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```									                                                 Stanford University
Psych 221 / EE 362
Winter 2002-2003

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
pmaeda@stanford.edu

Stanford University

Patrick Y. Maeda 3/10/03
Stanford University
Introduction and Motivation                                                    Psych 221 / EE 362
Winter 2002-2003

 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

Patrick Y. Maeda 3/10/03
Stanford University
Project Outline                                         Psych 221 / EE 362
Winter 2002-2003

 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

Patrick Y. Maeda 3/10/03
Stanford University
Coordinate Systems                                                                                          Psych 221 / EE 362
Winter 2002-2003

Object
Plane
y                                                  Pupil Coordinate System         Normalized Pupil Coordinate System
Optical
System                                                y                                    y
Object           x
h
Height                    y
           Image
Plane
Optical                 x                                          r                                    ρ
y
Axis                                                                      θ                                θ
x                                       x
x                       a                                    1

z
h’
Image
Height
x = r cos(θ)                         x = ρ cos(θ)
y = r sin(θ)                         y = ρ sin(θ)
θ = tan-1(x/y)                       θ = tan-1(x/y)
r = (x2+y2)1/2                       ρ = r/a = (x2+y2)1/2

Patrick Y. Maeda 3/10/03
Stanford University
Wave Aberration                                                                                Psych 221 / EE 362
Winter 2002-2003

2
2
Exit                                   1                     i W ( x, y ) 

Pupil                 PSF ( x, y )  2 2   FT  p( x, y )  e              
 d Ap
x         y
fx       , fy 
                                        d        d

Wave
FT PSF 
Aberration                        MTF ( s x , s y ) 
W(x,y)                                                 FT PSF  s x 0, s y 0

y

Aberrated                                                     Image
Reference                          Plane
z
Wavefront                                                                              x
Spherical
Wavefront
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.
Patrick Y. Maeda 3/10/03
Stanford University
Describing Optical Aberrations                              Psych 221 / EE 362
Winter 2002-2003

 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

Patrick Y. Maeda 3/10/03
Stanford University
Why Use Zernike Polynomials?                                                     Psych 221 / EE 362
Winter 2002-2003

 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

Patrick Y. Maeda 3/10/03
Stanford University
Mathematical Formulae 3                                                            Psych 221 / EE 362
Winter 2002-2003

The Zernike polynomial s are defined as 3 :

Z n (  , )  N n Rn (  ) cos(m )         for m  0 , 0    1 , 0    2
m              m     m

  N n Rn (  ) sin( m )         for m  0 , 0    1 , 0    2
m          m

for a given n : m can only take on values of  n,  n  2,  n  4,  , n

m
N n is the normalizat ion factor
2(n  1)
Nn 
m
 m 0  1 for m  0 ,  m 0  0 for m  0
1   m0
factorial.m
Rn (  ) is the radial polynomial
m
zernike.m
(n m ) 2
(1) s (n  s )!
R () 
n
m

s 0      s ! 0.5(n  m )  s ! 0.5(n  m )  s !
 n2 s

Patrick Y. Maeda 3/10/03
Stanford University
List of Zernike Polynomials 7, 9,10                                                              Psych 221 / EE 362
Winter 2002-2003

mode   order   frequency
j      n        m         Z n   , 
m
Meaning

0      0         0        1                               Constant term, or Piston
1      1       -1         2  sin ( )                    Tilt in y - direction, Distortion
2      1        1         2  cos( )                     Tilt in x - direction, Distortion
3      2       -2            6  2 sin (2 )              Astigmatis m with axis at  45 
4      2        0             
3 2 2  1                   Field curvature, Defocus
5      2        2           6  2 cos (2 )               Astigmatis m with axis at 0  or 90 
6      3       -3          8  3 sin (3 )
7      3       -1                             
8 3  3  2  sin( )          Coma along y - axis
8      3        1          8 3     3
 2  cos( )   Coma along x - axis
9      3        3          8  3 cos (3 )
10     4       -4          10  4 sin (4 )
11    4        -2                               
10 4  4  3  2 sin( 2 )     Secondary Astigmatis m
12     4        0            
5 6 4  6 2  1             Spherical Aberration , Defocus
13    4         2                                
10 4  4  3  2 cos(2 )      Secondary Astigmatis m
14    4         4          10  4 cos (4 )
                              
