Abstract

Zernike polynomials and their associated coefficients are commonly used to quantify the wavefront aberrations of the eye. When the aberrations of different eyes, pupil sizes, or corrections are compared or averaged, it is important that the Zernike coefficients have been calculated for the correct size, position, orientation, and shape of the pupil. We present the first complete theory to transform Zernike coefficients analytically with regard to concentric scaling, translation of pupil center, and rotation. The transformations are described both for circular and elliptical pupils. The algorithm has been implemented in MATLAB, for which the code is given in an appendix.

© 2007 Optical Society of America

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References

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  1. D. A. Atchison, "Recent advances in measurement of monochromatic aberrations of human eyes," Clin. Exp. Optom. 88, 5-27 (2005).
    [CrossRef] [PubMed]
  2. American National Standards Institute, "Methods for reporting optical aberrations of eyes," ANSI Z80.28-2004 (ANSI, 2004).
  3. L. N. Thibos, R. A. Applegate, J. T. Schwiegerling, R. Webb, and VSIA Standards Taskforce Members, "Standards for reporting the optical aberrations of eyes," J. Refract. Surg. 18, 652-660 (2002).
  4. D. Malacara, Optical Shop Testing (Wiley, 1992).
  5. G.-m. Dai, "Scaling Zernike expansion coefficients to smaller pupil sizes: a simpler formula," J. Opt. Soc. Am. A 23, 539-543 (2006).
    [CrossRef]
  6. C. E. Campbell, "Matrix method to find a new set of Zernike coefficients from an original set when the aperture radius is changed," J. Opt. Soc. Am. A 20, 209-217 (2003).
    [CrossRef]
  7. J. Schwiegerling, "Scaling Zernike expansion coefficients to different pupil sizes," J. Opt. Soc. Am. A 19, 1937-1945 (2002).
    [CrossRef]
  8. K. A. Goldberg and K. Geary, "Wave-front measurement errors from restricted concentric subdomains," J. Opt. Soc. Am. A 18, 2146-2152 (2001).
    [CrossRef]
  9. E. Donnenfeld, "The pupil is a moving target: centration, repeatability, and registration," J. Refract. Surg. 20, 593-596 (2004).
  10. M. A. Wilson, M. C. W. Campbell, and P. Simonet, "Change of pupil centration with change of illumination and pupil size," Optom. Vision Sci. 69, 129-136 (1992).
    [CrossRef]
  11. G. Walsh, "The effect of mydriasis on the pupillary centration of the human eye," Ophthalmic Physiol. Opt. 8, 178-182 (1988).
    [CrossRef] [PubMed]
  12. A. Guirao, D. R. Williams, and I. G. Cox, "Effect of rotation and translation on the expected benefit of an ideal method to correct the eye's higher-order aberrations," J. Opt. Soc. Am. A 18, 1003-1015 (2001).
    [CrossRef]
  13. S. Bará, J. Arines, J. Ares, and P. Prado, "Direct transformation of Zernike eye aberration coefficients between scaled, rotated and/or displaced pupils," J. Opt. Soc. Am. A 23, 2061-2066 (2006).
    [CrossRef]
  14. D. A. Atchison and D. H. Scott, "Monochromatic aberrations of human eyes in the horizontal visual field," J. Opt. Soc. Am. A 19, 2180-2184 (2002).
    [CrossRef]

2006 (2)

2005 (1)

D. A. Atchison, "Recent advances in measurement of monochromatic aberrations of human eyes," Clin. Exp. Optom. 88, 5-27 (2005).
[CrossRef] [PubMed]

2004 (1)

E. Donnenfeld, "The pupil is a moving target: centration, repeatability, and registration," J. Refract. Surg. 20, 593-596 (2004).

2003 (1)

2002 (3)

J. Schwiegerling, "Scaling Zernike expansion coefficients to different pupil sizes," J. Opt. Soc. Am. A 19, 1937-1945 (2002).
[CrossRef]

L. N. Thibos, R. A. Applegate, J. T. Schwiegerling, R. Webb, and VSIA Standards Taskforce Members, "Standards for reporting the optical aberrations of eyes," J. Refract. Surg. 18, 652-660 (2002).

D. A. Atchison and D. H. Scott, "Monochromatic aberrations of human eyes in the horizontal visual field," J. Opt. Soc. Am. A 19, 2180-2184 (2002).
[CrossRef]

2001 (2)

1992 (1)

M. A. Wilson, M. C. W. Campbell, and P. Simonet, "Change of pupil centration with change of illumination and pupil size," Optom. Vision Sci. 69, 129-136 (1992).
[CrossRef]

1988 (1)

G. Walsh, "The effect of mydriasis on the pupillary centration of the human eye," Ophthalmic Physiol. Opt. 8, 178-182 (1988).
[CrossRef] [PubMed]

Applegate, R. A.

