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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
[email protected]
Stanford University
Patrick Y. Maeda 3/10/03
Introduction and Motivation
Stanford University
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
Project Outline
Stanford University
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
Psych 221 / EE 362
Winter 2002-2003
Coordinate Systems
Object
Plane
y
Object
h
Height
x
Pupil Coordinate System
y
Optical
System
y
Image
Plane

Optical
Axis
Normalized Pupil Coordinate System
y
x
θ
x

h’
ρ
r
y
a
θ
x
x
1
z
Image
Height
x = r cos(θ)
y = r sin(θ)
θ = tan-1(x/y)
r = (x2+y2)1/2
x = ρ cos(θ)
y = ρ sin(θ)
θ = tan-1(x/y)
ρ = r/a = (x2+y2)1/2
Patrick Y. Maeda 3/10/03
Stanford University
Psych 221 / EE 362
Winter 2002-2003
Wave Aberration
Exit
Pupil
Wave
Aberration
W(x,y)
2
 i W ( x, y ) 

1
PSF ( x, y )  2 2
FT  p( x, y )  e 

 d Ap


MTF ( s x , s y ) 
2
fx 
x
y
, fy 
d
d
FT PSF 
FT PSF  s x 0, s y 0
y
Aberrated
Wavefront
Reference
Spherical
Wavefront
Image
Plane
x
z
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
Describing Optical Aberrations
Stanford University
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
Why Use Zernike Polynomials?
Stanford University
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
Mathematical Formulae 3
Stanford University
Psych 221 / EE 362
Winter 2002-2003
The Zernike polynomial s are defined as 3 :
Z nm (  , )  N nm Rn (  ) cos( m )
m
  N nm Rn (  ) sin( m )
m
for m  0 , 0    1 , 0    2
for m  0 , 0    1 , 0    2
for a given n : m can only take on values of  n,  n  2,  n  4, , n
N nm is the normalizat ion factor
N nm 
2(n  1)
1   m0
 m 0  1 for m  0 ,  m 0  0 for m  0
factorial.m
Rn (  ) is the radial polynomial
m
R () 
m
n
( n m ) 2

s 0
zernike.m
(1) s (n  s )!
 n2 s
s ! 0.5(n  m )  s ! 0.5(n  m )  s !
Patrick Y. Maeda 3/10/03
List of Zernike Polynomials 7, 9,10
Stanford University
Psych 221 / EE 362
Winter 2002-2003
mode
order
frequency
j
n
m
Z nm  , 
Meaning
0
1
2
0
1
1
0
-1
1
1
2  sin ( )
2  cos( )
Constant t erm, or Piston
Tilt in y - direction, Distortion
Tilt in x - direction, Distortion
3
2
-2
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
1
8 3  3  2  cos( )
9
3
3
8  3 cos (3 )
10
4
-4
10  4 sin (4 )
11
4
-2
12
4
0
13
4
2
10 4  4  3  2 cos( 2 )
14

4

4

10  4 cos (4 )

6  2 sin (2 )
8 3  3  2  sin(  )
10 4  4  3 2 sin( 2 )
5 6  4  6  2  1
Astigmatis m with axis at  45 
Coma along y - axis
Coma along x - axis
Secondary Astigmatis m
Spherical Aberration , Defocus
Secondary Astigmatis m
Patrick Y. Maeda 3/10/03
Wave Aberration Description
Stanford University
Psych 221 / EE 362
Winter 2002-2003
The wave aberration is expressed as a weighted sum of Zernike polynomial s 7 :
k
W (  , )  
n
W
n m n
m
n
Z nm (  , )
n
 1 m

