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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

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
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
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
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.
Margit Pap http://nuhag.eu

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
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.
Margit Pap http://nuhag.eu

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
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
Margit Pap http://nuhag.eu

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
Orthogonality of Zernike functions
Orthogonality of Zernike functions
1
π
∫ 2π0
∫ 10
Z `n(ρ, φ)Z`′n′(ρ, φ)ρdρdφ = δnn′δ``′ .
Margit Pap http://nuhag.eu

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
Orthogonality of Zernike functions
Orthogonality of Zernike functions
1
π
∫ 2π0
∫ 10
Z `n(ρ, φ)Z`′n′(ρ, φ)ρdρdφ = δnn′δ``′ .
Margit Pap http://nuhag.eu

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
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.
Margit Pap http://nuhag.eu

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
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.
Margit Pap http://nuhag.eu

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
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.
Margit Pap http://nuhag.eu

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
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.
Margit Pap http://nuhag.eu

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
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.
Margit Pap http://nuhag.eu

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
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.
Margit Pap http://nuhag.eu

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
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.
Margit Pap http://nuhag.eu

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
Discrete orthogonality
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 Cristoffelnumbers.
Margit Pap http://nuhag.eu

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
Discrete orthogonality
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 Cristoffelnumbers.
Margit Pap http://nuhag.eu

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
Discrete orthogonality
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 Cristoffelnumbers.
Margit Pap http://nuhag.eu

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
Discrete orthogonality
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).
Margit Pap http://nuhag.eu

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
Discrete orthogonality
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).
Margit Pap http://nuhag.eu

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
Discrete orthogonality
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).
Margit Pap http://nuhag.eu

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
Discrete orthogonality
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).
Margit Pap http://nuhag.eu

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
Discrete orthogonality
Theorem PapSchipp 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.
Margit Pap http://nuhag.eu

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
Discrete orthogonality
Theorem PapSchipp 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.
Margit Pap http://nuhag.eu

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
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 → +∞.
Margit Pap http://nuhag.eu

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
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 → +∞.
Margit Pap http://nuhag.eu

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
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 → +∞.
Margit Pap http://nuhag.eu

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
The discrete Zernike coefficients
LetGN(ρ, φ) =
∑2n+m52N−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
GN(ρ′, φ′)Zmn (ρ
′, φ′)dνN(ρ′, φ′).
Measuring on X we can compute the exact values.
Margit Pap http://nuhag.eu

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
The discrete Zernike coefficients
LetGN(ρ, φ) =
∑2n+m52N−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
GN(ρ′, φ′)Zmn (ρ
′, φ′)dνN(ρ′, φ′).
Measuring on X we can compute the exact values.
Margit Pap http://nuhag.eu

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
The discrete Zernike coefficients
LetGN(ρ, φ) =
∑2n+m52N−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
GN(ρ′, φ′)Zmn (ρ
′, φ′)dνN(ρ′, φ′).
Measuring on X we can compute the exact values.
Margit Pap http://nuhag.eu

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
The discrete Zernike coefficients
LetGN(ρ, φ) =
∑2n+m52N−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
GN(ρ′, φ′)Zmn (ρ
′, φ′)dνN(ρ′, φ′).
Measuring on X we can compute the exact values.
Margit Pap http://nuhag.eu

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
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, 455469, ISBN: 9789048187751
Computer implementations, experimental results onartificial corneallike surfaces.
Comparison of precision with the measurement methodsused by conventional topographers proved that theapproximations made using the discrete orthogonality arethe best.
Margit Pap http://nuhag.eu

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
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, 455469, ISBN: 9789048187751
Computer implementations, experimental results onartificial corneallike surfaces.
Comparison of precision with the measurement methodsused by conventional topographers proved that theapproximations made using the discrete orthogonality arethe best.
Margit Pap http://nuhag.eu

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
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, 455469, ISBN: 9789048187751
Computer implementations, experimental results onartificial corneallike surfaces.
Comparison of precision with the measurement methodsused by conventional topographers proved that theapproximations made using the discrete orthogonality arethe best.
Margit Pap http://nuhag.eu

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
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, 455469, ISBN: 9789048187751
Computer implementations, experimental results onartificial corneallike surfaces.
Comparison of precision with the measurement methodsused by conventional topographers proved that theapproximations made using the discrete orthogonality arethe best.
Margit Pap http://nuhag.eu

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
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 sightcorrecting operations.
Margit Pap http://nuhag.eu

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
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 sightcorrecting operations.
Margit Pap http://nuhag.eu

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
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 sightcorrecting operations.
Margit Pap http://nuhag.eu

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
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 sightcorrecting operations.
Margit Pap http://nuhag.eu

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
Connection to the voice transform
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 socalledvoice transform.
In the construction of the voicetransform the startingpoint will be a locally compact topological group (G , ·).
Let m be a leftinvariant 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.
Margit Pap http://nuhag.eu

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
Connection to the voice transform
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 socalledvoice transform.
In the construction of the voicetransform the startingpoint will be a locally compact topological group (G , ·).Let m be a leftinvariant Haar measure of G :∫
Gf (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.
Margit Pap http://nuhag.eu

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
Connection to the voice transform
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 socalledvoice transform.
In the construction of the voicetransform the startingpoint will be a locally compact topological group (G , ·).Let m be a leftinvariant Haar measure of G :∫
Gf (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.
Margit Pap http://nuhag.eu

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
Connection to the voice transform
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 socalledvoice transform.
In the construction of the voicetransform the startingpoint will be a locally compact topological group (G , ·).Let m be a leftinvariant Haar measure of G :∫
Gf (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.
Margit Pap http://nuhag.eu

