Abstract

Zernike polynomials have been widely used to describe the aberrations in wavefront sensing of the eye. The Zernike coefficients are often computed under different aperture sizes. For the sake of comparison, the same aperture diameter is required. Since no standard aperture size is available for reporting the results, it is important to develop a technique for converting the Zernike coefficients obtained from one aperture size to another size. By investigating the properties of Zernike polynomials, we propose a general method for establishing the relationship between two sets of Zernike coefficients computed with different aperture sizes.

© 2006 Optical Society of America

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References

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    [CrossRef]
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2004

2003

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]

C. W. Chong, P. Raveendran, and R. Mukundan, "A comparative analysis of algorithms for fast computation of Zernike moments," Pattern Recogn. 36, 731-742 (2003).
[CrossRef]

2002

J. Gu, H. Z. Shu, C. Toumoulin, and L. M. Luo, "A novel algorithm for fast computation of Zernike moments," Pattern Recogn. 35, 2905-2911 (2002).
[CrossRef]

D. R. Iskander, M. R. Morelande, M. J. Collins, and B. Davis, "Modeling of corneal surfaces with radial polynomials," IEEE Trans. Biomed. Eng. 49, 320-328 (2002).
[CrossRef] [PubMed]

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

2001

D. R. Iskander, M. J. Collins, and B. Davis, "Optimal modeling of corneal surfaces with Zernike polynomials," IEEE Trans. Biomed. Eng. 48, 85-97 (2001).
[CrossRef]

1998

1996

S. O. Belkasim, M. Ahmadi, and M. Shridhar, "Efficient algorithm for fast computation of Zernike moments," J. Franklin Inst. 333, 577-581 (1996).
[CrossRef]

R. R. Bailey and M. Srinath, "Orthogonal moment features for use with parametric and non-parametric classifiers," IEEE Trans. Pattern Anal. Mach. Intell. 18, 389-400 (1996).
[CrossRef]

1995

1994

1980

Ahmadi, M.

S. O. Belkasim, M. Ahmadi, and M. Shridhar, "Efficient algorithm for fast computation of Zernike moments," J. Franklin Inst. 333, 577-581 (1996).
[CrossRef]

Bailey, R. R.

R. R. Bailey and M. Srinath, "Orthogonal moment features for use with parametric and non-parametric classifiers," IEEE Trans. Pattern Anal. Mach. Intell. 18, 389-400 (1996).
[CrossRef]

Belkasim, S. O.

S. O. Belkasim, M. Ahmadi, and M. Shridhar, "Efficient algorithm for fast computation of Zernike moments," J. Franklin Inst. 333, 577-581 (1996).
[CrossRef]

Bille, J. F.

Burns, S. A.

Campbell, C. E.

Carroll, J. P.

J. P. Carroll, "A method to describe corneal topography," Optom. Vision Sci. 71, 259-264 (1994).
[CrossRef]

Chong, C. W.

C. W. Chong, P. Raveendran, and R. Mukundan, "A comparative analysis of algorithms for fast computation of Zernike moments," Pattern Recogn. 36, 731-742 (2003).
[CrossRef]

Collins, M. J.

D. R. Iskander, M. R. Morelande, M. J. Collins, and B. Davis, "Modeling of corneal surfaces with radial polynomials," IEEE Trans. Biomed. Eng. 49, 320-328 (2002).
[CrossRef] [PubMed]

D. R. Iskander, M. J. Collins, and B. Davis, "Optimal modeling of corneal surfaces with Zernike polynomials," IEEE Trans. Biomed. Eng. 48, 85-97 (2001).
[CrossRef]

Coppens, J.

Davis, B.

D. R. Iskander, M. R. Morelande, M. J. Collins, and B. Davis, "Modeling of corneal surfaces with radial polynomials," IEEE Trans. Biomed. Eng. 49, 320-328 (2002).
[CrossRef] [PubMed]

D. R. Iskander, M. J. Collins, and B. Davis, "Optimal modeling of corneal surfaces with Zernike polynomials," IEEE Trans. Biomed. Eng. 48, 85-97 (2001).
[CrossRef]

Goelz, S.

Greivenkamp, J. E.

