CNR-INOA
Polinomi di Zernike
Bibliografia: roorda e Maeda
CNR-INOA
Sistema di coordinate
Optical
System
Optical
Axis
y
z
xy
x
Object
Plane
Image
Plane
y
x
!
"
Object
Height
h’
h
Image
Height
Optical
System
Optical
Axis
y
z
xy
x
Object
Plane
Image
Plane
y
x
!
"
Object
Height
h’
h
Image
Height
_
y
x
r
a
x = r cos(_)
y = r sin( _)
_ = tan -1(x/y)
r = (x 2+y2)1/2
_
y
x
_
1
x = _ cos(_)
y = _ sin( _)
_ = tan -1(x/y)
_ = r/a = (x 2+y2 )1/2
Normalized Pupil Coordinate SystemPupil Coordinate System
_
y
x
r
a
x = r cos(_)
y = r sin( _)
_ = tan -1(x/y)
r = (x 2+y2)1/2
_
y
x
_
1
x = _ cos(_)
y = _ sin( _)
_ = tan -1(x/y)
_ = r/a = (x 2+y2 )1/2
Normalized Pupil Coordinate SystemPupil Coordinate System
CNR-INOA
Sistema di coordinate perl’occhio
CNR-INOA
Aberrazione d’onda
L’aberrazione del fronte d’onda, W(x,y), è la distanza, in termini di camminoottico OPD (prodotto tra indice di rifrazione e cammino fisico), tra la sfera diriferimento e il fronte d’onda “reale”.
y
zx
Pupillad’uscita
PianoImmagine
Fronted’onda
aberratoFronte d’onda
sferico diriferimento
Aberrazioned’ondaW(x,y)
{ }
{ } 0,0
2
,
),(2
22
),(
),(1
),(
==
==
!
=
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(=
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ss
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xf
yxWi
p
PSFFT
PSFFTssMTF
eyxpFTAd
yxPSF))
)
*
)
CNR-INOA
I passaggi matematici
CNR-INOA
Sviluppo polinomiale
• L’aberrazione del fronte d’onda W(x,y)può essere sviluppata in termini dipolinomi di Zernike
• Il contributo di ogni termine èindipendente dall’altro
CNR-INOA
Formule matematiche
[ ] [ ]sn
mn
s
s
m
n
m
n
mm
m
m
n
m
n
m
n
m
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mmn
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polynomial radial theis )(
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:as defined are spolynomial ZernikeThe
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+=
+!+!!
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factorial.m
zernike.m
CNR-INOA
Lista dei polinomi di Zernike( )
( )
( )( )
( )( )( )
MMMM
oo
o
)4(c10 4 4 14
mAstigmatisSecondary )2cos(3410 2 4 13
Defocus ,Aberration Spherical 1665 0 4 12
mAstigmatisSecondary )2sin(3410 2- 4 11
)4(sin10 4- 4 10
)3(c8 3 3 9
axis- xalong Coma )cos(238 1 3 8
axis-y along Coma )sin(238 1- 3 7
)3(sin8 3- 3 6
90or 0at axis with mAstigmatis )2(c6 2 2 5
Defocus curvature, Field 123 0 2 4
45at axis with mAstigmatis )2(sin6 2- 2 3
Distortion direction,-in xTilt )cos(2 1 1 2
Distortion direction,-yin Tilt )(sin2 1- 1 1
Pistonor erm,Constant t 1 0 0 0
Meaning ,
frequencyorder mode
4
24
24
24
4
3
3
3
3
2
2
2
!"
!""
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!""
!"
!"
!""
!""
!"
!"
"
!"
!"
!"
!"
os
os
os
Zmnjm
n
#
+#
#
#
#
#
±
CNR-INOA
Aberrazione del fronte d’onda
),(),(
))cos()(())sin()((
),(),(
:spolynomial Zernikeof sum weighteda as expressed is aberration waveThe
max
0
0
1
7
!
! !!
! !
