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Stanford UniversityPsych 221 / EE 362Winter 2002-2003
Patrick Y. Maeda 3/10/03
Zernike Polynomials and Their Use in Describing the Wavefront Aberrations of the
Human Eye
Psych 221/EE362 Applied Vision and Imaging Systems
Course Project, Winter 2003
Patrick Y. Maedapmaeda@stanford.edu
Stanford University
Stanford UniversityPsych 221 / EE 362Winter 2002-2003
Patrick Y. Maeda 3/10/03
Introduction and Motivation
Great interest in correcting higher order aberrations of the eye Laser eye surgery (PRK, LASIK)
– Currently, only defocus and astigmatism being corrected (2nd order aberrations)
– Improve vision better than 20/20
– Correct problems caused or induced by current generation of laser surgery
Imaging of the retina and other structures in the eye using adaptive optics
Correction requires measurement of optical aberrations Defocus and astigmatism can be determined using sets of lenses Measurement of higher orders require more sophisticated techniques
– Measurement of the wavefront aberration with Shack-Hartmann Wavefront Sensor
Mathematical description of the aberrations needed Accurate description of wave aberration function Accurate estimation of wave aberration function from measurement data
Stanford UniversityPsych 221 / EE 362Winter 2002-2003
Patrick Y. Maeda 3/10/03
Project Outline
Introduction/Motivation
General Optical System Description
Monochromatic Wavefront Aberrations
PSF and MTF calculations
Why Use Zernike Polynomials?
Definition of Zernike Polynomials
Describing Wave Aberrations using Zernike Polynomials
Simulating the Effects of Wave Aberrations
Wavefront Measurement and Data Fitting with Zernike Polynomials
Conclusion, References, Source Code Appendix
Stanford UniversityPsych 221 / EE 362Winter 2002-2003
Patrick Y. Maeda 3/10/03
Coordinate Systems
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
θ
y
x
r
a
x = r cos(θ)y = r sin(θ)θ = tan-1(x/y)r = (x2+y2)1/2
θ
y
x
ρ
1
x = ρ cos(θ)y = ρ sin(θ)θ = tan-1(x/y)ρ = r/a = (x2+y2)1/2
Normalized Pupil Coordinate SystemPupil Coordinate System
θ
y
x
r
a
x = r cos(θ)y = r sin(θ)θ = tan-1(x/y)r = (x2+y2)1/2
θ
y
x
ρ
1
x = ρ cos(θ)y = ρ sin(θ)θ = tan-1(x/y)ρ = r/a = (x2+y2)1/2
Normalized Pupil Coordinate SystemPupil Coordinate System
Stanford UniversityPsych 221 / EE 362Winter 2002-2003
Patrick Y. Maeda 3/10/03
Wave Aberration
The wavefront aberration, W(x,y), is the distance, in optical path length (product of the refractive index and path length), from the reference sphere to the wavefront in the exit pupil measured along the ray as a function of the transverse coordinates (x,y) of the ray intersection with the reference sphere. It is not the wavefront itself but it is the departure of the wavefront from the reference sphere.
y
zx
ExitPupil
ImagePlane
AberratedWavefront Reference
Spherical Wavefront
Wave Aberration
W(x,y)
0,0
2
,
),(2
22
),(
),(1),(
yx
yx
ss
yx
dy
fdx
f
yxWi
p
PSFFT
PSFFTssMTF
eyxpFTAd
yxPSF
Stanford UniversityPsych 221 / EE 362Winter 2002-2003
Patrick Y. Maeda 3/10/03
Describing Optical Aberrations
Optical system aberrations have historically been described, characterized, and catalogued by power series expansions
Many optical systems have circular pupils
Application of experimental results typically require data fitting
It is, therefore, desirable to expand the wave aberration in terms of a complete set of basis functions that are orthogonal over the interior of a circle
Stanford UniversityPsych 221 / EE 362Winter 2002-2003
Patrick Y. Maeda 3/10/03
Why Use Zernike Polynomials?
