zernike polynomials - university of arizona...also, air force or university of arizona other single...

1

Zernike Polynomials

• Fitting irregular and non-rotationally symmetric surfaces over a circular region.

• Atmospheric Turbulence.• Corneal Topography• Interferometer measurements.• Ocular Aberrometry

Background• The mathematical functions

were originally described by Frits Zernike in 1934.

• They were developed to describe the diffracted wavefront in phase contrast imaging.

• Zernike won the 1953 Nobel Prize in Physics for developing Phase Contrast Microscopy.

2

Phase Contrast Microscopy

Transparent specimens leave the amplitude of the illuminationvirtually unchanged, but introduces a change in phase.

Applications• Typically used to fit a wavefront or

surface sag over a circular aperture.

• Astronomy - fitting the wavefront entering a telescope that has been distorted by atmospheric turbulence.

• Diffraction Theory - fitting the wavefront in the exit pupil of a system and using Fourier transform properties to determine the Point Spread Function.

Source:http://salzgeber.at/astro/moon/seeing.html

3

Applications

• Ophthalmic Optics - fitting corneal topography and ocular wavefront data.

• Optical Testing - fitting reflected and transmitted wavefront data measured interferometically.

Surface Fitting

• Reoccurring Theme: Fitting a complex, non-rotationally symmetric surfaces (phase fronts) over a circular domain.

• Possible goals of fitting a surface:– Exact fit to measured data points?– Minimize “Error” between fit and data points?– Extract Features from the data?

4

1D Curve Fitting

0

5

10

15

20

25

-0.1 0.1 0.3 0.5 0.7 0.9 1.1 1.3 1.5

Low-order Polynomial Fity = 9.9146x + 2.3839

R2 = 0.9383

0

5

10

15

20

25

-0.1 0.1 0.3 0.5 0.7 0.9 1.1 1.3 1.5

In this case, the error is the vertical distance between the line andthe data point. The sum of the squares of the error is minimized.

5

High-order Polynomial Fit

0

5

10

15

20

25

-0.1 0.1 0.3 0.5 0.7 0.9 1.1 1.3 1.5-400

-200

0

200

400

600

800

1000

1200

1400

1600

1800

-0.1 0.1 0.3 0.5 0.7 0.9 1.1 1.3 1.5

y = a0 + a1x + a2x2 + … a16x16

Cubic Splines

0

5

10

15

20

25

-0.1 0.1 0.3 0.5 0.7 0.9 1.1 1.3 1.5

Piecewise definition of the function.

6

Fitting Issues• Know your data. Too many terms in the fit can be numerically

unstable and/or fit noise in the data. Too few terms may miss real trends in the surface.

• Typically want “nice” properties for the fitting function such as smooth surfaces with continuous derivatives. For example, cubic splines have continuous first and second derivatives.

• Typically want to represent many data points with just a few terms of a fit. This gives compression of the data, but leaves some residual error. For example, the line fit represents 16 data points with two numbers: a slope and an intercept.

Why Zernikes?

• Zernike polynomials have nice mathematical properties.– They are orthogonal over the continuous unit circle.– All their derivatives are continuous.– They efficiently represent common errors (e.g. coma,

spherical aberration) seen in optics.– They form a complete set, meaning that they can

represent arbitrarily complex continuous surfaces given enough terms.

7

Orthogonal Functions

• Orthogonal functions are sets of surfaces which have some nice mathematical properties for surface fitting.

• These functions satisfy the property

∫∫⎩⎨⎧ =

=A

jji Otherwise

ji

0C

dxdy)y,x(V)y,x(V

where Cj is a constant for a given j

Orthogonality and Expansion Coefficients

∑=i

ii )y,x(Va)y,x(W

)y,x(V)y,x(Va)y,x(V)y,x(W ji

iij ∑=

dxdy)y,x(V)y,x(Vadxdy)y,x(V)y,x(Wi

jA

iiA

j ∑ ∫∫∫∫ =

dxdy)y,x(V)y,x(WC1a

Aj

jj ∫∫=

Linear Expansion

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Orthogonality - 1D Example

( ) ( )∫π2

0

integers are m' and m wheredxx'mcosmxsin

( ) ( )[ ]∫π

−++=2

0

dxx'mmsinx'mmsin21

Consider the integral

( ) ( ) π=

=⎥⎦⎤

⎢⎣⎡

−−

+++

−=2x

0x'mmx'mmcos

'mmx'mmcos

21

0=


( ) ( )∫π2

0

integers are m' and m wheredxx'msinmxsin

( ) ( )[ ]∫π

−−+−=2

0

dxx'mmcosx'mmcos21

Two sine terms

( ) ( ) π=

=⎥⎦⎤

⎢⎣⎡

−−

−++

−=2x

0x'mmx'mmsin

'mmx'mmsin

21

0= if m ≠ m’ !!!