Patrick Y. Maeda 3/10/03
Stanford University
Wave Aberration Description                                               Psych 221 / EE 362
Winter 2002-2003

The wave aberration is expressed as a weighted sum of Zernike polynomial s 7 :

k       n
W (  , )            W   n
m
Z n (  , )
m

n m n

 1 m
k                               n

    Wn ( N n Rn (  ) sin( m ))   Wnm ( N n Rn (  ) cos(m ))
m   m                             m   m

n m   n                            m 0                          

j max
W ( x, y )    W Z
j 0
j   j   ( x, y )

WaveAberration.m

Patrick Y. Maeda 3/10/03
Stanford University
Double-Index Zernike Polynomials                                                 Psych 221 / EE 362
Winter 2002-2003

Order, n   -6      -5   -4    -3 -2 -1      0   1 2     3   4   5   6         Common Names7

  , 
0                                                                               Piston
m                                      ZernikePolynomial.m
Z   n
1                                                                                Tilt

Astigmatism (m=-2,2),
2                                                                            Defocus(m=0)

Coma (m=-1,1),
3                                                                           Trefoil(m=-3,3)

Spherical Aberration
4                                                                               (m=0)

Secondary Coma
5                                                                              (m=-1,1)

Secondary Spherical
6                                                                         Aberration (m=0)
Patrick Y. Maeda 3/10/03
Stanford University
Double-Index Zernike Polynomial PSFs                                            Psych 221 / EE 362
Winter 2002-2003

Order, n   -6      -5   -4    -3 -2 -1      0   1 2     3   4   5   6         Common Names7

  , 
0                                                                              Piston
m                                     ZernikePolynomialPSF.m
Z   n
1                                                                               Tilt

Astigmatism (m=-2,2),
2                                                                         Defocus(m=0)

Coma (m=-1,1),
3                                                                          Trefoil(m=-3,3)

Spherical Aberration
4                                                                             (m=0)

Secondary Coma
5                                                                              (m=-1,1)

Secondary Spherical
6                                                                       Aberration (m=0)
Patrick Y. Maeda 3/10/03
Stanford University
Double-Index Zernike Polynomial MTFs                                                                                                                                                                                                                                                                                                                                                                                                                                                    Psych 221 / EE 362
Winter 2002-2003

Order, n                   -6                             -5                                -4                                 -3 -2 -1      0   1 2     3                                                                                                                                                                                                                               4                                       5                             6                     Common Names7
1

0.9

0.8

0.7

Pupil Diameter = 4 mm                                                                                                                              Piston
0       ZernikePolynomialMTF.m
0.6

0.5

0.4

0.3

0.2

0.1
0 to 50 cycles/degree
 = 570 nm
0
0        10        20   30             40        50

  , 
1                                                                    1

0.9                                                                   0.9

m
0.8                                                                   0.8

Z                                                                                                                                                                                                                                                                                                                RMS wavefront error = 0.2
0.7                                                                   0.7

1
0.6

0.5

0.4
0.6

0.5

0.4
Tilt
n                                                                                                                                                0.3

0.2
0.3

0.2

MTFy
0.1                                                                   0.1

0                                                                    0
0        10        20   30             40        50                   0        10        20   30             40        50

1                                                                     1                                                                     1

0.9

0.8

0.7
0.9

0.8

0.7
0.9

0.8

0.7
MTFx                                                             Astigmatism (m=-2,2),
2
0.6                                                                   0.6                                                                   0.6

0.5                                                                   0.5                                                                   0.5

0.4

0.3

0.2

0.1
0.4

0.3

0.2

0.1
0.4

0.3

0.2

0.1
Defocus(m=0)
0                                                                     0                                                                     0
0        10        20   30             40        50                   0        10        20   30             40        50                   0        10        20   30             40        50