=
=
"
"=
"=
=
#$%
&'(
+"=
=
j
j
jj
k
n
n
m
m
n
m
n
m
n
nm
m
n
m
n
m
n
k
n
n
nm
m
n
m
n
yxZWyxW
mRNWmRNW
ZWW
)*)*
)*)*
Calcola_aberrazione_onda.m
CNR-INOA
Utilizzo di un solo indice
Talvolta si ricorre all’utilizzo di un solo indice j. Inquesto caso questa tabella indica l’equivalenza tra ledue notazioni
CNR-INOA
Polinomi di Zernike a doppio indiceAzimuthal Frequency, m
-6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6Radial
Order, n
0
1
2
3
4
5
6
Common Names7
Piston
Tilt
Astigmatism (m=-2,2),Defocus(m=0)
Coma (m=-1,1),Trefoil(m=-3,3)
Spherical Aberration(m=0)
Secondary Coma(m=-1,1)
Secondary SphericalAberration (m=0)
( ) ,!"m
nZ
ZernikePolynomial.m
CNR-INOA
PSF per i vari termini di ZernikeAzimuthal Frequency, m
-6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6Radial
Order, n
0
1
2
3
4
5
6
Common Names7
Piston
Tilt
Astigmatism (m=-2,2),Defocus(m=0)
Coma (m=-1,1),Trefoil(m=-3,3)
Spherical Aberration(m=0)
Secondary Coma(m=-1,1)
Secondary SphericalAberration (m=0)
( ) ,!"m
nZ
ZernikePolynomialPSF.m
CNR-INOA
Zernike Polynomials
CNR-INOA
Double-Index Zernike Polynomial MTFs
( ) ,!"m
nZ
Azimuthal Frequency, m-6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6
RadialOrder, n
0
1
2
3
4
5
6
0 10 20 30 40 500
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0 10 20 30 40 500
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0 10 20 30 40 500
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0 10 20 30 40 500
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0 10 20 30 40 500
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0 10 20 30 40 500
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0 10 20 30 40 500
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0 10 20 30 40 500
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0 10 20 30 40 500
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0 10 20 30 40 500
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0 10 20 30 40 500
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0 10 20 30 40 500
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0 10 20 30 40 500
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0 10 20 30 40 500
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0 10 20 30 40 500
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0 10 20 30 40 500
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0 10 20 30 40 500
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0 10 20 30 40 500
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0 10 20 30 40 500
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0 10 20 30 40 500
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0 10 20 30 40 500
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0 10 20 30 40 500
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0 10 20 30 40 500
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0 10 20 30 40 500
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0 10 20 30 40 500
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0 10 20 30 40 500
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0 10 20 30 40 500
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0 10 20 30 40 500
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
MTFyMTFx
Common Names7
Piston
Tilt
Astigmatism (m=-2,2),Defocus(m=0)
Coma (m=-1,1),Trefoil(m=-3,3)
Spherical Aberration(m=0)
Secondary Coma(m=-1,1)
Secondary SphericalAberration (m=0)
Pupil Diameter = 4 mm0 to 50 cycles/degreeλ = 570 nmRMS wavefront error = 0.2λ
ZernikePolynomialMTF.m
CNR-INOA
Double-Index ZernikePolynomial MTFs
Azimuthal Frequency, m-6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6
RadialOrder, n
0
1
2
3
4
5
6
MTFyMTFx
Common Names7
Piston
Tilt
Astigmatism (m=-2,2),Defocus(m=0)
Coma (m=-1,1),Trefoil(m=-3,3)
Spherical Aberration(m=0)
Secondary Coma(m=-1,1)
Secondary SphericalAberration (m=0)
0 10 20 30 40 500
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0 10 20 30 40 500
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0 10 20 30 40 500
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0 10 20 30 40 500
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0 10 20 30 40 500
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0 10 20 30 40 500
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0 10 20 30 40 500
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0 10 20 30 40 500
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0 10 20 30 40 500
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0 10 20 30 40 500
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0 10 20 30 40 500
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0 10 20 30 40 500
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0 10 20 30 40 500
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0 10 20 30 40 500
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0 10 20 30 40 500
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0 10 20 30 40 500
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0 10 20 30 40 500
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0 10 20 30 40 500
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0 10 20 30 40 500
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0 10 20 30 40 500
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0 10 20 30 40 500
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0 10 20 30 40 500
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0 10 20 30 40 500
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0 10 20 30 40 500
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0 10 20 30 40 500
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0 10 20 30 40 500
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0 10 20 30 40 500
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0 10 20 30 40 500
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Pupil Diameter = 7.3 mm0 to 50 cycles/degreeλ = 570 nmRMS wavefront error = 0.2λ( ) ,!"m
nZ
ZernikePolynomialMTF.m
CNR-INOA
Simulation based on HumanEye Data
0 10 20 30 40 500
0.2
0.4
0.6
0.8
1
sx (cycle/deg)
MTF of Zero Aberration System, 5.4mm pupil
0 10 20 30 40 500
0.2
0.4
0.6
0.8
1
sy (cycle/deg)
MTF of Zero Aberration System, 5.4mm pupil
0 10 20 30 40 500
0.2
0.4
0.6
0.8
1
sx (cycle/deg)
MTF of Aberrated System, Wrms = 0.85012!