Grimm, W.

Gu, J.

J. Gu, H. Z. Shu, C. Toumoulin, and L. M. Luo, "A novel algorithm for fast computation of Zernike moments," Pattern Recogn. 35, 2905-2911 (2002).
[CrossRef]

He, J. C.

Iskander, D. R.

D. R. Iskander, M. R. Morelande, M. J. Collins, and B. Davis, "Modeling of corneal surfaces with radial polynomials," IEEE Trans. Biomed. Eng. 49, 320-328 (2002).
[CrossRef] [PubMed]

D. R. Iskander, M. J. Collins, and B. Davis, "Optimal modeling of corneal surfaces with Zernike polynomials," IEEE Trans. Biomed. Eng. 48, 85-97 (2001).
[CrossRef]

Liang, J.

Luo, L. M.

J. Gu, H. Z. Shu, C. Toumoulin, and L. M. Luo, "A novel algorithm for fast computation of Zernike moments," Pattern Recogn. 35, 2905-2911 (2002).
[CrossRef]

Marcos, S.

Miller, J. K.

Morelande, M. R.

D. R. Iskander, M. R. Morelande, M. J. Collins, and B. Davis, "Modeling of corneal surfaces with radial polynomials," IEEE Trans. Biomed. Eng. 49, 320-328 (2002).
[CrossRef] [PubMed]

Mukundan, R.

C. W. Chong, P. Raveendran, and R. Mukundan, "A comparative analysis of algorithms for fast computation of Zernike moments," Pattern Recogn. 36, 731-742 (2003).
[CrossRef]

Petkovsek, M.

M. Petkovsek, H. S. Wilf, and D. Zeilberger, A=B (AK Peters, 1996) (available on line at the University of Pennsylvania).

Raveendran, P.

C. W. Chong, P. Raveendran, and R. Mukundan, "A comparative analysis of algorithms for fast computation of Zernike moments," Pattern Recogn. 36, 731-742 (2003).
[CrossRef]

Schwiegerling, J.

Shridhar, M.

S. O. Belkasim, M. Ahmadi, and M. Shridhar, "Efficient algorithm for fast computation of Zernike moments," J. Franklin Inst. 333, 577-581 (1996).
[CrossRef]

Shu, H. Z.

J. Gu, H. Z. Shu, C. Toumoulin, and L. M. Luo, "A novel algorithm for fast computation of Zernike moments," Pattern Recogn. 35, 2905-2911 (2002).
[CrossRef]

Sicam, V. A.

Silva, D. E.

Srinath, M.

R. R. Bailey and M. Srinath, "Orthogonal moment features for use with parametric and non-parametric classifiers," IEEE Trans. Pattern Anal. Mach. Intell. 18, 389-400 (1996).
[CrossRef]

Teague, M. R.

Toumoulin, C.

J. Gu, H. Z. Shu, C. Toumoulin, and L. M. Luo, "A novel algorithm for fast computation of Zernike moments," Pattern Recogn. 35, 2905-2911 (2002).
[CrossRef]

van den Berg, T. P.

van der Heijde, R. L.

Wang, J. Y.

Webb, R. H.

Wilf, H. S.

M. Petkovsek, H. S. Wilf, and D. Zeilberger, A=B (AK Peters, 1996) (available on line at the University of Pennsylvania).

Zeilberger, D.

M. Petkovsek, H. S. Wilf, and D. Zeilberger, A=B (AK Peters, 1996) (available on line at the University of Pennsylvania).

Appl. Opt.

IEEE Trans. Biomed. Eng.

D. R. Iskander, M. J. Collins, and B. Davis, "Optimal modeling of corneal surfaces with Zernike polynomials," IEEE Trans. Biomed. Eng. 48, 85-97 (2001).
[CrossRef]

D. R. Iskander, M. R. Morelande, M. J. Collins, and B. Davis, "Modeling of corneal surfaces with radial polynomials," IEEE Trans. Biomed. Eng. 49, 320-328 (2002).
[CrossRef] [PubMed]

IEEE Trans. Pattern Anal. Mach. Intell.