1                                                                     1                                                                     1                                                                     1

0.9                                                                   0.9                                                                   0.9                                                                   0.9

0.8

0.7
0.8

0.7
0.8

0.7
0.8

0.7
Coma (m=-1,1),
3
0.6                                                                   0.6                                                                   0.6                                                                   0.6

0.5                                                                   0.5                                                                   0.5                                                                   0.5

0.4

0.3

0.2

0.1
0.4

0.3

0.2

0.1
0.4

0.3

0.2

0.1
0.4

0.3

0.2

0.1
Trefoil(m=-3,3)
0                                                                     0                                                                     0                                                                     0
0        10        20   30             40        50                   0        10        20   30             40        50                   0        10        20   30             40        50                   0        10        20   30             40        50

1                                                                    1                                                                     1                                                                     1                                                                     1

0.9                                                                   0.9                                                                   0.9                                                                   0.9                                                                   0.9

0.8

0.7
0.8

0.7
0.8

0.7
0.8

0.7
0.8

0.7
Spherical Aberration
4
0.6                                                                   0.6                                                                   0.6                                                                   0.6                                                                   0.6

0.5                                                                   0.5                                                                   0.5                                                                   0.5                                                                   0.5

0.4

0.3

0.2

0.1
0.4

0.3

0.2

0.1
0.4

0.3

0.2

0.1
0.4

0.3

0.2

0.1
0.4

0.3

0.2

0.1
(m=0)
0                                                                    0                                                                     0                                                                     0                                                                     0
0        10        20   30             40        50                   0        10        20   30             40        50                   0        10        20   30             40        50                   0        10        20   30             40        50                   0        10        20   30             40        50

1                                                                      1                                                                    1                                                                     1                                                                     1                                                                     1

0.9                                                                   0.9                                                                   0.9                                                                   0.9                                                                   0.9                                                                   0.9

0.8

0.7
0.8

0.7
0.8

0.7
0.8

0.7
0.8

0.7
0.8

0.7
Secondary Coma
5
0.6                                                                   0.6                                                                   0.6                                                                   0.6                                                                   0.6                                                                   0.6

0.5                                                                   0.5                                                                   0.5                                                                   0.5                                                                   0.5                                                                   0.5

0.4

0.3

0.2

0.1
0.4

0.3

0.2

0.1
0.4

0.3

0.2

0.1
0.4

0.3

0.2

0.1
0.4

0.3

0.2

0.1
0.4

0.3

0.2

0.1
(m=-1,1)
0                                                                      0                                                                    0                                                                     0                                                                     0                                                                     0
0        10        20   30             40        50                   0        10        20   30             40        50                   0        10        20   30             40        50                   0        10        20   30             40        50                   0        10        20   30             40        50                   0        10        20   30             40        50

1                                                           1                                                                     1                                                                     1                                                                     1                                                                     1                                                                     1

0.9                                                         0.9                                                                   0.9                                                                   0.9                                                                   0.9                                                                   0.9                                                                   0.9

0.8

0.7
0.8

0.7
0.8

0.7
0.8

0.7
0.8

0.7
0.8

0.7
0.8

0.7
Secondary Spherical
6
0.6                                                         0.6                                                                   0.6                                                                   0.6                                                                   0.6                                                                   0.6                                                                   0.6

0.5                                                         0.5                                                                   0.5                                                                   0.5                                                                   0.5                                                                   0.5                                                                   0.5

0.4

0.3

0.2

0.1
0.4

0.3

0.2

0.1
0.4

0.3

0.2

0.1
0.4

0.3

0.2

0.1
0.4

0.3

0.2

0.1
0.4

0.3

0.2

0.1
0.4

0.3

0.2

0.1
Aberration (m=0)
0                                                           0                                                                     0                                                                     0                                                                     0                                                                     0                                                                     0
0   10   20   30             40        50                   0        10        20   30             40        50                   0        10        20   30             40        50                   0        10        20   30             40        50                   0        10        20   30             40        50                   0        10        20   30             40        50                   0        10        20   30   40   50

