

∫ 2π

0Z `n(ρ, φ)Z `′

n′(ρ, φ)ρdρdφ = δnn′δ``′ .

Table ofContents

Zenikefunctions

Fritz Zernike

Problems

Properties

References

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

∫ 2π

0G (ρ, φ)Z `

n(ρ, φ)ρdρdφ.

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Zenikefunctions

Fritz Zernike

Problems

Properties

References

A`nZ `n(ρ, φ)

A`n =1

∫ 2π

0G (ρ, φ)Z `

n(ρ, φ)ρdρdφ.

Table ofContents

Zenikefunctions

Fritz Zernike

Problems

Properties

References

A`nZ `n(ρ, φ)

A`n =1

∫ 2π

0G (ρ, φ)Z `

n(ρ, φ)ρdρdφ.

Table ofContents

Zenikefunctions

Fritz Zernike

Problems

Properties

References

A`nZ `n(ρ, φ)

A`n =1

∫ 2π

0G (ρ, φ)Z `

n(ρ, φ)ρdρdφ.

Table ofContents

Zenikefunctions

Fritz Zernike

Problems

Properties

References

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

Problems

Table ofContents

Zenikefunctions

Fritz Zernike

Problems

Properties

References

Problems

Table ofContents

Zenikefunctions

Fritz Zernike

Problems

Properties

References

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`Nj (x)dx ,

the corresponding Cristoffel-numbers.

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Zenikefunctions

Fritz Zernike

Problems

Properties

References

ANj :=

−1`Nj (x)dx ,

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Zenikefunctions

Fritz Zernike

Problems

Properties

References

ANj :=

−1`Nj (x)dx ,

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Zenikefunctions

Fritz Zernike

Problems

Properties

References

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 ,

4N + 1

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

and let define

ν(zjk) :=AN

2(4N + 1).

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Zenikefunctions

Fritz Zernike

Problems

Properties

References

ρNk :=

√1 + λNk

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

X := {zjk :=

(ρNk ,

4N + 1

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

and let define

ν(zjk) :=AN

2(4N + 1).

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Zenikefunctions

Fritz Zernike

Problems

Properties

References

ρNk :=

√1 + λNk

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

X := {zjk :=

(ρNk ,

4N + 1

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

and let define

ν(zjk) :=AN

2(4N + 1).

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Zenikefunctions

Fritz Zernike

Problems

Properties

References

Let introduce the following discrete integral∫X

f (ρ, φ)dνN :=N∑

4N∑j=0

f (ρNk ,2πj

4N + 1)AN

2(4N + 1).

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Zenikefunctions

Fritz Zernike

Problems

Properties

References

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)Zm′

n′ (ρ, φ)dνN = δnn′δmm′ .

For all f ∈ C (D)

limN→∞

fdνN =1

∫ 2π

0f (ρ, φ)ρdρdφ.

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Zenikefunctions

Fritz Zernike

Problems

Properties

References

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)Zm′

n′ (ρ, φ)dνN = δnn′δmm′ .

For all f ∈ C (D)

limN→∞

fdνN =1

∫ 2π

0f (ρ, φ)ρdρdφ.

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Zenikefunctions

Fritz Zernike

Problems

Properties

References

The discrete Zernike coefficients

A′`n =

G (ρ, φ)Z `n(ρ, φ)dνN(ρ, φ) =

N∑k=1

4N∑j=0

G (ρNk ,2πj

4N + 1)Z `

n(ρNk ,2πj

4N + 1)AN

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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Zenikefunctions

Fritz Zernike

Problems

Properties

References

A′`n =

N∑k=1

4N∑j=0

G (ρNk ,2πj

4N + 1)Z `

n(ρNk ,2πj

4N + 1)AN

2(4N + 1)

Table ofContents

Zenikefunctions

Fritz Zernike

Problems

Properties

References

A′`n =

N∑k=1

4N∑j=0

G (ρNk ,2πj

4N + 1)Z `

n(ρNk ,2πj

4N + 1)AN

2(4N + 1)

Table ofContents

Zenikefunctions

Fritz Zernike

Problems

Properties

References

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

∫ 2π

0GN(ρ′, φ′)Zm

n (ρ′, φ′)ρ′dρ′dφ′,

GN(ρ′, φ′)Zmn (ρ′, φ′)dνN(ρ′, φ′).