9


( ) ( )∫π2

0

integers are m' and m wheredxx'msinmxsin

( )∫π

=2

0

2 dxmxsin

Two sine terms with m = m’

( ) π=

=⎥⎦⎤

⎢⎣⎡ −=

2x

0xm4mx2sin

2x

π= Similar arguments for two cosine terms


odd jeven j

x

21jsin

x2jcos

Vj

⎪⎪⎩

⎪⎪⎨

⎧

⎟⎠⎞

⎜⎝⎛ +

⎟⎠⎞

⎜⎝⎛

=

From the previous arguments, we can define

( ) ( ) π== ∫∫ππ

2dxdxxVxV2

0

2

000

Note that when j = 0, V0 = 1 and

so the constant Cj = 2π for j = 0, and Cj = π for all other values of j.

10

Orthogonality - 1D ExampleThe functions satisfy the orthogonality condition over [0, 2π]

( ) ( ) ( )Otherwise

ji

01

dxxVxV 0i2

0ji

=

⎩⎨⎧ πδ+

=∫π

where δ is the Kronecker delta function defined as

Otherwiseji

01

ij

=

⎩⎨⎧

=δ

Fit to W(x) = x

010-0.49

08-0.57

06-0.66665

04-1302-21π0

ajj

-1

0

1

2

3

4

5

6

7

0 1 2 3 4 5 6 7

W(x)1 term3 terms11 terms

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Extension to Two Dimensions

In many cases, wavefronts take on a complex shape defined over a circular region and we wish to fit this surface to a series of simpler components.

Wavefront Fitting

=

-0.003 x

+ 0.002 x

+ 0.001 x

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Unit Circle

x

y

ρ

θ1

Divide the realradial coordinateby the maximum radiusto get a normalizedcoordinate ρ

Orthogonal Functions on the Unit Circle

• Taylor polynomials (i.e. 1, x, y, x2, xy, y2,….) are not orthogonal on the unit circle.

⎩⎨⎧ =

=θρρθρθρ∫ ∫π

Otherwiseji

0

Cdd),(V),(V j

j

2

0

1

0i

where Cj is a constant for a given j

Many solutions, but let’s try something with the form

( ) ( )θΘρ=θρ iii R),(V

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Orthogonal Functions on the Unit Circle

• The orthogonality condition separates into the product of two 1D integrals

⎩⎨⎧ =

=⎥⎦

⎤⎢⎣

⎡θθΘθΘ⎥

⎦

⎤⎢⎣

⎡ρρρρ ∫∫

π

Otherwiseji

0

Cd)( )(d)(R)(R j

j

2

0ij

1

0i

This looks like the 1D Example,so sin(mθ) and cos(mθ) a possibility

This has extra ρ in it, so we need different functions

ANSI Standard Zernikes

⎪⎩

⎪⎨⎧

<θρ−≥θρ

=θρ 0 mfor ; msin)(RN

0mfor ; mcos)(RN),(Z mn

mn

mn

mnm

n

Double Indexn is radial orderm is azimuthal frequency

Normalization

RadialComponent

AzimuthalComponent

ANSI Z80.28-2004 Methods for Reporting Optical Aberrations of Eyes.

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only dependson |m| (i.e. samefor both sine &cosine terms

Powers of ρ

[ ] [ ]s2n

2/)mn(

0s

smn ! s)mn(5.0! s)mn(5.0!s

)!sn()1()(R −−

=

ρ−−−+

−−=ρ ∑

Constant that dependson n and m

2n2d)(R)(R 'nnm

'n

1

0

mn +

δ=ρρρρ∫

The Radial polynomials satisfy the orthogonality equation


constant thatdepends on n & m

0m

mn 1

2n2Nδ++

=

15

Orthogonality

⎩⎨⎧ ==

=θρρθρθρ∫∫π

Otherwisem'm;n'n for

0

Cdd),(Z),(Z mn,m

n

2

0

1

0

'm'n

( )( )

( )( )

⎩⎨⎧ ==

=ρρρρ×

θ⎭⎬⎫

⎩⎨⎧

θ−θ

⎭⎬⎫

⎩⎨⎧

θ−θ

∫

∫π

Otherwisem'm;n'n for

0

Cd)(R)(R

dmsin

mcos'msin

'mcosNN

mn,mn

1

0

'm'n

2

0

mn

'm'n

Orthogonality

( )[ ]⎩⎨⎧ ==

=⎥⎦⎤

⎢⎣⎡

+δ

πδ+δ Otherwise

m'm;n'n for

0C

2n2 1NN mn,n'n

0mm'mmn

'm'n

This equals zerounless m=m’ This equals zero

unless n=n’So this portion issatisfied

What happens when n=n’ and m=m’?