R. R. Bailey and M. Srinath, "Orthogonal moment features for use with parametric and non-parametric classifiers," IEEE Trans. Pattern Anal. Mach. Intell. 18, 389-400 (1996).
[CrossRef]

J. Franklin Inst.

S. O. Belkasim, M. Ahmadi, and M. Shridhar, "Efficient algorithm for fast computation of Zernike moments," J. Franklin Inst. 333, 577-581 (1996).
[CrossRef]

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

Optom. Vision Sci.

J. P. Carroll, "A method to describe corneal topography," Optom. Vision Sci. 71, 259-264 (1994).
[CrossRef]

Pattern Recogn.

J. Gu, H. Z. Shu, C. Toumoulin, and L. M. Luo, "A novel algorithm for fast computation of Zernike moments," Pattern Recogn. 35, 2905-2911 (2002).
[CrossRef]

C. W. Chong, P. Raveendran, and R. Mukundan, "A comparative analysis of algorithms for fast computation of Zernike moments," Pattern Recogn. 36, 731-742 (2003).
[CrossRef]

Other

M. Petkovsek, H. S. Wilf, and D. Zeilberger, A=B (AK Peters, 1996) (available on line at the University of Pennsylvania).

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Tables (2)

Tables Icon

Table 1 Coefficient Conversion Relationships for Zernike Polynomial Expansions up to Order 8

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Table 2 Coefficient Conversion Relationships for Different Values of m and K Where N = m + 2 K = 7

Equations (48)