L. N. Thibos, R. A. Applegate, J. T. Schwiegerling, R. Webb, and VSIA Standards Taskforce Members, "Standards for reporting the optical aberrations of eyes," J. Refract. Surg. 18, 652-660 (2002).

Ares, J.

Arines, J.

Atchison, D. A.

D. A. Atchison, "Recent advances in measurement of monochromatic aberrations of human eyes," Clin. Exp. Optom. 88, 5-27 (2005).
[CrossRef] [PubMed]

D. A. Atchison and D. H. Scott, "Monochromatic aberrations of human eyes in the horizontal visual field," J. Opt. Soc. Am. A 19, 2180-2184 (2002).
[CrossRef]

Bará, S.

Campbell, C. E.

Campbell, M. C. W.

M. A. Wilson, M. C. W. Campbell, and P. Simonet, "Change of pupil centration with change of illumination and pupil size," Optom. Vision Sci. 69, 129-136 (1992).
[CrossRef]

Cox, I. G.

Dai, G.-m.

Donnenfeld, E.

E. Donnenfeld, "The pupil is a moving target: centration, repeatability, and registration," J. Refract. Surg. 20, 593-596 (2004).

Geary, K.

Goldberg, K. A.

Guirao, A.

Malacara, D.

D. Malacara, Optical Shop Testing (Wiley, 1992).

Prado, P.

Schwiegerling, J.

Schwiegerling, J. T.

L. N. Thibos, R. A. Applegate, J. T. Schwiegerling, R. Webb, and VSIA Standards Taskforce Members, "Standards for reporting the optical aberrations of eyes," J. Refract. Surg. 18, 652-660 (2002).

Scott, D. H.

Simonet, P.

M. A. Wilson, M. C. W. Campbell, and P. Simonet, "Change of pupil centration with change of illumination and pupil size," Optom. Vision Sci. 69, 129-136 (1992).
[CrossRef]

Thibos, L. N.

L. N. Thibos, R. A. Applegate, J. T. Schwiegerling, R. Webb, and VSIA Standards Taskforce Members, "Standards for reporting the optical aberrations of eyes," J. Refract. Surg. 18, 652-660 (2002).

Walsh, G.

G. Walsh, "The effect of mydriasis on the pupillary centration of the human eye," Ophthalmic Physiol. Opt. 8, 178-182 (1988).
[CrossRef] [PubMed]

Webb, R.

L. N. Thibos, R. A. Applegate, J. T. Schwiegerling, R. Webb, and VSIA Standards Taskforce Members, "Standards for reporting the optical aberrations of eyes," J. Refract. Surg. 18, 652-660 (2002).

Williams, D. R.

Wilson, M. A.

M. A. Wilson, M. C. W. Campbell, and P. Simonet, "Change of pupil centration with change of illumination and pupil size," Optom. Vision Sci. 69, 129-136 (1992).
[CrossRef]

Clin. Exp. Optom. (1)

D. A. Atchison, "Recent advances in measurement of monochromatic aberrations of human eyes," Clin. Exp. Optom. 88, 5-27 (2005).
[CrossRef] [PubMed]

J. Opt. Soc. Am. A (7)

J. Refract. Surg. (2)

E. Donnenfeld, "The pupil is a moving target: centration, repeatability, and registration," J. Refract. Surg. 20, 593-596 (2004).

L. N. Thibos, R. A. Applegate, J. T. Schwiegerling, R. Webb, and VSIA Standards Taskforce Members, "Standards for reporting the optical aberrations of eyes," J. Refract. Surg. 18, 652-660 (2002).

Ophthalmic Physiol. Opt. (1)

G. Walsh, "The effect of mydriasis on the pupillary centration of the human eye," Ophthalmic Physiol. Opt. 8, 178-182 (1988).
[CrossRef] [PubMed]

Optom. Vision Sci. (1)

M. A. Wilson, M. C. W. Campbell, and P. Simonet, "Change of pupil centration with change of illumination and pupil size," Optom. Vision Sci. 69, 129-136 (1992).
[CrossRef]

Other (2)

D. Malacara, Optical Shop Testing (Wiley, 1992).

American National Standards Institute, "Methods for reporting optical aberrations of eyes," ANSI Z80.28-2004 (ANSI, 2004).

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Figures (2)

Fig. 1
Fig. 1

The four possible coordinate transformations: (upper left) scaling from the original pupil size, r 0 , to the new radius, r s ; (upper right) scaling combined with translation by r t and θ t ; (lower left) rotation by the angle θ r ; (lower right) transformation to an ellipse, which is rotated by an angle θ e , with the major radius, r m a , equal to the original radius and a reduced minor radius, r m i . The angles are positive counter clockwise. Dotted lines and circles show the original coordinate axes (x and y) and wavefronts, solid lines are the new axes ( x and y ), and striped areas are the new wavefronts.