0 10 20 30 40 500
0.2
0.4
0.6
0.8
1
sy (cycle/deg)
MTF of Aberrated System, Wrms = 0.85012!
Mode j Coefficient (µm) RMS Coefficient (µm)
0 0 0
1 0 0
2 0 0
3 1.02 0.416413256
4 0 0
5 0.33 0.134721936
6 0.21 0.074246212
7 -0.26 -0.091923882
8 0.03 0.010606602
9 -0.34 -0.120208153
10 -0.12 -0.037947332
11 0.05 0.015811388
12 0.19 0.084970583
13 -0.19 -0.060083276
14 0.15 0.047434165
Total RMS Wavefront Error (µm) 0.484608089
WaveAberration.m WaveAberrationPSF.m
WaveAberrationMTF.m
CNR-INOA
What are ZernikePolynomials?
• set of basic shapes that are used to fitthe wavefront
• analogous to the parabolic x2 shape thatcan be used to fit 2D data
CNR-INOA
Properties of ZernikePolynomials• orthogonal
– terms are not similar in any way, so the weighting of oneterms does not depend on whether or not other terms arebeing fit also
• normalized– the RMS wave aberration can be simply calculated as the
vector of all or a subset of coefficients
• efficient– Zernike shapes are very similar to typical aberrations found
in the eye
CNR-INOA
Measurement SetupPupil
Retina
Iris
RealAberratedWavefront
IdealPlanar
Wavefront
y
zx
IncomingLight Beam
CNR-INOA
Shack-Hartmann SensorLayout
CCDPupil Relay OpticsPBS
LightSource
LensletArray
CNR-INOA
Shack-Hartmann WavefrontSensor
rr’ r”
f f f f
p p’
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Perfect eye
Wavefront Lens Array CCD Array
Aberrated (typical) eye
Shack-Hartmann WavefrontSensor
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Lenslet Array
CNR-INOA
Alcune immagini
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BD KW SMShack-Hartmann Images
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WavefrontWavefront MapsMaps(at best focal plane)(at best focal plane)
BD KW SM
0.33 DS –0.17 DC X 87
6.42 DS –0.6 DC X 126
0 DS –1 DC X 3
CNR-INOA
Wavefront sensor imageWavefront sensor image Wavefront aberrationWavefront aberration
Aberrations of an RK patientAberrations of an RK patient
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Aberrations of a LASIK patientAberrations of a LASIK patient
Wavefront sensor imageWavefront sensor image Wavefront aberrationWavefront aberration
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Post - RKPost - RK Post - LASIKPost - LASIK
CNR-INOA
Cheratocono
CNR-INOA
LAC per cheratoconounaided eye custom contact lens
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PSF (pupilla 5 mm)unaided eye custom contact lens
rms = 4.16strehl ratio = 0.0008
1 degree1 degree
rms = 1.48strehl ratio = 0.004
CNR-INOA
Variazione dellamappacorneale alpassare deltempo conocchio aperto
W.CharmanContact Lens& Anterior Eye28 (2005)75-92
CNR-INOA
PSF through-focus (5 mmpupil)
0
2
4
6
8
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6 6.5 7
defocus [D]
rms
wav
e ab
erra
tion
(mic
rons
)
0.00000
0.00100
0.00200
0.00300
0.00400
0.00500
strehl ratio
The highest strehl ratio does not correlate withrms when aberrations are high
CNR-INOA
Simmetria tra i due occhi
Junzhong Liang andDavid R. Williams
CNR-INOA
Metrics to Define Image Quality
CNR-INOA
-2 -1 0 1 2-2.5
-2
-1.5
-1
-0.5
0
0.5
1
1.5
2
Wave Aberration Contour MapWave Aberration Contour Map
mm (right-left)
mm
(sup
erio
r-inf
erio
r)
CNR-INOA
Breakdown of Zernike Terms
-0.5 0 0.5 1 1.5 2123456789
1011121314151617181920
Zern
ike
term
Coefficient value (microns)
astig.defocus
astig.trefoilcomacomatrefoil
spherical aberration
2nd order
3rd order
4th order
5th order
CNR-INOA
Root Mean Square( ) ( )( )
( )
( )
21, ,
pupil area
, wave aberration
, average wave aberration
RMS W x y W x y dxdyA
A
W x y
W x y
= !
!
!
!