Patrick Y. Maeda 3/10/03
Stanford University
Double-Index Zernike Polynomial MTFs                                                                                                                                                                                                                                                                                                                                                                                                                                                    Psych 221 / EE 362
Winter 2002-2003

Order, n                   -6                             -5                                -4                                 -3 -2 -1      0   1 2     3                                                                                                                                                                                                                               4                                       5                             6                     Common Names7
1

0.9

0.8

0.7

Pupil Diameter = 7.3 mm                                                                                                                            Piston
0       ZernikePolynomialMTF.m
0.6

0.5

0.4

0.3

0.2

0.1
0 to 50 cycles/degree
 = 570 nm
0
0        10        20   30             40        50

  , 
1                                                                     1

0.9                                                                   0.9

m
0.8                                                                   0.8

Z                                                                                                                                                                                                                                                                                                                RMS wavefront error = 0.2
0.7                                                                   0.7

1
0.6

0.5

0.4
0.6

0.5

0.4
Tilt
n                                                                                                                                                0.3

0.2
0.3

0.2

MTFy
0.1                                                                   0.1

0                                                                     0
0        10        20   30             40        50                   0        10        20   30             40        50

1                                                                    1                                                                     1

0.9

0.8

0.7
0.9

0.8

0.7
0.9

0.8

0.7
MTFx                                                             Astigmatism (m=-2,2),
2
0.6                                                                   0.6                                                                   0.6

0.5                                                                   0.5                                                                   0.5

0.4

0.3

0.2

0.1
0.4

0.3

0.2

0.1
0.4

0.3

0.2

0.1
Defocus(m=0)
0                                                                    0                                                                     0
0        10        20   30             40        50                   0        10        20   30             40        50                   0        10        20   30             40        50

1                                                                    1                                                                     1                                                                     1

0.9                                                                   0.9                                                                   0.9                                                                   0.9

0.8

0.7
0.8

0.7
0.8

0.7
0.8

0.7
Coma (m=-1,1),
3
0.6                                                                   0.6                                                                   0.6                                                                   0.6

0.5                                                                   0.5                                                                   0.5                                                                   0.5

0.4

0.3

0.2

0.1
0.4

0.3

0.2

0.1
0.4

0.3

0.2

0.1
0.4

0.3

0.2

0.1
Trefoil(m=-3,3)
0                                                                    0                                                                     0                                                                     0
0        10        20   30             40        50                   0        10        20   30             40        50                   0        10        20   30             40        50                   0        10        20   30             40        50

1                                                                     1                                                                      1                                                                    1                                                                     1

0.9                                                                   0.9                                                                   0.9                                                                   0.9                                                                   0.9

0.8

0.7
0.8

0.7
0.8

0.7
0.8

0.7
0.8

0.7
Spherical Aberration
4
0.6                                                                   0.6                                                                   0.6                                                                   0.6                                                                   0.6

0.5                                                                   0.5                                                                   0.5                                                                   0.5                                                                   0.5

0.4

0.3

0.2

0.1
0.4

0.3

0.2

0.1
0.4

0.3

0.2

0.1
0.4

0.3

0.2

0.1
0.4

0.3

0.2

0.1
(m=0)
0                                                                     0                                                                      0                                                                    0                                                                     0
0        10        20   30             40        50                   0        10        20   30             40        50                   0        10        20   30             40        50                   0        10        20   30             40        50                   0        10        20   30             40        50

1                                                                     1                                                                     1                                                                     1                                                                     1                                                                     1

0.9                                                                   0.9                                                                   0.9                                                                   0.9                                                                   0.9                                                                   0.9

0.8

0.7
0.8

0.7
0.8

0.7
0.8

0.7
0.8

0.7
0.8

0.7
Secondary Coma
5
0.6                                                                   0.6                                                                   0.6                                                                   0.6                                                                   0.6                                                                   0.6