Fig. 2
Fig. 2

Coupling of Zernike coefficients when the wavefront is transformed. The square boxes of the pyramids represent real, standard Zernike coefficients with the radial order, n, increasing downward and the azimuthal frequency, m, going from negative to positive values from left to right. The crosses denote the single original Zernike coefficient before the transformation, and the filled circles denote the coefficients to which the original coefficient couple. The large, upper pyramid shows as an example how the coefficient with n = 10 and m = 6 is transformed when the wavefront is arbitrarily translated and scaled; the unfilled circles denote the additional coefficients if rotation also is included. The six small pyramids show the transformation of spherical aberration: Ta, arbitrary translation; Th, horizontal translation; Tv, vertical translation; S, concentric scaling; R, rotation; E, elliptical scaling.

Equations (41)

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W ( ρ , θ ) = n m c n m Z n m ( ρ , θ ) = Z c ,
ρ = p h y s i c a l r a d i a l c o o r d i n a t e m a x i m u m v a l u e o f r a d i a l c o o r d i n a t e = r r 0 ,
Z n m ( ρ , θ ) = N n R n m ( ρ ) M m ( θ ) ,
N n = n + 1 ,
R n m ( ρ ) = s = 0 n m 2 A n , s m ρ n 2 s ,
A n , s m = ( 1 ) s ( n s ) ! s ! [ 0.5 ( n + m ) s ] ! [ 0.5 ( n m ) s ] ! ,
M m ( θ ) = e i m θ ,
0 1 0 2 π Z n m ( ρ , θ ) Z j k * ( ρ , θ ) ρ d θ d ρ = π δ m k δ n j , δ a b = { 1 a = b 0 a b .
c n m = ( c n m i c n m ) 2 ,
c n 0 = c n 0 ,
c n m = ( c n m + i c n m ) 2 , m positive .
c n m = ( c n m + c n m ) 2 ,
c n 0 = c n 0 ,
c n m = i ( c n m c n m ) 2 , m positive .
Z = [ Z n m a x n m a x Z n m a x 1 ( n m a x 1 ) Z n m a x 2 ( n m a x 2 ) Z n m a x ( n m a x 2 ) Z n m a x n m a x ] .
Z = ρ M [ R ] [ N ] .
ρ M = [ ρ n m a x e i n m a x θ ρ n m a x 1 e i ( n m a x 1 ) θ ρ n m a x 2 e i ( n m a x 2 ) θ ρ n m a x e i ( n m a x 2 ) θ ρ n m a x e i n m a x θ ] .
[ R ] = [ A n m a x , 0 n m a x 0 0 0 0 0 A n m a x 1 , 0 ( n m a x 1 ) 0 0 0 0 0 A ( n m a x 2 ) , 0 ( n m a x 2 ) A n m a x , 1 ( n m a x 2 ) 0 0 0 0 A n m a x , 0 ( n m a x 2 ) 0 0 0 0 0 A n m a x , 0 n m a x ] ,
[ N ] = [ N n m a x 0 0 0 0 0 N n m a x 1 0 0 0 0 0 N ( n m a x 2 ) 0 0 0 0 0 N n m a x 0 0 0 0 0 N n m a x ] .
ρ M [ R ] [ N ] c = ρ M [ R ] [ N ] c .
ρ M = ρ M [ η ] .
c = [ C ] c .
[ η ] [ R ] [ N ] = [ R ] [ N ] [ C ] [ C ] = [ N ] 1 [ R ] 1 [ η ] [ R ] [ N ] .
ρ n e i m θ = ρ n m ( ρ e i θ ) m = ( ρ e i θ ρ e i θ ) ( n m ) 2 ( ρ e i θ ) m = ( ρ e i θ ) ( n + m ) 2 ( ρ e i θ ) ( n m ) 2 ,
ρ e i θ = η s ρ e i θ .
ρ n e i m θ = η s n ρ n e i m θ ,
ρ e i θ = η s ρ e i θ + η t e i θ t .
( a + b ) j = k = 0 j ( j k ) a j k b k