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Orthogonality

( ) ( )[ ] mn,0m2m

n C2n2

1 1 N =⎥⎦⎤

⎢⎣⎡

+πδ+

When n=n’ and m=m’

( )[ ] m,n0mm0

C2n2

1 1 1

22n=⎥⎦

⎤⎢⎣⎡

+πδ+⎥

⎦

⎤⎢⎣

⎡δ++

π=m,nC

Orthogonality

m'mn'nmn

2

0

1

0

'm'n dd),(Z),(Z δπδ=θρρθρθρ∫∫

π

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The First Few Zernike Polynomials( )( )( )( ) ( )( ) ( )( ) ( )θρ=θρ

−ρ=θρ

θρ=θρ

θρ=θρ

θρ=θρ

=θρ

−

−

2cos6,Z

123,Z

2sin6,Z

cos,Z

sin,Z

1,Z

222

202

222

11

11

00

Zernike Polynomials

Azimuthal Frequency, θ

Rad

ialP

olyn

omia

l, ρ

Z00

Z11Z1

1−

Z20

Z31− Z3

1

Z40 Z4

2

Z22

Z42−

Z33− Z3

3

Z44Z4

4−

Z22−

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Caveats to the Definition ofZernike Polynomials

• At least six different schemes exist for the Zernike polynomials.• Some schemes only use a single index number instead of n and m.

With the single number, there is no unique ordering or definition for the polynomials, so different orderings are used.

• Some schemes set the normalization to unity for all polynomials.• Some schemes measure the polar angle in the clockwise direction from

the y axis.• The expansion coefficients depend on pupil size, so the maximum

radius used must be given.• Some groups fit OPD, other groups fit Wavefront Error.• Make sure which set is being given for a specific application.

Another Coordinate System

x

y

ρφ

Normalized Polar Coordinates:

⎟⎟⎠

⎞⎜⎜⎝

⎛=φ

=ρ

−

yxtan

rr

1

max

1

ρ ranges from [0, 1]φ ranges from [-180°, 180°]

NON-STANDARD

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Zernike Polynomials - Single Index

Azimuthal Frequency, θ

Rad

ialP

olyn

omia

l, ρ

Z0

Z1

Z4 Z5Z3

Z9Z8Z7Z6

Z10 Z11 Z12 Z13 Z14

Z2

ANSI STANDARD

Starts at 0Left-to-RightTop-to-Bottom

Other Single Index Schemes

Z1

Z3

Z4 Z6Z5

Z10Z8Z7 Z9

Z15Z13Z11 Z12 Z14

Z2

NON-STANDARD

Starts at 1cosines are even termssines are odd terms

Noll, RJ. Zernike polynomials and atmospheric turbulence. J Opt Soc Am 66; 207-211 (1976).

Also Zemax “Standard Zernike Coefficients”

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Z1

Z3

Z4 Z5 Z6

Z10Z7 Z8 Z11

Z18Z13Z9 Z12 Z17

Z2

NON-STANDARD

Starts at 1increases along diagonalcosine terms first35 terms plus two extraspherical aberration termsNo Normalization!!!

Zemax “Zernike Fringe Coefficients”

Also, Air Force or University of Arizona


• Born & Wolf• Malacara• Others??? Plus mixtures of non-normalized, coordinate

systems.

NON-STANDARD

Use two indices n, m to unambiguously define polynomials.Use a single standard index if needed to avoid confusion.

21

ExamplesExample 1:

4m

0.25 D of myopia for a 4 mm pupil (rmax = 2 mm)

( ) ( ) ( )θρ+θρ=ρ

=ρ

== ,Z34000

1,Z4000

120008000

28000

rW 02

00

222

4mm

ExamplesExample 2:

1m

1.00 D of myopia for a 2 mm pupil (rmax = 1 mm)

( ) ( )θρ+θρ=ρ

== ,Z34000

1,Z4000

120002000

rW 02

00

22

2mm

Same Zernike Expansion as Example 1, but different rmax.

Always need to give pupil size with Zernike coefficients!!

22

RMS Wavefront Error

• RMS Wavefront Error is defined as

( )∑∫∫

∫∫>

=φρρ

φρρφρ=

m all,1n

2m,n

2

WFE add

dd)),(W(RMS

Zeroth Order Zernike Polynomials

Z00

This term is called Piston and is usually ignored.The surface is constant over the entire circle, sono error or variance exists.