Zernike polynomials form a complete set of functions or modes that are orthogonal over a circle of unit radius Convenient for serving as a set of basis functions Expressible in polar coordinates or Cartesian coordinates Scaled so that non-zero order modes have zero mean and unit variance
– Puts modes in a common reference frame for meaningful relative comparison
Other power series descriptions are not orthogonal
Wave aberrations in an optical system with a circular pupil accurately described by a weighted sum of Zernike polynomials
The Orthonormal set of Zernike polynomials is recommended for describing wave aberration functions and for data fitting of experimental measurements for the eye7
– Terms are normalized so that the coefficient of a particular term or mode is the RMS contribution of that term
Stanford UniversityPsych 221 / EE 362Winter 2002-2003
Patrick Y. Maeda 3/10/03
Mathematical Formulae 3
sn
mn
s
sm
n
mn
mmm
mn
mn
mn
mn
mn
mn
mn
smnsmnssnR
R
mmnN
N
nnnnmn
mmRN
mmRNZ
22)(
0
000
3
!)(5.0!)(5.0!)!()1()(
polynomial radial theis )(
0for 0 , 0for 11
)1(2
factorion normalizat theis
,,4,2, of on values only takecan :given afor
20 , 10 , 0for )sin()(
20 , 10 , 0for )cos()(),(
:as defined are spolynomial ZernikeThe
factorial.m
zernike.m
Stanford UniversityPsych 221 / EE 362Winter 2002-2003
Patrick Y. Maeda 3/10/03
List of Zernike Polynomials 7, 9,10
)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
os
os
os
Zmnj mn
Stanford UniversityPsych 221 / EE 362Winter 2002-2003
Patrick Y. Maeda 3/10/03
Wave Aberration Description
),(),(
))cos()(())sin()((
),(),(
:spolynomial Zernikeof sum weighteda as expressed is aberration waveThe
max
0
0
1
7
j
jjj
k
n
n
m
mn
mn
mn
nm
mn
mn
mn
k
n
n
nm
mn
mn
yxZWyxW
mRNWmRNW
ZWW
WaveAberration.m
Stanford UniversityPsych 221 / EE 362Winter 2002-2003
Patrick Y. Maeda 3/10/03
Double-Index Zernike Polynomials
Azimuthal Frequency, m-6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6
Radial 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 Spherical Aberration (m=0)
,mnZ ZernikePolynomial.m
Stanford UniversityPsych 221 / EE 362Winter 2002-2003
Patrick Y. Maeda 3/10/03
Double-Index Zernike Polynomial PSFs
Azimuthal Frequency, m-6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6
Radial 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 Spherical Aberration (m=0)
,mnZ ZernikePolynomialPSF.m
Stanford UniversityPsych 221 / EE 362Winter 2002-2003
Patrick Y. Maeda 3/10/03
Double-Index Zernike Polynomial MTFs
,mnZ
Azimuthal Frequency, m-6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6
Radial Order, n
0
1
2
3
4
5
6
0 10 20 30 40 500
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MTFy
MTFx
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 Spherical Aberration (m=0)
Pupil Diameter = 4 mm0 to 50 cycles/degree = 570 nmRMS wavefront error = 0.2
ZernikePolynomialMTF.m
Stanford UniversityPsych 221 / EE 362Winter 2002-2003
Patrick Y. Maeda 3/10/03
Double-Index Zernike Polynomial MTFs
Azimuthal Frequency, m-6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6
Radial Order, n
0
1
2
3
4
5
6
MTFy
MTFx
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 Spherical Aberration (m=0)
0 10 20 30 40 500
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Pupil Diameter = 7.3 mm0 to 50 cycles/degree = 570 nmRMS wavefront error = 0.2 ,m
nZ
ZernikePolynomialMTF.m
Stanford UniversityPsych 221 / EE 362Winter 2002-2003
Patrick Y. Maeda 3/10/03
Simulation based on Human Eye Data
0 10 20 30 40 500
0.2
0.4
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1
sx (cycle/deg)
MTF of Zero Aberration System, 5.4mm pupil
0 10 20 30 40 500
0.2
0.4
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0.8
1
sy (cycle/deg)
MTF of Zero Aberration System, 5.4mm pupil
0 10 20 30 40 500
0.2
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1
sx (cycle/deg)
MTF of Aberrated System, Wrms = 0.85012
0 10 20 30 40 500
0.2
0.4
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sy (cycle/deg)
MTF of Aberrated System, Wrms = 0.85012
Mode j Coefficient (m) RMS Coefficient (m)
