Stanford University Psych 221 / EE 362 Winter 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. Maeda [email protected]Stanford University
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Stanford University Psych 221 / EE 362 Winter 2002-2003 Stanford University Psych 221 / EE 362 Winter 2002-2003 Patrick Y. Maeda 3/10/03 Zernike Polynomials.
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Stanford University
Psych 221 / EE 362
Winter 2002-2003
Stanford University
Psych 221 / EE 362
Winter 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
Normalized Pupil Coordinate SystemPupil Coordinate System
Stanford University
Psych 221 / EE 362
Winter 2002-2003
Stanford University
Psych 221 / EE 362
Winter 2002-2003
Patrick Y. Maeda 3/10/03
Wave AberrationWave 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.
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 University
Psych 221 / EE 362
Winter 2002-2003
Stanford University
Psych 221 / EE 362
Winter 2002-2003
Patrick Y. Maeda 3/10/03
Why Use Zernike Polynomials?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 University
Psych 221 / EE 362
Winter 2002-2003
Stanford University
Psych 221 / EE 362
Winter 2002-2003
Patrick Y. Maeda 3/10/03
Mathematical Formulae 3Mathematical Formulae 3
sn
mn
s
sm
n
mn
mmm
mn
mn
mn
mn
mn
mn
mn
smnsmns
snR
R
mmn
N
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 University
Psych 221 / EE 362
Winter 2002-2003
Stanford University
Psych 221 / EE 362
Winter 2002-2003
Patrick Y. Maeda 3/10/03
List of Zernike Polynomials 7, 9,10List of Zernike Polynomials 7, 9,10
Data Fitting with Zernike PolynomialsData Fitting with Zernike Polynomials
(15) ),(),(
(14) ),(),(
),(),(
),(),(
modefor that error wavefrontrms the toequal is
expansion in the mode theoft coefficien theis
),(),(
),(),(
),(),(
y
yxZW
f
yxy
x
yxZW
f
yxx
y
yxZW
y
yxW
x
yxZW
x
yxW
W
ZW
yxZWyxW
f
yxy
y
yxW
f
yxx
x
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 University
Psych 221 / EE 362
Winter 2002-2003
Stanford University
Psych 221 / EE 362
Winter 2002-2003
Patrick Y. Maeda 3/10/03
Least-squares Estimation 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
W
W
yxhyxhyxh
yxhyxhyxh
yxhyxhyxh
yxgyxgyxg
yxgyxgyxg
yxgyxgyxg
yxc
yxc
yxc
yxb
yxb
yxb
yxhy
yxZyxg
x
yxZ
yxcf
yxyyxb
f
yxx
Stanford University
Psych 221 / EE 362
Winter 2002-2003
Stanford University
Psych 221 / EE 362
Winter 2002-2003
Patrick Y. Maeda 3/10/03
Benefits of Orthogonality 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 University
Psych 221 / EE 362
Winter 2002-2003
Stanford University
Psych 221 / EE 362
Winter 2002-2003
Patrick Y. Maeda 3/10/03
ConclusionsConclusions
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 University
Psych 221 / EE 362
Winter 2002-2003
Stanford University
Psych 221 / EE 362
Winter 2002-2003
Patrick Y. Maeda 3/10/03
ReferencesReferences
[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.