0.5                                                                   0.5                                                                   0.5                                                                   0.5                                                                   0.5                                                                   0.5

0.4

0.3

0.2

0.1
0.4

0.3

0.2

0.1
0.4

0.3

0.2

0.1
0.4

0.3

0.2

0.1
0.4

0.3

0.2

0.1
0.4

0.3

0.2

0.1
(m=-1,1)
0                                                                     0                                                                     0                                                                     0                                                                     0                                                                     0
0        10        20   30             40        50                   0        10        20   30             40        50                   0        10        20   30             40        50                   0        10        20   30             40        50                   0        10        20   30             40        50                   0        10        20   30             40        50

1                                                           1                                                                     1                                                                      1                                                                    1                                                                      1                                                                    1

0.9                                                         0.9                                                                   0.9                                                                   0.9                                                                   0.9                                                                   0.9                                                                   0.9

0.8

0.7
0.8

0.7
0.8

0.7
0.8

0.7
0.8

0.7
0.8

0.7
0.8

0.7
Secondary Spherical
6
0.6                                                         0.6                                                                   0.6                                                                   0.6                                                                   0.6                                                                   0.6                                                                   0.6

0.5                                                         0.5                                                                   0.5                                                                   0.5                                                                   0.5                                                                   0.5                                                                   0.5

0.4

0.3

0.2

0.1
0.4

0.3

0.2

0.1
0.4

0.3

0.2

0.1
0.4

0.3

0.2

0.1
0.4

0.3

0.2

0.1
0.4

0.3

0.2

0.1
0.4

0.3

0.2

0.1
Aberration (m=0)
0                                                           0                                                                     0                                                                      0                                                                    0                                                                      0                                                                    0
0   10   20   30             40        50                   0        10        20   30             40        50                   0        10        20   30             40        50                   0        10        20   30             40        50                   0        10        20   30             40        50                   0        10        20   30             40        50                   0        10        20   30   40   50

Patrick Y. Maeda 3/10/03
Stanford University
Simulation based on Human Eye Data                                                                                                    Psych 221 / EE 362
Winter 2002-2003

Mode j     Coefficient (m)      RMS Coefficient (m)
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                               MTF of Zero Aberration System, 5.4mm pupil   MTF of Zero Aberration System, 5.4mm pupil
7             -0.26               -0.091923882                                1                                            1

8              0.03               0.010606602                                0.8                                          0.8
9             -0.34               -0.120208153
0.6                                          0.6
10            -0.12               -0.037947332
11             0.05               0.015811388                                0.4                                          0.4
12             0.19               0.084970583
0.2                                          0.2
13            -0.19               -0.060083276
14             0.15               0.047434165                                  0                                            0
0   10       20     30     40   50           0    10       20     30     40   50
s x (cycle/deg)                               s y (cycle/deg)
Total RMS Wavefront Error (m)       0.484608089
MTF of Aberrated System, Wrms = 0.85012 MTF of Aberrated System, Wrms = 0.85012
1                                        1

0.8                                          0.8

0.6                                          0.6

0.4                                          0.4

0.2                                          0.2

0                                            0
0   10       20     30     40   50           0    10       20     30     40   50
s x (cycle/deg)                               s y (cycle/deg)

WaveAberrationMTF.m

WaveAberration.m                               WaveAberrationPSF.m
Patrick Y. Maeda 3/10/03
Stanford University
Measurement Setup                              Psych 221 / EE 362
Winter 2002-2003

y
Pupil
Iris
z
x

Incoming
Light Beam

Retina                   Ideal      Real
Planar   Aberrated
Wavefront Wavefront

Patrick Y. Maeda 3/10/03
Stanford University
Shack-Hartmann Sensor Layout              Psych 221 / EE 362
Winter 2002-2003

PBS   Pupil Relay Optics                 CCD

Lenslet
Array
Light
Source

Patrick Y. Maeda 3/10/03
Stanford University
Shack-Hartmann Wavefront Sensor                       Psych 221 / EE 362
Winter 2002-2003