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Z n m ( ρ , θ ) = { N n m R n m ( ρ ) cos ( m θ ) for m 0 N n m R n m ( ρ ) sin ( m θ ) for m < 0 } , m n , n m even ,
R n m ( ρ ) = s = 0 ( n m ) 2 ( 1 ) s ( n s ) ! s ! [ ( n + m ) 2 s ] ! [ ( n m ) 2 s ] ! ρ n 2 s ,
N n m = 2 ( n + 1 ) 1 + δ m , 0 .
R m + 2 k m ( ρ ) = s = 0 k ( 1 ) s ( m + 2 k s ) ! s ! ( k s ) ! ( m + k s ) ! ρ m + 2 k 2 s = s = k 0 ( 1 ) k s ( m + k + s ) ! s ! ( k s ) ! ( m + s ) ! ρ m + 2 s ( making the change of variables = k s ) = s = 0 k c k , s m ρ m + 2 s ,
C k , s m = ( 1 ) k s ( m + k + s ) ! s ! ( k s ) ! ( m + s ) ! .
W ( r , θ ) = n = 0 N m a n , m Z n m ( r r max , θ ) ,
a n , m = 0 r max 0 2 π Z n m ( r r max , θ ) W ( r , θ ) r d r d θ .
W ( r , θ ) = n = 0 N m a n , m Z n m ( r , θ ) ,
W ( r , θ ) = n = 0 N m b n , m Z n m ( λ r , θ ) ,
W m ( r , θ ) = { ( k = 0 K a m + 2 k , m N m + 2 k m R m + 2 k m ( r ) ) cos ( m θ ) , for m 0 ( k = 0 K a m + 2 k , m N m + 2 k m R m + 2 k m ( r ) ) sin ( m θ ) , for m < 0 ) ,
K = { ( N m ) 2 , if N and m have the same parity ( N 1 m ) 2 , otherwise } .
W m ( r , θ ) = { ( k = 0 K b m + 2 k , m N m + 2 k m R m + 2 k m ( λ r ) ) cos ( m θ ) , for m 0 ( k = 0 K b m + 2 k , m N m + 2 k m R m + 2 k m ( λ r ) ) sin ( m θ ) , for m < 0 . )
k = 0 K b m + 2 k , m N m + 2 k m R m + 2 k m ( λ r ) = k = 0 K a m + 2 k , m N m + 2 k m R m + 2 k m ( r ) .
R ¯ m + 2 k m ( r ) = N m + 2 k m R m + 2 k m ( r ) .
k = 0 K b m + 2 k , m R ¯ m + 2 k m ( λ r ) = k = 0 K a m + 2 k , m R ¯ m + 2 k m ( r ) .
f ( r ) = n = 0 K a n P n ( r ) = n = 0 K b n P n ( λ r ) ,
P n ( r ) = k = 0 n c n , k r k , c n , n 0 ;
b i = 1 λ i [ a i + n = i + 1 K ( k = i n c n , k d k , i λ k i ) a n ] , i = 0 , 1 , 2 , , K ,
P n m ( r ) = P m + q k m ( r ) = s = 0 k c k , s m r m + q s , k = 0 , 1 , 2 , , K .
f ( r ) = k = 0 K a m + q k , m P m + q k m ( r ) = k = 0 K b m + q k , m P m + q k m ( λ r ) ;
b m + q k = 1 λ m + q k [ a m + q k + i = k + 1 K ( j = k i c i , j m d j , k m λ ( j k ) q ) a m + q i ] , k = 0 , 1 , 2 , , K ,
d k , s m = ( m + 2 s + 1 ) k ! ( m + k ) ! ( k s ) ! ( m + k + s + 1 ) ! .
R ¯ m + 2 k m ( r ) = N m + 2 k m R m + 2 k m ( r ) = 2 ( m + 2 k + 1 ) 1 + δ m , 0 R m + 2 k m ( r ) = s = 0 k c ¯ k , s m r m + 2 s ,
c ¯ k , s m = 2 ( m + 2 k + 1 ) 1 + δ m , 0 c k , s m = ( 1 ) k s 2 ( m + 2 k + 1 ) 1 + δ m , 0 ( m + k + s ) ! s ! ( k s ) ! ( m + s ) ! .
d ¯ k , s m = 1 + δ m , 0 2 ( m + 2 s + 1 ) d k , s m = ( 1 + δ m , 0 ) ( m + 2 s + 1 ) 2 k ! ( m + k ) ! ( k s ) ! ( m + k + s + 1 ) ! .
b m + 2 k , m = 1 λ m + 2 k [ a m + 2 k , m + i = k + 1 K j = k i ( c ¯ i , j m d ¯ j , k m λ 2 ( j k ) ) a m + 2 i , m ] = 1 λ m + 2 k [ a m + 2 k , m + i = k + 1 K C ( m , k , i ) a m + 2 i , m ] , k = 0 , 1 , , K ,
C ( m , k , i ) = ( m + 2 i + 1 ) ( m + 2 k + 1 ) × j = k i ( 1 ) i j λ 2 ( j k ) ( m + i + j ) ! ( i j ) ! ( j k ) ! ( m + j + k + 1 ) ! for i = k + 1 , k + 2 , , 0 .
C ( m , k , i ) = C ( m + 2 l , k l , i l ) .
