Table of Contents Motivation – CORNEA project Zenike functions Fritz Zernike Orthogonality of Zernike functions Representation of the cornea surface Problems Discrete orthogonality The discrete Zernike coefficients Computing the discrete Zernike coefficients Zernike representation of some test surfaces Connection to the voice transform Unitary Numerical Harmonic Analysis Group Connection between Zernike functions, corneal topography and the voice transform Margit Pap 1 [email protected], [email protected]November 11, 2010 1 University of P´ ecs, Hungary, NuHAG Margit Pap [email protected], [email protected]Connection between Zernike functions, corneal topography and
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Connection between Zernike functions, corneal topography and the voice transform
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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
1University of Pecs, 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)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
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
∫ 2π
0
∫ 1
0Z `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
∫ 2π
0
∫ 1
0Z `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
∫ 2π
0
∫ 1
0G (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
∫ 2π
0
∫ 1
0G (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
∫ 2π
0
∫ 1
0G (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
∫ 2π
0
∫ 1
0G (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,
be the corresponding fundamental polynomials ofLagrange interpolation.
ANj :=
∫ 1
−1`Nj (x)dx ,
the corresponding Cristoffel-numbers.
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) :=AN
k
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) :=AN
k
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) :=AN
k
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)AN
k
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 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→∞
∫X
fdνN =1
π
∫ 2π
0
∫ 1
0f (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 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→∞
∫X
fdνN =1
π
∫ 2π
0
∫ 1
0f (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(ρNk ,2πj
4N + 1)AN
k
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(ρNk ,2πj
4N + 1)AN
k
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(ρNk ,2πj
4N + 1)AN
k
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+|m|52N−1
AmnZmn (ρ, φ)
be an arbitrary linear combination of Zernike polynomialsof degree less than 2N.
The coefficients Amn can be expressed in the following twoways:
Amn =1
π
∫ 2π
0
∫ 1
0GN(ρ′, φ′)Zm
n (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+|m|52N−1
AmnZmn (ρ, φ)
be an arbitrary linear combination of Zernike polynomialsof degree less than 2N.
The coefficients Amn can be expressed in the following twoways:
Amn =1
π
∫ 2π
0
∫ 1
0GN(ρ′, φ′)Zm
n (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+|m|52N−1
AmnZmn (ρ, φ)
be an arbitrary linear combination of Zernike polynomialsof degree less than 2N.
The coefficients Amn can be expressed in the following twoways:
Amn =1
π
∫ 2π
0
∫ 1
0GN(ρ′, φ′)Zm
n (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+|m|52N−1
AmnZmn (ρ, φ)
be an arbitrary linear combination of Zernike polynomialsof degree less than 2N.
The coefficients Amn can be expressed in the following twoways:
Amn =1
π
∫ 2π
0
∫ 1
0GN(ρ′, φ′)Zm
n (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, 455-469, ISBN: 978-90-481-8775-1
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, 455-469, ISBN: 978-90-481-8775-1
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, 455-469, ISBN: 978-90-481-8775-1
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, 455-469, ISBN: 978-90-481-8775-1
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 sight-correcting 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 sight-correcting 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 sight-correcting 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 sight-correcting 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. 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) =
∫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. 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) =
∫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. 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) =
∫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. 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) =
∫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 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.
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 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.
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 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.
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 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.
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(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.
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(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.
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(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.
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(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.
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
Gabor transform
Gabor transform
The Gabor transform is a voice transform generated bythe representation of the Weyl-Heisenberg group:
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− |b|2[(Lbρ∗f∗)(b)+(Lb
ρ∗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
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).
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
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.
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
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.
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.
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
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.
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.
Schipp F., Wade W. R., Transforms on normed Fields,Leaflets in Mathematics, Janus Pannonius University Pecs,1995.
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
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.
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
END
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