m
m
m
    Wn ( N n Rn (  ) sin( m ))   Wnm ( N nm Rn (  ) cos( m )) 
n m   n
m 0

k
W ( x, y ) 
j max
W Z
j 0
j
j
( x, y )
WaveAberration.m
Patrick Y. Maeda 3/10/03
Stanford University
Psych 221 / EE 362
Winter 2002-2003
Double-Index Zernike Polynomials
Radial
Order, n
0
-6
Z
m
n
-5
-4
  , 
Azimuthal Frequency, m
-3 -2 -1
0
1 2
3
4
5
6
ZernikePolynomial.m
Common Names7
Piston
1
Tilt
2
Astigmatism (m=-2,2),
Defocus(m=0)
3
Coma (m=-1,1),
Trefoil(m=-3,3)
4
Spherical Aberration
(m=0)
5
Secondary Coma
(m=-1,1)
6
Secondary Spherical
Aberration (m=0)
Patrick Y. Maeda 3/10/03
Stanford University
Psych 221 / EE 362
Winter 2002-2003
Double-Index Zernike Polynomial PSFs
Radial
Order, n
0
-6
Z
m
n
-5
-4
  , 
Azimuthal Frequency, m
-3 -2 -1
0
1 2
3
4
5
Common Names7
6
ZernikePolynomialPSF.m
Piston
1
Tilt
2
Astigmatism (m=-2,2),
Defocus(m=0)
3
Coma (m=-1,1),
Trefoil(m=-3,3)
4
Spherical Aberration
(m=0)
5
Secondary Coma
(m=-1,1)
6
Secondary Spherical
Aberration (m=0)
Patrick Y. Maeda 3/10/03
Double-Index Zernike Polynomial MTFs
Radial
Order, n
-6
-5
Azimuthal Frequency, m
-3 -2 -1
0
1 2
3
-4
4
5
Stanford University
Psych 221 / EE 362
Winter 2002-2003
Common Names7
6
1
0.9
0.8
Pupil Diameter = 4 mm
0 to 50 cycles/degree
 = 570 nm
RMS wavefront error = 0.2
0.7
0
ZernikePolynomialMTF.m
0.6
0.5
0.4
0.3
0.2
0.1
0
Z
1
  , 
m
n
20
30
0.8
0.8
0.7
0.7
0.6
0.6
0.5
0.5
0.4
0.4
0.3
0.3
0.2
0.2
4
50
10
20
30
40
0
50
0
10
20
30
40
1
1
1
0.9
0.9
0.8
0.8
0.8
0.7
0.7
0.7
0.6
0.6
0.6
0.5
0.5
0.5
0.4
0.4
0.4
0.3
0.3
0.3
0.2
0.2
0.1
0.1
0
10
20
MTFy
MTFx
0.1
0
0.9
0
3
40
1
0.9
0.1
2
30
40
0
50
50
0.1
10
20
30
40
0
50
0
10
20
30
40
1
1
1
1
0.9
0.9
0.9
0.9
0.8
0.8
0.8
0.8
0.7
0.7
0.7
0.7
0.6
0.6
0.6
0.6
0.5
0.5
0.5
0.5
0.4
0.4
0.4
0.4
0.3
0.3
0.3
0.3
0.2
0.2
0.2
0.1
0.1
0.1
0
0
0
0
10
20
30
40
50
0
10
20
30
40
50
50
Coma (m=-1,1),
Trefoil(m=-3,3)
0.2
0.1
0
10
20
30
40
0
50
0
10
20
30
40
1
1
1
1
0.9
0.9
0.9
0.9
0.9
0.8
0.8
0.8
0.8
0.8
0.7
0.7
0.7
0.7
0.7
0.6
0.6
0.6
0.6
0.6
0.5
0.5
0.5
0.5
0.5
0.4
0.4
0.4
0.4
0.4
0.3
0.3
0.3
0.3
0.3
0.2
0.2
0.2
0.2
0.1
0.1
0.1
0.1
0
10
20
30
40
0
50
0
10
20
30
40
0
50
0
10
20
30
40
0
50
50
Spherical Aberration
(m=0)
0.2
0.1
0
10
20
30
40
0
50
0
10
20
30
40
1
1
1
1
1
1
0.9
0.9
0.9
0.9
0.9
0.9
0.8
0.8
0.8
0.8
0.8
0.8
0.7
0.7
0.7
0.7
0.7
0.7
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
0.4
0.4
0.4
0.4
0.4
0.3
0.3
0.3
0.3
0.3
0.3
0.2
0.2
0.2
0.2
0.2
0.1
0.1
0.1
0.1
0.1
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
50
Secondary Coma
(m=-1,1)
0.2
0.1
0
10
20
30
40
0
50
0
10
20
30
40
1
1
1
1
1
1
1
0.9
0.9
0.9
0.9
0.9
0.9
0.9
0.8
0.8
0.8
0.8
0.8
0.8
0.8
0.7
0.7
0.7
0.7
0.7
0.7
0.7