Measuring on X we can compute the exact values.

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Zenikefunctions

Fritz Zernike

Problems

Properties

References

LetGN(ρ, φ) =

∑2n+|m|52N−1

AmnZmn (ρ, φ)

Amn =1

∫ 2π

0GN(ρ′, φ′)Zm

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Zenikefunctions

Fritz Zernike

Problems

Properties

References

LetGN(ρ, φ) =

∑2n+|m|52N−1

AmnZmn (ρ, φ)

Amn =1

∫ 2π

0GN(ρ′, φ′)Zm

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Zenikefunctions

Fritz Zernike

Problems

Properties

References

LetGN(ρ, φ) =

∑2n+|m|52N−1

AmnZmn (ρ, φ)

Amn =1

∫ 2π

0GN(ρ′, φ′)Zm

Table ofContents

Zenikefunctions

Fritz Zernike

Problems

Properties

References

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

References

Table ofContents

Zenikefunctions

Fritz Zernike

Problems

Properties

References

Table ofContents

Zenikefunctions

Fritz Zernike

Problems

Properties

References

Table ofContents

Zenikefunctions

Fritz Zernike

Problems

Properties

References

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

Properties

References

Classical methods

Table ofContents

Zenikefunctions

Fritz Zernike

Problems

Properties

References

Classical methods

Table ofContents

Zenikefunctions

Fritz Zernike

Problems

Properties

References

Classical methods

Table ofContents

Zenikefunctions

Fritz Zernike

Problems

Properties

References

H. G. Feichtinger and K. H. Grochenig unified the theoryof Gabor 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) =

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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Zenikefunctions

Fritz Zernike

Problems

Properties

References

f (x) dm(x) =

f (a−1 · x) dm(x), (a ∈ G ).

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Zenikefunctions

Fritz Zernike

Problems

Properties

References

f (x) dm(x) =

f (a−1 · x) dm(x), (a ∈ G ).

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Zenikefunctions

Fritz Zernike

Problems

Properties

References

f (x) dm(x) =

f (a−1 · x) dm(x), (a ∈ G ).

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Zenikefunctions

Fritz Zernike

Problems

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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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Zenikefunctions

Fritz Zernike

Problems

Properties

References

i) Ux ·y = Ux ◦ Uy (x , y ∈ G ),

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Zenikefunctions

Fritz Zernike

Problems

Properties

References

i) Ux ·y = Ux ◦ Uy (x , y ∈ G ),

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Zenikefunctions

Fritz Zernike

Problems

Properties

References

i) Ux ·y = Ux ◦ Uy (x , y ∈ G ),

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Zenikefunctions

Fritz Zernike

Problems

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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 Gabor transform is a voice transform of theHeisenberg group.

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Zenikefunctions

Fritz Zernike

Problems

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References

Definition

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

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Zenikefunctions

Fritz Zernike

Problems

Properties

References

Definition

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

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Zenikefunctions

Fritz Zernike

Problems

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References

Definition

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

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Zenikefunctions

Fritz Zernike

Problems

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References

Gabor transform

The Gabor 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- Gabor transform

Vϕf (a, ω) =

f (t)ϕ(t − a)e−2πiωtdt = 〈f ,MωTaϕ〉

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

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Zenikefunctions

Fritz Zernike

Problems

Properties

References

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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Zenikefunctions

Fritz Zernike

Problems

Properties

References

The voice transform of the Blaschke group

The Blaschke group Let us denote by

Ba(z) := εz − b

1− bz(z ∈ C, a = (b, ε) ∈ 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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Fritz Zernike

Problems

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Ba(z) := εz − b

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

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

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Fritz Zernike

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Ba(z) := εz − b

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

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

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Zenikefunctions

Fritz Zernike

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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 , εj), j ∈ {1, 2} anda := (b, ε) =: a1 ◦ a2 then

b =b1ε2 + b2

1 + b1b2ε2, ε = ε1

ε2 + b1b2

1 + ε2b1b2

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

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Zenikefunctions

Fritz Zernike

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b =b1ε2 + b2

1 + b1b2ε2, ε = ε1

ε2 + b1b2

1 + ε2b1b2

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Zenikefunctions

Fritz Zernike

Problems

Properties

References

b =b1ε2 + b2

1 + b1b2ε2, ε = ε1

ε2 + b1b2

1 + ε2b1b2

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Zenikefunctions

Fritz Zernike

Problems

Properties

References

b =b1ε2 + b2

1 + b1b2ε2, ε = ε1

ε2 + b1b2

1 + ε2b1b2

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Zenikefunctions

Fritz Zernike

Problems

Properties

References

b =b1ε2 + b2

1 + b1b2ε2, ε = ε1

ε2 + b1b2

1 + ε2b1b2

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Fritz Zernike

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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) =