ρ n e i m θ = p = 0 ( n + m ) 2 q = 0 ( n m ) 2 ( n + m 2 p ) ( n m 2 q ) η s n p q η t p + q e i ( p q ) θ t ρ ( n p q ) e i ( m p + q ) θ .
ρ e i θ = ρ e i ( θ + θ r ) ,
ρ n e i m θ = e i m θ r ρ n e i m θ ,
ρ e i θ = e i θ e ( ρ cos ( θ θ e ) + i ρ sin ( θ θ e ) ) = e i θ e ( η e ρ cos ( θ θ e ) + i ρ sin ( θ θ e ) ) = η e + 1 2 ρ e i θ + η e 1 2 e i 2 θ e ρ e i θ .
ρ n e i m θ = 1 2 n p = 0 ( n + m ) 2 q = 0 ( n m ) 2 ( n + m 2 p ) ( n m 2 q ) ( η e + 1 ) n p q ( η e 1 ) p + q e i 2 ( p q ) θ e ρ n e i ( m 2 p + 2 q ) θ .
Z = [ Z 3 3 Z 2 2 Z 1 1 Z 3 1 Z 0 0 Z 2 0 Z 1 1 Z 3 1 Z 2 2 Z 3 3 ] = [ 2 ρ 3 e i 3 θ 3 ρ 2 e i 2 θ 2 ρ e i θ 2 ( 3 ρ 3 2 ρ ) e i θ 1 3 ( 2 ρ 2 1 ) 2 ρ e i θ 2 ( 3 ρ 3 2 ρ ) e i θ 3 ρ 2 e i 2 θ 2 ρ 3 e i 3 θ ] ,
ρ M = [ ρ 3 e i 3 θ ρ 2 e i 2 θ ρ e i θ ρ 3 e i θ 1 ρ 2 ρ e i θ ρ 3 e i θ ρ 2 e i 2 θ ρ 3 e i 3 θ ] ,
[ R ] = [ 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 2 0 0 0 0 0 0 0 0 0 3 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 2 0 0 0 0 0 0 0 0 0 0 1 2 0 0 0 0 0 0 0 0 0 3 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 ] ,
[ N ] = [ 2 0 0 0 0 0 0 0 0 0 0 3 0 0 0 0 0 0 0 0 0 0 2 0 0 0 0 0 0 0 0 0 0 2 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 3 0 0 0 0 0 0 0 0 0 0 2 0 0 0 0 0 0 0 0 0 0 2 0 0 0 0 0 0 0 0 0 0 3 0 0 0 0 0 0 0 0 0 0 2 ] ,
c = [ c 3 3 c 2 2 c 1 1 c 3 1 c 0 0 c 2 0 c 1 1 c 3 1 c 2 2 c 3 3 ] = [ ( c 3 3 + i c 3 3 ) 2 ( c 2 2 + i c 2 2 ) 2 ( c 1 1 + i c 1 1 ) 2 ( c 3 1 + i c 3 1 ) 2 c 0 0 c 2 0 ( c 1 1 i c 1 1 ) 2 ( c 3 1 i c 3 1 ) 2 ( c 2 2 i c 2 2 ) 2 ( c 3 3 i c 3 3 ) 2 ] ,
[ η ] s c a l e = [ η s 3 0 0 0 0 0 0 0 0 0 0 η s 2 0 0 0 0 0 0 0 0 0 0 η s 0 0 0 0 0 0 0 0 0 0 η s 3 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 η s 2 0 0 0 0 0 0 0 0 0 0 η s 0 0 0 0 0 0 0 0 0 0 η s 3 0 0 0 0 0 0 0 0 0 0 η s 2 0 0 0 0 0 0 0 0 0 0 η s 3 ] ,
[ η ] t r a n s l a t e = [ η s 3 0 0 0 0 0 0 0 0 0 3 η s 2 η t e i θ t η s 2 0 η s 2 η t e i θ t 0 0 0 0 0 0 3 η s η t 2 e i 2 θ t 2 η s η t e i θ t η s 2 η s η t 2 0 η s η t e i θ t 0 η s η t 2 e i 2 θ t 0 0 0 0 0 η s 3 0 0 0 0 0 0 η t 3 e 3 i θ t η t 2 e 2 i θ t η t e i θ t η t 3 e i θ t 1 η t 2 η t e i θ t η t 3 e i θ t η t 2 e i 2 θ t η t 3 e i 3 θ t 0 0 0 2 η s 2 η t e i θ t 0 η s 2 0 2 η s 2 η t e i θ t 0 0 0 0 0 η s η t 2 e i 2 θ t 0 η s η t e i θ t η s 2 η s η t 2 2 η s η t e i θ t 3 η s η t 2 e i 2 θ t 0 0 0 0 0 0 0 η s 3 0 0 0 0 0 0 0 0 0 η s 2 η t e i θ t η s 2 3 η s 2 η t e i θ t 0 0 0 0 0 0 0 0 0 η s 3 ] ,
[ η ] r o t a t e = [ e i 3 θ r 0 0 0 0 0 0 0 0 0 0 e i 2 θ r 0 0 0 0 0 0 0 0 0 0 e i θ r 0 0 0 0 0 0 0 0 0 0 e i θ r 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 e i θ r 0 0 0 0 0 0 0 0 0 0 e i θ r 0 0 0 0 0 0 0 0 0 0 e i 2 θ r 0 0 0 0 0 0 0 0 0 0 e i 3 θ r ] .

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