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
0.4
0.4
0.4
0.4
0.4
0.4
0.3
0.3
0.3
0.3
0.3
0.3
0.3
0.2
0.2
0.2
0.2
0.2
0.2
0.1
0.1
0.1
0.1
0.1
0.1
0
0
10
20
30
40
50
0
0
10
20
30
40
50
0
0
10
20
30
40
50
0
Tilt
Astigmatism (m=-2,2),
Defocus(m=0)
0.2
0
1
0
6
10
0.9
0
5
0
1
Piston
0
10
20
30
40
50
0
0
10
20
30
40
50
0
50
Secondary Spherical
Aberration (m=0)
0.2
0.1
0
10
20
30
40
50
0
0
10
20
30
40
50
Patrick Y. Maeda 3/10/03
Double-Index Zernike Polynomial MTFs
Radial
Order, n
-6
-5
Azimuthal Frequency, m
-3 -2 -1
0
1 2
3
-4
4
5
Stanford University
Psych 221 / EE 362
Winter 2002-2003
Common Names7
6
1
0.9
0.8
Pupil Diameter = 7.3 mm
0 to 50 cycles/degree
 = 570 nm
RMS wavefront error = 0.2
0.7
0
ZernikePolynomialMTF.m
0.6
0.5
0.4
0.3
0.2
0.1
0
Z
1
  , 
m
n
20
30
0.9
0.8
0.8
0.7
0.7
0.6
0.6
0.5
0.5
0.4
0.4
0.3
0.3
0.2
0.2
4
50
10
20
30
40
0
50
0
10
20
30
40
1
1
1
0.9
0.9
0.8
0.8
0.8
0.7
0.7
0.7
0.6
0.6
0.6
0.5
0.5
0.5
0.4
0.4
0.4
0.3
0.3
0.3
0.2
0.2
0.1
0.1
0
10
20
MTFy
MTFx
0.1
0
0.9
0
3
40
0.9
0.1
2
30
40
0
50
50
0.1
10
20
30
40
0
50
0
10
20
30
40
1
1
1
1
0.9
0.9
0.9
0.9
0.8
0.8
0.8
0.8
0.7
0.7
0.7
0.7
0.6
0.6
0.6
0.6
0.5
0.5
0.5
0.5
0.4
0.4
0.4
0.4
0.3
0.3
0.3
0.3
0.2
0.2
0.2
0.1
0.1
0.1
0
0
0
0
10
20
30
40
50
0
10
20
30
40
50
50
Coma (m=-1,1),
Trefoil(m=-3,3)
0.2
0.1
0
10
20
30
40
0
50
0
10
20
30
40
1
1
1
1
0.9
0.9
0.9
0.9
0.9
0.8
0.8
0.8
0.8
0.8
0.7
0.7
0.7
0.7
0.7
0.6
0.6
0.6
0.6
0.6
0.5
0.5
0.5
0.5
0.5
0.4
0.4
0.4
0.4
0.4
0.3
0.3
0.3
0.3
0.3
0.2
0.2
0.2
0.2
0.1
0.1
0.1
0.1
0
10
20
30
40
0
50
0
10
20
30
40
0
50
0
10
20
30
40
0
50
50
Spherical Aberration
(m=0)
0.2
0.1
0
10
20
30
40
0
50
0
10
20
30
40
1
1
1
1
1
1
0.9
0.9
0.9
0.9
0.9
0.9
0.8
0.8
0.8
0.8
0.8
0.8
0.7
0.7
0.7
0.7
0.7
0.7
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
0.4
0.4
0.4
0.4
0.4
0.3
0.3
0.3
0.3
0.3
0.3
0.2
0.2
0.2
0.2
0.2
0.1
0.1
0.1
0.1
0.1
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
50
Secondary Coma
(m=-1,1)
0.2
0.1
0
10
20
30
40
0
50
0
10
20
30
40
1
1
1
1
1
1
1
0.9
0.9
0.9
0.9
0.9
0.9
0.9
0.8
0.8
0.8
0.8
0.8
0.8
0.8
0.7
0.7
0.7
0.7
0.7
0.7
0.7
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
0.4
0.4
0.4
0.4
0.4
0.4
0.3
0.3
0.3
0.3
0.3
0.3
0.3
0.2
0.2
0.2
0.2
0.2
0.2
0.1
0.1
0.1
0.1
0.1
0.1
0
0
10
20
30
40
50
0
0
10
20
30
40
50
0
0
10
20
30
40
50
Tilt
Astigmatism (m=-2,2),
Defocus(m=0)
0.2
0
1
0
6
10
1
0
5
0
1
Piston
0
0
10
20
30
40
50
0
0
10
20
30
40
50
0
50
Secondary Spherical
Aberration (m=0)
0.2
0.1
0
10
20
30
40
50
0
0
10
20
30
40
50
Patrick Y. Maeda 3/10/03
Stanford University
Psych 221 / EE 362
Winter 2002-2003
Simulation based on Human Eye Data
Mode j
Coefficient (m)
RMS Coefficient (m)
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