23

First Order Zernike Polynomials

Z11Z1

1−

These terms represent a tilt in the wavefront.

max11

max11

1111

1111

1111

rxa

rya

cosasina),(Za),(Za

+=

θρ+θρ=θρ+θρ

−

−

−−Combining these terms results in a

general equation for a plane, thusby changing the coefficients, a planeat any orientation can be created.This rotation of the pattern is truefor the sine/cosine pairs of Zernikes

Second Order Zernike Polynomials

Z20 Z2

2Z22−

These wavefronts are what you would expect from Jackson crossedcylinder J0 and J45 and a spherical lens. Thus, combining these termsgives any arbitrary spherocylindrical refractive error.

24

Third Order Zernike Polynomials

Z31− Z3

1Z33− Z3

3

The inner two terms are coma and the outer two terms are trefoil. These terms represent asymmetric aberrations that cannot be corrected with convention spectacles or contact lenses.

Fourth Order Zernike Polynomials

Z40 Z4

2Z42− Z4

4Z44−

SphericalAberration

4th orderAstigmatism

4th orderAstigmatismQuadroil Quadroil

These terms represent more complex shapes of the wavefront.Spherical aberration can be corrected by aspheric lenses.

25

Discrete data

• Up to this point, the data has been continuous, so we can mathematically integrate functions to get expansion coefficients.

• Real-world data is sampled at discrete points.• The Zernike polynomials are not orthogonal for discrete

points, but for high sampling densities they are almost orthogonal.

Speed• The long part of calculating Zernike polynomials is

calculating factorial functions.

⎪⎩

⎪⎨⎧

<θρ−≥θρ

=θρ 0 mfor ; msin)(RN

0mfor ; mcos)(RN),(Z mn

mn

mn

mnm

n

[ ] [ ]s2n

2/)mn(

0s

smn ! s)mn(5.0! s)mn(5.0!s

)!sn()1()(R −−

=

ρ−−−+

−−=ρ ∑

26

Speed• Chong et al.* developed a recurrence relationship that

avoids the need for calculating the factorials.• The results give a blazing fast algorithm for calculating

Zernike expansion coefficents using orthogonality.

*Pattern Recognition, 36;731-742 (2003).

Least Squares Fit

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛

=

⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛−

−

−

−

)y,x(f

)y,x(f)y,x(f

a

aaa

)y,x(Z)y,x(Z)y,x(Z)y,x(Z

)y,x(Z)y,x(Z)y,x(Z)y,x(Z)y,x(Z)y,x(Z)y,x(Z)y,x(Z

NN

22

11

nm

11

11

00

NNmnNN

11NN

11NN

00

22mn22

1122

1122

00

11mn11

1111

1111

00

MM

L

MLMMM

L

L

( ) FZZZA

FZZAZFZA

T1T

TT

−=

=

=

27

Gram-Schmidt Orthogonalization• Examines set of discrete data and creates a series of

functions which are orthogonal over the data set.• Orthogonality is used to calculate expansion coefficients.• These surfaces can then be converted to a standard set of

surfaces such as Zernike polynomials.Advantages• Numerically stable, especially for low sampling density.Disadvantages• Can be slow for high-order fits• Orthogonal functions depend upon data set, so a new set

needs to be calculated for every fit.

Elevation Fit Comparison

0.125 SecondsChong Algorithm

10 SecondsGram-Schmidt

32 Orders or 560 total polynomials

28

Shack-Hartmann Wavefront Sensor

Plane WavefrontsEye

Lenslet Array

Aberrated WavefrontsEye

Lenslet Array

Perfect wavefronts give a uniform grid of points, whereas aberratedwavefronts distort the grid pattern.

Least Squares Fit

( ) FZZZA

FZZAZFZA

T1T

TT

−=

=

= Again, conceptually easy to understand, although this can be relatively slow for high order fits.

⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

=

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

=

dy/)y,x(dW

dy/)y,x(dWdy/)y,x(dWdx/)y,x(dW

dx/)y,x(dWdx/)y,x(dW

a

aa

dy/)y,x(dVdy/)y,x(dVdy/)y,x(dV

dy/)y,x(dVdy/)y,x(dVdy/)y,x(dVdy/)y,x(dVdy/)y,x(dVdy/)y,x(dVdx/)y,x(dVdx/)y,x(dVdx/)y,x(dV

dx/)y,x(dVdx/)y,x(dVdx/)y,x(dVdx/)y,x(dVdx/)y,x(dVdx/)y,x(dV

Z

NN

22

11

NN

22

11

J

2

1

NNJNN2NN1

22J222221

11J112111

NNJNN2NN1

22J222221

11J112111

M

M

M

L

MMMM

L

L

L

MMMM

L

L

29

Example Image

Wavefront Reconstruction

PSF

zernike polynomials - university of arizona...also, air force or university of arizona other single...

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