0 0 01 0 02 0 03 1.02 0.4164132564 0 05 0.33 0.1347219366 0.21 0.0742462127 -0.26 -0.0919238828 0.03 0.0106066029 -0.34 -0.12020815310 -0.12 -0.03794733211 0.05 0.01581138812 0.19 0.08497058313 -0.19 -0.06008327614 0.15 0.047434165
Total RMS Wavefront Error (m) 0.484608089
WaveAberration.m WaveAberrationPSF.m
WaveAberrationMTF.m
Stanford UniversityPsych 221 / EE 362Winter 2002-2003
Patrick Y. Maeda 3/10/03
Measurement Setup
Pupil
Retina
Iris
RealAberrated Wavefront
IdealPlanar
Wavefront
y
zx
IncomingLight Beam
Stanford UniversityPsych 221 / EE 362Winter 2002-2003
Patrick Y. Maeda 3/10/03
Shack-Hartmann Sensor Layout
CCDPupil Relay OpticsPBS
Light Source
Lenslet Array
Stanford UniversityPsych 221 / EE 362Winter 2002-2003
Patrick Y. Maeda 3/10/03
Shack-Hartmann Wavefront Sensor
Lenslet Array
Focal Length f
y(x1, y1)
y(x1, y2)
y(x1, y3)
y(x1, y4)
Aberrated Wavefront
Stanford UniversityPsych 221 / EE 362Winter 2002-2003
Patrick Y. Maeda 3/10/03
Data Fitting with Zernike Polynomials
(15) ),(),(
(14) ),(),(
),(),(
),(),(
modefor that error wavefrontrms the toequal is
expansion in the mode theoft coefficien theis
),(),(
),(),(
),(),(
yyxZ
Wf
yxy
xyxZ
Wf
yxx
yyxZ
Wy
yxW
xyxZ
Wx
yxW
W
ZW
yxZWyxWf
yxyy
yxWf
yxxx
yxW
j
jj
j
jj
j
jj
j
jj
j
jj
jjj
Equations (14) and (15) can be used to determine the Wj’s using Least-squares Estimation
Stanford UniversityPsych 221 / EE 362Winter 2002-2003
Patrick Y. Maeda 3/10/03
Least-squares Estimation
of nsposematrix tra theis where
)(
:bygiven is of estimate squares-Least The
2
),(),(),(
),(),(),(),(),(),(),(),(),(
),(),(),(),(),(),(
),(
),(),(),(
),(),(
formmatrix in expressed becan (15) and (14) Equations
),(),(
and ),(),(
),(),( and ),(),(
Let,
1LS
max2
max
2
1
max21
21max212211
11max112111
max21
21max212211
11max112111
21
11
21
11
T
TT
j
kkjkkkk
j
j
kkjkkkk
j
j
kk
kk
jj
jj
or
jk
W
WW
yxhyxhyxh
yxhyxhyxhyxhyxhyxhyxgyxgyxg
yxgyxgyxgyxgyxgyxg
yxc
yxcyxcyxb
yxbyxb
yxhy
yxZyxg
xyxZ
yxcf
yxyyxbf
yxx
Stanford UniversityPsych 221 / EE 362Winter 2002-2003
Patrick Y. Maeda 3/10/03
Benefits of Orthogonality
1
22LS
11
)( matrix, dconditione-ill
an ofinversion in theresult may functions basis ofset orthogonal-non a that Note
matrix diagonal aby tion multiplica and spolynomial Zernike theof sderivative partial theonto data
theof projectionby obtained are tscoefficien aberration waveThe matrix diagonal a is where
elements diagonal zero-nonh matrix wit diagonal a is ere wh
orthogonal are in columns theTherefore
orthogonal arey in sderivative partialTheir orthogonal arein x sderivative partialTheir
:orthogonal are s' theSince
T
T
T
j
DD
DD
Z
Stanford UniversityPsych 221 / EE 362Winter 2002-2003
Patrick Y. Maeda 3/10/03
Conclusions
Zernike Polynomials well suited for Describing wave aberration functions of optical systems with circular
pupils Estimation of wave aberration coefficients from wavefront measurements
Able to integrate Psych 221 learning with material from optical systems and Fourier optics courses
Linear systems theory make image formation and image quality evaluation straightforward
Suggestions for future work Extend simulation to incorporate chromatic effects Investigate the how wave aberration changes with accommodation Conduct simulations on a wide set of patient data Simulate the higher order aberrations induced by the PRK and LASIK Research some of the new wavefront technologies like implantable lenses
Stanford UniversityPsych 221 / EE 362Winter 2002-2003
Patrick Y. Maeda 3/10/03
References
[1] MacRae, S. M., Krueger, R. R., Applegate, A. A., (2001), Customized Corneal Ablation, The Quest for SuperVision, Slack Incorporated.
[2] Williams, D., Yoon, G. Y., Porter, J., Guirao, A., Hofer, H., Cox, I., (2000), “Visual Benefits of Correcting Higher 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 of the 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 the Human 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.
Stanford UniversityPsych 221 / EE 362Winter 2002-2003
Patrick Y. Maeda 3/10/03
Appendix I
Matlab Source Code Files:
zernike.mZernikePolynomial.mZernikePolynomialPSF.mZernikePolynomialMTF.mWaveAberration.mWaveAberrationPSF.mWaveAberrationMTF.m
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