0
0
0
1.02
0
0.33
0.21
-0.26
0.03
-0.34
-0.12
0.05
0.19
-0.19
0.15
0
0
0
0.416413256
0
0.134721936
0.074246212
-0.091923882
0.010606602
-0.120208153
-0.037947332
0.015811388
0.084970583
-0.060083276
0.047434165
Total RMS Wavefront Error (m)
MTF of Zero Aberration System, 5.4mm pupil
1
0.8
0.8
0.6
0.6
0.4
0.4
0.2
0.2
0
0.484608089
MTF of Zero Aberration System, 5.4mm pupil
1
0
10
20
30
s x (cycle/deg)
40
50
0
0
10
20
30
s y (cycle/deg)
40
50
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
10
20
30
s x (cycle/deg)
40
50
0
0
10
20
30
s y (cycle/deg)
40
50
WaveAberrationMTF.m
WaveAberration.m
WaveAberrationPSF.m
Patrick Y. Maeda 3/10/03
Stanford University
Psych 221 / EE 362
Winter 2002-2003
Measurement Setup
y
Pupil
Iris
x
z
Incoming
Light Beam
Retina
Ideal
Real
Planar
Aberrated
Wavefront Wavefront
Patrick Y. Maeda 3/10/03
Shack-Hartmann Sensor Layout
PBS
Stanford University
Psych 221 / EE 362
Winter 2002-2003
Pupil Relay Optics
CCD
Lenslet
Array
Light
Source
Patrick Y. Maeda 3/10/03
Stanford University
Psych 221 / EE 362
Winter 2002-2003
Shack-Hartmann Wavefront Sensor
Lenslet Array
y(x1, y1)
Aberrated
Wavefront
y(x1, y2)
y(x1, y3)
y(x1, y4)
Focal Length f
Patrick Y. Maeda 3/10/03
Data Fitting with Zernike Polynomials
Stanford University
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
Z j ( x, y )
W ( x, y )
 W j
x
x
j
Z j ( x, y )
W ( x, y )
 W j
y
y
j
Z j ( x, y )
x( x, y )
 W j
f
x
j
(14)
Z j ( x, y )
y ( x, y )
 W j
f
y
j
(15)
Equations (14) and (15) can be used to determine the Wj’s using Least-squares Estimation
Patrick Y. Maeda 3/10/03
Stanford University
Psych 221 / EE 362
Winter 2002-2003
Least-squares Estimation
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 )
 b( x , y )   g ( x , y ) g ( x , y )
2
1
2
 1 2   1 1 2



 


 
b( x k , y k )   g1 ( x k , y k ) g 2 ( x k , y k )
 c( x1 , y1 )   h1 ( x1 , y1 ) h2 ( x1 , y1 )

 
c
(
x
,
y
)
 1 2   h1 ( x1 , y 2 ) h2 ( x1 , y 2 )

 




 
c( x k , y k )   h1 ( x k , y k ) h2 ( x k , y k )
 g j max ( x1 , y1 ) 
 g j max ( x1 , y 2 ) 
  W1 



 g j max ( x k , y k )  W2 

 h j max ( x1 , y1 )    
 

 h j max ( x1 , y 2 )  W j max 




 h j max ( x k , y k ) 
2k 2  j max
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
Benefits of Orthogonality
Stanford University
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
where 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
Conclusions
Stanford University
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
References
Stanford University
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
Appendix I
Stanford University
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