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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Fritz Zernike

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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) =

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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Fritz Zernike

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The representation of the Blaschke group on H2(T)

Denote by εn(t) = e int (t ∈ I = [0, 2π], n ∈ N), letconsider the Hilbert space H = H2(T), the closure inL2(T)-norm of the set

span{εn, n ∈ N}.

The inner product is given by

〈f , g〉 :=1

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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Fritz Zernike

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span{εn, n ∈ N}.

〈f , g〉 :=1

), f ∈ H2(T):

(Ua−1f )(z) :=

√e iθ(1− |b|2)

1− bz

)Margit Pap http://nuhag.eu

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Zenikefunctions

Fritz Zernike

Problems

Properties

References

span{εn, n ∈ N}.

〈f , g〉 :=1

), f ∈ H2(T):

(Ua−1f )(z) :=

√e iθ(1− |b|2)

1− bz

)Margit Pap http://nuhag.eu

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Zenikefunctions

Fritz Zernike

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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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Fritz Zernike

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References

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

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Zenikefunctions

Fritz Zernike

Problems

Properties

References

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

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Zenikefunctions

Fritz Zernike

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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) := 〈εn,Ua−1εm〉 ofrepresentation U with respect to the basis {εn : 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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Fritz Zernike

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vmn(a−1) =

√1− r2√

m + n + 1e−i(m+1/2)ψ(−1)mZ

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Fritz Zernike

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vmn(a−1) =

√1− r2√

m + n + 1e−i(m+1/2)ψ(−1)mZ

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Zenikefunctions

Fritz Zernike

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vmn(a−1) =

√1− r2√

m + n + 1e−i(m+1/2)ψ(−1)mZ

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Fritz Zernike

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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) =

(−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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Zenikefunctions

Fritz Zernike

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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) =

(−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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Zenikefunctions

Fritz Zernike

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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) =

√ε(1− |b|2)

(1− bz)

(ε(z − b)

1− bz

Let Vεm f (a−1) = 〈f ,Ua−1εm〉 and let define the followingprojection operator

Pf (a, z) :=∞∑

(Vεm 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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Zenikefunctions

Fritz Zernike

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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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Zenikefunctions

Fritz Zernike

Problems

Properties

References

Properties

Infinite series representation of voice – transform

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

⟨〈F ,G 〉

rF (re iϕ)G (re iϕ)

1− r2dϕdr . (1)

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

∑∞n=0 |bn| <∞, then for all

f ∈ H2(T) and a = (re iϕ, 1) ∈ B

Vρf (a−1) = (Vρf )(re iϕ) =√

1− r2∞∑

`=−∞r |`|e−i`ϕ

∞∑n=0

c`nP`n(r2),

where the infinite series is absolute convergent and convergentin norm induced by (1).

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Zenikefunctions

Fritz Zernike

Problems

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References

Properties

Coefficients

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

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

furthermore

⟨〈Vρf ,Vρf 〉

∞∑m=0

∞∑n=0

|〈f , εn〉|2|〈ρ, εm〉|2

n + m + 1.

c`n =2n + `+ 1

∫I(Vρf )(re iϕ)r |`|e i`ϕP`

n(r2)r√

1− r2dϕdr (` ∈ Z, n ∈ N).

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Zenikefunctions

Fritz Zernike

Problems

Properties

References

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

(αnbf )(n).

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Zenikefunctions

Fritz Zernike

Problems

Properties

References

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).Margit Pap http://nuhag.eu

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Zenikefunctions

Fritz Zernike

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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)+(Lb

ρ∗f∗)(b)] (a = (b, 1) ∈ B).

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Zenikefunctions

Fritz Zernike

Problems

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Properties

Admissible functions

Theorem Let suppose that ρ is a real trigonometricpolynomial and ρ∗ is an odd or even algebraic polynomialwhich vanishes in 0, namely

b0 = 0, bk = b−k (N ∈ N∗, k 5 N), ρ∗(−z) = ±ρ∗(z) (z ∈ D).