C ( m + 2 l , k l , i l ) = ( m + 2 i + 1 ) ( m + 2 k + 1 ) j = k t i l ( 1 ) i l j λ 2 ( j k + l ) ( m + l + i + j ) ! ( i l j ) ! ( j k + l ) ! ( m + l + j + k + 1 ) ! = ( m + 2 i + 1 ) ( m + 2 k + 1 ) j = k l i ( 1 ) i j λ 2 ( j k ) ( m + i + j ) ! ( i j ) ! ( j k ) ! ( m + j + k + 1 ) ! .
b m + 2 ( K l ) , m = 1 λ N 2 l [ a m + 2 ( K l ) , m + i = K l + 1 K C ( m , K l , i ) a m + 2 i , m ] = 1 λ N 2 l [ a m + 2 ( K l ) , m + i = 0 l 1 C ( m , K l , i + K l + 1 ) a N + 2 i 2 l + 2 , m ] , l = 0 , 1 , , K ,
b m + 2 ( K l ) , m = 1 λ N 2 l [ a m + 2 ( K l ) , m + i = K l + 1 K C ( m , K l , i ) a m + 2 i , m ] = 1 λ N 2 l [ a m + 2 ( K l ) , m + i = 0 l 1 C ( m , K l , i + K l + 1 ) a N + 2 i 2 l + 2 , m ] , l = 0 , 1 , , K .
C ( m , K l , i + K l + 1 ) = C ( m , K l , i + K l + 1 ) .
for i = 0 , 1 , , l 1 , l = 0 , 1 , , min ( K , K ) .
C ( m , K l , i + K l + 1 ) = ( m + 2 i + 2 K 2 l + 3 ) ( m + 2 K 2 l + 1 ) × j = k l i + K l + 1 ( 1 ) i + K + 1 l j λ 2 ( j K + l ) ( m + i + K l + j + 1 ) ! ( i + K l + 1 j ) ! ( j K + l ) ! ( m + j + K l + 1 ) ! = ( N + 2 i 2 l + 3 ) ( N 2 l + 1 ) j = 0 i + 1 ( 1 ) i + 1 j λ 2 j ( N + i 2 l + j + 1 ) ! j ! ( i + 1 j ) ! ( N + j l + 1 ) ! .
C ( m , K l , i + K l + 1 ) = ( m + 2 i + 2 K 2 l + 3 ) ( m + 2 K 2 l + 1 ) × j = k l i + K l + 1 ( 1 ) i + K + 1 l j λ 2 ( j K + l ) ( m + i + K l + j + 1 ) ! ( i + K l + 1 j ) ! ( j K + l ) ! ( m + j + K l + 1 ) ! = ( N + 2 i 2 l + 3 ) ( N 2 l + 1 ) j = 0 i + 1 ( 1 ) i + 1 j λ 2 j ( N + i 2 l + j + 1 ) ! j ! ( i + 1 j ) ! ( N + j l + 1 ) ! .
f ( r ) = ( a 0 , a 1 , a 2 , , a K ) [ P 0 ( r ) P 1 ( r ) P 2 ( r ) P K ( r ) ] = ( b 0 , b 1 , b 2 , , b K ) [ P 0 ( λ r ) P 1 ( λ r ) P 2 ( λ r ) P K ( λ r ) ] .
[ P 0 ( r ) P 1 ( r ) P 2 ( r ) P K ( r ) ] = C K [ 1 r r 2 r K ] ,
[ P 0 ( λ r ) P 1 ( λ r ) P 2 ( λ r ) P K ( λ r ) ] = C K [ 1 λ r λ 2 r 2 λ K r K ] = C K diag ( 1 , λ , λ 2 , , λ K ) [ 1 r r 2 r K ] .
( a 0 , a 1 , a 2 , , a K ) C K [ 1 r r 2 r K ] = ( b 0 , b 1 , b 2 , , b K ) C K diag ( 1 , λ , λ 2 , , λ K ) [ 1 r r 2 r K ] .
( b 0 , b 1 , b 2 , , b K ) = ( a 0 , a 1 , a 2 , , a K ) C K ( diag ( 1 , λ , λ 2 , , λ K ) ) 1 C K 1 = ( a 0 , a 1 , a 2 , , a K ) C K diag ( 1 , λ 1 , λ 2 , , λ K ) D K .
s = l k c k , s m d s , l m = δ k , l , 0 l k K .
c k , k m d k , k m = ( m + 2 k ) ! k ! ( m + k ) ! × ( m + 2 k + 1 ) k ! ( m + k ) ! ( m + 2 k + 1 ) ! = 1 .
s = l k c k , s m d s , l m = s = l k ( 1 ) k s ( m + 2 l + 1 ) ( m + k + s ) ! ( s l ) ! ( k s ) ! ( m + s + l + 1 ) ! = ( 1 ) k ( m + 2 l + 1 ) s = l k F ( m , k , l , s ) ,
F ( m , k , l , s ) = ( 1 ) s ( m + k + s ) ! ( s l ) ! ( k s ) ! ( m + s + l + 1 ) ! .
G ( m , k , l , s ) = ( 1 ) s + 1 ( m + k + s ) ! ( s l ) ! ( k + 1 s ) ! ( m + l + s ) ! ( k + 1 s ) ( s l ) ( k l ) ( m + k + l + 1 ) .
F ( m , k , l , s ) = G ( m , k , l , s + 1 ) G ( m , k , l , s ) .
s = l k F ( m , k , l , s ) = s = l k [ G ( m , k , l , s + 1 ) G ( m , k , l , s ) ] = G ( m , k , l , k + 1 ) G ( m , k , l , l ) = 0 .
s = l k c k , s m d s , l m = 0 for l < k .

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