Lenslet Array

y(x1, y1)
Aberrated
y(x1, y2)
Wavefront

y(x1, y3)

y(x1, y4)

Focal Length f
Patrick Y. Maeda 3/10/03
Stanford University
Data Fitting with Zernike Polynomials                                                 Psych 221 / EE 362
Winter 2002-2003

W ( x, y ) x( x, y )

x            f
W ( x, y ) y ( x, y )

y            f
W ( x, y )   W j Z j ( x, y )
j

W j is the coefficien t of the Z j mode in the expansion
W j is equal to the rms wavefront error for that mode
W ( x, y )        Z j ( x, y )
 W j
x          j        x
W ( x, y )        Z j ( x, y )
 W j
y          j        y
x( x, y )        Z j ( x, y )
 W j                      (14)
f        j        x
y ( x, y )        Z j ( x, y )
 W j                     (15)
f         j        y

Equations (14) and (15) can be used to determine the Wj’s using Least-squares Estimation

Patrick Y. Maeda 3/10/03
Stanford University
Least-squares Estimation                                                                                         Psych 221 / EE 362
Winter 2002-2003

Let,
x( x, y )                      y ( x, y )
 b( x, y ) and                 c ( x, y )
f                               f
Z j ( x, y )                     Z j ( x, y )
 g j ( x, y ) and                 h j ( x, y )
x                                 y
Equations (14) and (15) can be expressed in matrix form
 b( x1 , y1 )   g1 ( x1 , y1 ) g 2 ( x1 , y1 )            g j max ( x1 , y1 ) 
 b( x , y )   g ( x , y ) g ( x , y )                     g j max ( x1 , y 2 ) 
 1 2   1 1 2                         2     1    2                                
                                                                              W1 
                                                                                 
b( x k , y k )   g1 ( x k , y k ) g 2 ( x k , y k )      g j max ( x k , y k )  W2 
         2k 2  j max
 c( x1 , y1 )   h1 ( x1 , y1 ) h2 ( x1 , y1 )             h j max ( x1 , y1 )      
                                                                                         
 c( x1 , y 2 )   h1 ( x1 , y 2 ) h2 ( x1 , y 2 )          h j max ( x1 , y 2 )  W j max 
      
                                                                            
                                                                                 
c( x k , y k )   h1 ( x k , y k ) h2 ( x k , y k )
                                                          h j max ( x k , y k ) 

or
  
The Least - squares estimate of  is given by :
 LS  ( T  ) 1 T 
where  T is the matrix tra nspose of 
Patrick Y. Maeda 3/10/03
Stanford University
Benefits of Orthogonality                                                         Psych 221 / EE 362
Winter 2002-2003

Since the Z j ' s are orthogonal :
Their partial derivative s in x are orthogonal
Their partial derivative s in y are orthogonal

Therefore the columns in  are orthogonal
  T   D1            ere
wh D1 is a diagonal matrix wit h non - zero diagonal elements
  LS  D2 T  where D2 is a diagonal matrix
 The wave aberration coefficien ts are obtained by projection of the
data onto the partial derivative s of the Zernike polynomial s and
multiplica tion by a diagonal matrix

Note that a non - orthogonal set of basis functions may result in the inversion of an
ill - conditione d matrix, ( T  ) 1

Patrick Y. Maeda 3/10/03
Stanford University
Conclusions                                                           Psych 221 / EE 362
Winter 2002-2003

 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

Patrick Y. Maeda 3/10/03
Stanford University
References                                                                                                    Psych 221 / EE 362
Winter 2002-2003

[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.

Patrick Y. Maeda 3/10/03
Stanford University
Appendix I                  Psych 221 / EE 362
Winter 2002-2003

Matlab Source Code Files:

zernike.m
ZernikePolynomial.m
ZernikePolynomialPSF.m
ZernikePolynomialMTF.m
WaveAberration.m
WaveAberrationPSF.m
WaveAberrationMTF.m

Patrick Y. Maeda 3/10/03

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