Then ρ is an admissible function for the representation Ua

which means that Vρρ ∈ L2m(B).

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Zenikefunctions

Fritz Zernike

Problems

Properties

References

Born, M., Wolf, E., Principles of Optics 5th ed.,Pergamon, New York, 1975.

De Carvalho, L. A. V., De Castro, J. C., Preliminaryresults of an Instrument for Measuring the OpticalAberrations of the Human Eye, Brasilian Journal ofPhysics, Vol. 33, no. 1, March 2003.

Evangelista G., Cavaliere S., Discrete frequency wrapedwavelets: theory and applications, IEEE Transactions onSignal Processing, vol. 46, 4.April. 1998.

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Zenikefunctions

Fritz Zernike

Problems

Properties

References

Feichtinger H. G., Grochenig K. , A unified approach toatomic decompositions trough integrable grouprepresentations , Functions Spaces and Applications, M.Cwinkel e. all. eds. Lecture Notes in Math. 1302,Springer-Verlag (1989), 307-340.

Feichtinger H. G., Grochenig K., Banach spaces relatedto integrable group representations and their atomicdecomposition I., Journal of Functional Analysis, Vol. 86,No. 2,(1989), 307-340.

Genberg, V., Optical Surface Evaluation, SPIE, Proc.Vol. 450, paper 450-08, Nov., (1983)

Genberg V., Michels, G.,Opto-mechanical analysis ofsegmented adaptive optics, SPIE, Vol. 4444 (10), SanDiego, Aug. (2001).

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Zenikefunctions

Fritz Zernike

Problems

Properties

References

Grossman A., Morlet J., Decomposition of Hardyfunctions into squere integrable wavelets of constant shape, SIAM J. Math. Anal.,Vol. 15 (1984) 723-736.

Grossman A., Morlet J., Paul T., Transforms associatedto squere integrable group representationsI, Generalresults, J. Math Physics, Vol. 26 No. 10, (1985)2473-2479.

Heil C. E., Walnut D. F., Continuous and discretewavelet transforms, SIAM Review, Vol. 31, No. 4,December (1989), 628-666.

Kisil V. V., Wavelets in Applied and pure Mathematics,http://maths.leeds.ac.uk/ kisilv/courses/wavelets.htlm.

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Zenikefunctions

Fritz Zernike

Problems

Properties

References

Padilla-Vivenco, A., Martinez- Ramirez, F. Granados,A., Digital Image Reconstruction by using ZernikeMoments, SPIEUSE, V. 15237-33, pp. 1-9.

Pap, M., Schipp, F., Discrete orthogonality of Zernikefunctions Mathematica Pannonica 16/1, (2005), 137-144.

Schipp F., Wade W. R., Transforms on normed Fields,Leaflets in Mathematics, Janus Pannonius University Pecs,1995.

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Zenikefunctions

Fritz Zernike

Problems

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References

A. Soumelidis, Z. Fazakas, M. Pap, F. Schipp:Discrete orthogonality of Zernike functions and itsapplication to corneal topography, Electronic Engineringand Computing Thechnology, Lecture notes in electricalenginering, 2010, Vol 60, 455-469, ISBN:978-90-481-8775-1A. Warzynczyk, Group Representation and SpecialFunctions, D. Reidel Publishing Company, Kluwer Acad.Publ. Gr., Polish Sci. Publ., Warszawa (1984).Wyant,J. C., Creath, K., Basic Wavefront AberrationTheory for Optical Metrology, Applied Optics and OpticalEngineering, Vol XI, Academic Press (1992).Zernike, F., Beugungstheorie des Schneidenverfharensund seiner verbesserten Form , derPhasenkontrastmethode, Physica (1934) 1, 689-704.

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Zenikefunctions

Fritz Zernike

Problems

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References

Laszlo Lovasz reminded us that ”Mathematics is queen andservant of science” (Eric Temple Bell).The different branches of science get increasingly more distancefrom each other with the deepening of the knowledge, thedialogue becomes increasingly heavier between them. For theworld’s accurate understanding at the same time is necessarythe contact of disciplines.03. 11. 2010, Hungarian Academy of Sciences

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connection between zernike functions, corneal topography and the voice transform

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