""
CNR-INOA
Root Mean Square( ) ( ) ( ) ( )
2 2 2 22 0 2 1
2 2 2 3.......RMS Z Z Z Z
! != + + +
astig
matism
term
defoc
us te
rm
astig
matism
term
trefoi
l term ……
Include the terms for which you want to determine theirimpact (eg defocus and astigmatism only, third orderterms or high order terms etc.)
CNR-INOA
Problemi nel RMS
J.S. McLellan et al.Vis. Res. 46 (2006)3009-3016
CNR-INOA
Point Spread Function
CNR-INOA
Strehl Ratiodiffraction-limited PSF
Hdl
Heye
actual PSF
Strehl Ratio = eye
dl
H
H
CNR-INOA
Typical Values for Wave Aberration
• Strehl ratios are about 5% for a 5 mm pupil that hasbeen corrected for defocus and astigmatism.
• Strehl ratios for small (~ 1 mm) pupils approach 1,but the image quality is poor due to diffraction.
Strehl Ratio
CNR-INOA
Typical Values for Wave AberrationPopulation Statistics
spherical aberration
comacomatrefoil
trefoil
CNR-INOA
Typical Values for Wave Aberration
Iglesias et al, 1998Navarro et al, 1998Liang et al, 1994Liang and Williams, 1997Liang et al, 1997Walsh et al, 1984He et al, 1999Calver et al, 1999Calver et al, 1999Porter et al., 2001He et al, 2002He et al, 2002Xu et al, 2003Paquin et al, 2002Paquin et al, 2002Carkeet et al, 2002Cheng et al, 2004
0
0.2
0.4
0.6
0.8
1
1.2
0 1 2 3 4 5 6 7 8 9pupil size (mm)
rms
wav
e ab
erra
tion
(mic
rons
) Shack-Hartmann MethodsOther Methods
Change in aberrations with pupil size
CNR-INOA
Typical Values for Wave AberrationChange in aberrations with age
Monochromatic Aberrations as a Function of Age, from Childhood to Advanced AgeIsabelle Brunette,1 Juan M. Bueno,2 Mireille Parent,1,3 Habib Hamam,3 and Pierre Simonet3
CNR-INOA
Convolution
CNR-INOA
( , ) ( , ) ( , )PSF x y O x y I x y! =
Convolution
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Simulated Images
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20/20 letters
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“I have never experienced any inconvenience from thisimperfection, nor did I ever discover it till I made theseexperiments; and I believe I can examine minute objectswith as much accuracy as most of those whose eyes aredifferently formed”
Thomas Young (1801) on his own aberrations.
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References[1] MacRae, S. M., Krueger, R. R., Applegate, A. A., (2001), Customized Corneal Ablation, The Quest forSuperVision, Slack Incorporated.
[2] Williams, D., Yoon, G. Y., Porter, J., Guirao, A., Hofer, H., Cox, I., (2000), “Visual Benefits of CorrectingHigher Order Aberrations of the Eye,” Journal of Refractive Surgery, Vol. 16, September/October 2000,S554-S559.
[3] Thibos, L., Applegate, R.A., Schweigerling, J.T., Webb, R., VSIA Standards Taskforce Members (2000),"Standards for Reporting the Optical Aberrations of Eyes," OSA Trends in Optics and Photonics Vol. 35,Vision Science and its Applications, Lakshminarayanan,V. (ed) (Optical Society of America, Washington,DC), pp: 232-244.
[4] Goodman, J. W. (1968). Introduction to Fourier Optics. San Francisco: McGraw Hill
[5] Gaskill, J. D. (1978). Linear Systems, Fourier Transforms, Optics. New York: Wiley
[6] Fischer, R. E. (2000). Optical System Design. New York: McGraw Hill
[7] Thibos, L. N.(1999), Handbook of Visual Optics, Draft Chapter on Standards for Reporting Aberrations ofthe Eye. http://research.opt.indiana.edu/Library/HVO/Handbook.html
[8] Bracewell, R. N. (1986). The Fourier Transform and Its Applications. McGraw Hill
[9] Mahajan, V. N. (1998). Optical Imaging and Aberrations, Part I Ray Geometrical Optics, SPIE Press
[10] Liang, L., Grimm, B., Goelz, S., Bille, J., (1994), “Objective Measurement of Wave Aberrations of theHuman Eye with the use of a Hartmann-Shack Wave-front Sensor,” J. Opt. Soc. Am. A, Vol. 11, No. 7,1949-1957.
[11] Liang, L., Williams, D. R., (1997), “Aberration and Retinal Image Quality of the Normal Human Eye,” J. Opt.Soc. Am. A, Vol. 14, No. 11, 2873-2883.