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
Unitary representation
Unitary representation of the group (G , ·): Let usconsider a Hilbertspace (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.
Margit Pap http://nuhag.eu

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
Unitary representation
Unitary representation of the group (G , ·): Let usconsider a Hilbertspace (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.
Margit Pap http://nuhag.eu

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
Unitary representation
Unitary representation of the group (G , ·): Let usconsider a Hilbertspace (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.
Margit Pap http://nuhag.eu

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
Unitary representation
Unitary representation of the group (G , ·): Let usconsider a Hilbertspace (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.
Margit Pap http://nuhag.eu

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
Definition of the voice transform
Definition
The voice transform of f ∈ H generated by therepresentation U and by the parameter ρ ∈ H is the(complexvalued) 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.
Margit Pap http://nuhag.eu

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
Definition of the voice transform
Definition
The voice transform of f ∈ H generated by therepresentation U and by the parameter ρ ∈ H is the(complexvalued) 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.
Margit Pap http://nuhag.eu

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
Definition of the voice transform
Definition
The voice transform of f ∈ H generated by therepresentation U and by the parameter ρ ∈ H is the(complexvalued) 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.
Margit Pap http://nuhag.eu

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
Definition of the voice transform
Definition
The voice transform of f ∈ H generated by therepresentation U and by the parameter ρ ∈ H is the(complexvalued) 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.
Margit Pap http://nuhag.eu

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
Gábor transform
Gábor transform
The Gábor transform is a voice transform generated bythe representation of the WeylHeisenberg 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

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
Affine wavelet transform
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

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
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 11 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.
Margit Pap http://nuhag.eu

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
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 11 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.
Margit Pap http://nuhag.eu

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
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 11 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.
Margit Pap http://nuhag.eu

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
The voice transform of the Blaschke group
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�, �).
Margit Pap http://nuhag.eu

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
The voice transform of the Blaschke group
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�, �).
Margit Pap http://nuhag.eu

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
The voice transform of the Blaschke group
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�, �).
Margit Pap http://nuhag.eu

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
The voice transform of the Blaschke group
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�, �).
Margit Pap http://nuhag.eu

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
The voice transform of the Blaschke group
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�, �).
Margit Pap http://nuhag.eu

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
The voice transform of the Blaschke group
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− b2)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.
Margit Pap http://nuhag.eu

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
The voice transform of the Blaschke group
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− b2)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.
Margit Pap http://nuhag.eu

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
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− b2)(1− bz)
f(e iθ(z − b)
1− bz)
Margit Pap http://nuhag.eu

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
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− b2)(1− bz)
f(e iθ(z − b)
1− bz)
Margit Pap http://nuhag.eu

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
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− b2)(1− bz)
f(e iθ(z − b)
1− bz)
Margit Pap http://nuhag.eu

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
The voice transform of the Blaschke group
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 34, pp. 387395.
Pap M., Schipp F., The voice transform on the Blaschkegroup II., Annales Univ. Sci. (Budapest), Sect. Comput.,29 (2008) 157173.
Margit Pap http://nuhag.eu

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
The voice transform of the Blaschke group
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 34, pp. 387395.
Pap M., Schipp F., The voice transform on the Blaschkegroup II., Annales Univ. Sci. (Budapest), Sect. Comput.,29 (2008) 157173.
Margit Pap http://nuhag.eu

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
The voice transform of the Blaschke group
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 34, pp. 387395.
Pap M., Schipp F., The voice transform on the Blaschkegroup II., Annales Univ. Sci. (Budapest), Sect. Comput.,29 (2008) 157173.
Margit Pap http://nuhag.eu

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
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−nmin{n,m}(r , ϕ).
An important consequence of this connection is theaddition formula for Zernike functions.
Margit Pap http://nuhag.eu

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
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−nmin{n,m}(r , ϕ).
An important consequence of this connection is theaddition formula for Zernike functions.
Margit Pap http://nuhag.eu

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
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−nmin{n,m}(r , ϕ).
An important consequence of this connection is theaddition formula for Zernike functions.
Margit Pap http://nuhag.eu

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
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−nmin{n,m}(r , ϕ).
An important consequence of this connection is theaddition formula for Zernike functions.
Margit Pap http://nuhag.eu

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
The matrix elements of representations of theBlaschke group
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−mmin{m,n}(r , ϕ) =
∑k
(−1)ke−i(m+1/2)ψ1e−i(k+1/2)ψ2√(m + k + 1)(n + k + 1)
Zk−mmin{m,k}(r1, ϕ1)Z
n−kmin{k,n}(r2, ϕ2),
Margit Pap http://nuhag.eu

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
The matrix elements of representations of theBlaschke group
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−mmin{m,n}(r , ϕ) =
∑k
(−1)ke−i(m+1/2)ψ1e−i(k+1/2)ψ2√(m + k + 1)(n + k + 1)
Zk−mmin{m,k}(r1, ϕ1)Z
n−kmin{k,n}(r2, ϕ2),
Margit Pap http://nuhag.eu

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
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− b2)(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

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
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.
Margit Pap http://nuhag.eu

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
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

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
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).
Margit Pap http://nuhag.eu

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
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).
Margit Pap http://nuhag.eu

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
Properties
Representation of the voice – transform as a differentialoperator
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

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
Properties
Representation of the voice – transform as a differentialoperator
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− b2[(Lbρ∗f ∗)(b)+(Lbρ∗f∗)(b)] (a = (b, 1) ∈ B).
Margit Pap http://nuhag.eu

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
Properties
Admissible functions
Th