Abstract

In optics, Zernike polynomials are widely used in testing, wavefront sensing, and aberration theory. This unique set of radial polynomials is orthogonal over the unit circle and finite on its boundary. This Letter presents a recursive formula to compute Zernike radial polynomials using a relationship between radial polynomials and Chebyshev polynomials of the second kind. Unlike the previous algorithms, the derived recurrence relation depends neither on the degree nor on the azimuthal order of the radial polynomials. This leads to a reduction in the computational complexity.

© 2013 Optical Society of America

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References

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  1. F. Zernike, Physica 1, 689 (1934).
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  11. A. Prata and W. V. T. Rusch, Appl. Opt. 28, 749 (1989).
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    [CrossRef]

2012

2011

2010

2009

2007

A. J. E. M. Janssen and P. Dirksen, J. Eur. Opt. Soc. 2, 07012 (2007).
[CrossRef]

2006

G.-M. Dai, J. Opt. Soc. Am. A 23, 539 (2006).
[CrossRef]

A. J. E. M. Janssen and P. Dirksen, J. Microlith. Microfab. Microsyst. 5, 030501 (2006).
[CrossRef]

2003

C. W. Chong, P. Raveendran, and R. Mukundan, Pattern Recogn. 36, 731 (2003).
[CrossRef]

1989

1976

E. C. Kintner, Opt. Acta 23, 679 (1976).
[CrossRef]

1934

F. Zernike, Physica 1, 689 (1934).
[CrossRef]

Arines, J.

Chong, C. W.

C. W. Chong, P. Raveendran, and R. Mukundan, Pattern Recogn. 36, 731 (2003).
[CrossRef]

Dai, G.-M.

Dirksen, P.

A. J. E. M. Janssen and P. Dirksen, J. Eur. Opt. Soc. 2, 07012 (2007).
[CrossRef]

A. J. E. M. Janssen and P. Dirksen, J. Microlith. Microfab. Microsyst. 5, 030501 (2006).
[CrossRef]

Gao, X.

X. Gao, Q. Wang, X. Li, D. Tao, and K. Zhang, IEEE Trans. Image Process. 20, 2738 (2011).
[CrossRef]

Janssen, A. J. E. M.

A. J. E. M. Janssen and P. Dirksen, J. Eur. Opt. Soc. 2, 07012 (2007).
[CrossRef]

A. J. E. M. Janssen and P. Dirksen, J. Microlith. Microfab. Microsyst. 5, 030501 (2006).
[CrossRef]

Kintner, E. C.

E. C. Kintner, Opt. Acta 23, 679 (1976).
[CrossRef]

Lee, H.

Li, Q.

Li, X.

X. Gao, Q. Wang, X. Li, D. Tao, and K. Zhang, IEEE Trans. Image Process. 20, 2738 (2011).
[CrossRef]

Liu, Z. J.

Mukundan, R.

C. W. Chong, P. Raveendran, and R. Mukundan, Pattern Recogn. 36, 731 (2003).
[CrossRef]

Navarro, R.

Nijboer, B. R. A.

B. R. A. Nijboer, “The diffraction theory of aberrations,” Ph.D. thesis (University of Groningen, 1942).

Prata, A.

Raveendran, P.

C. W. Chong, P. Raveendran, and R. Mukundan, Pattern Recogn. 36, 731 (2003).
[CrossRef]

Rivera, R.

Rusch, W. V. T.

Schwiegerling, J.

Tao, D.

X. Gao, Q. Wang, X. Li, D. Tao, and K. Zhang, IEEE Trans. Image Process. 20, 2738 (2011).
[CrossRef]

Wang, Q.

Z. J. Liu, Q. Li, ZW. Xia, and Q. Wang, Appl. Opt. 51, 7529 (2012).
[CrossRef]

X. Gao, Q. Wang, X. Li, D. Tao, and K. Zhang, IEEE Trans. Image Process. 20, 2738 (2011).
[CrossRef]

Xia, ZW.

Zernike, F.

F. Zernike, Physica 1, 689 (1934).
[CrossRef]

Zhang, K.

X. Gao, Q. Wang, X. Li, D. Tao, and K. Zhang, IEEE Trans. Image Process. 20, 2738 (2011).
[CrossRef]

Appl. Opt.

IEEE Trans. Image Process.

X. Gao, Q. Wang, X. Li, D. Tao, and K. Zhang, IEEE Trans. Image Process. 20, 2738 (2011).
[CrossRef]

J. Eur. Opt. Soc.

A. J. E. M. Janssen and P. Dirksen, J. Eur. Opt. Soc. 2, 07012 (2007).
[CrossRef]

J. Microlith. Microfab. Microsyst.

A. J. E. M. Janssen and P. Dirksen, J. Microlith. Microfab. Microsyst. 5, 030501 (2006).
[CrossRef]

J. Opt. Soc. Am. A

Opt. Acta

E. C. Kintner, Opt. Acta 23, 679 (1976).
[CrossRef]

Opt. Express

Opt. Lett.

Pattern Recogn.

C. W. Chong, P. Raveendran, and R. Mukundan, Pattern Recogn. 36, 731 (2003).
[CrossRef]

Physica

F. Zernike, Physica 1, 689 (1934).
[CrossRef]

Other

B. R. A. Nijboer, “The diffraction theory of aberrations,” Ph.D. thesis (University of Groningen, 1942).

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Figures (2)

Fig. 1.
Fig. 1.

Flow of proposed method to compute Zernike radial polynomials Rnm(r).

Fig. 2.
Fig. 2.

Number of operations required to compute Zernike radial polynomials up to degree n.

Tables (1)

Tables Icon

Table 1. Complexity Analysis of Radial Polynomials, Rnm(r), up to Degree n in Terms of A, Where A=((3/2)(n/2)3)(3/2)(n/2)+n/22+n/2+2n+2

Equations (18)

Equations on this page are rendered with MathJax. Learn more.

Znm(r,θ)=NnmRnm(r)Θ(mθ),
Nnm=2(n+1)1+δm0,
Rnm(r)=k=0n|m|2(1)k(nk)!k!(n+m2k)!(nm2k)!rn2k.
Θ(mθ)={cos(mθ);m0sin(mθ);m<0.
01Rnm(r)Rkm(r)rdr=δnk2(n+1).
Rnm(r)=1K1[(K2r2+K3)Rn2m(r)+K4Rn4m(r)],n=m+4,m+6,
K1=(n+m)(nm)(n2)2,K2=2n(n1)(n2),K3=m2(n1)n(n1)(n2),K4=n(n+m2)(nm2)2.
Rmm(r)=rm,Rm+2m(r)=(m+2)rm+2(m+1)rm.
Rnm(r)=rL1Rn1|m1|(r)+L2Rn2m(r),
L1=2nm+n,L2=mnm+n.
Rnm4(r)=H1Rnm(r)+(H2+H3r2)Rnm2(r),
H1=m(m1)2mH2+H3(n+m+2)(nm)8,H2=H3(n+m)(nm+2)4(m1)+(m2),H3=4(m2)(m3)(n+m2)(nm+4).
Rnm(r)=12π02πUn(rcosθ)cos(mθ)dθ,
Un(x)=2xUn1(x)Un2(x),
Un(rcosθ)cos(mθ)=(2rcosθ)Un1(rcosθ)cos(mθ)Un2(rcosθ)cos(mθ).
Un(rcosθ)cos(mθ)=r[cos(|m1|θ)+cos((m+1)θ)]×Un1(rcosθ)cos(mθ)Un2(rcosθ)cos(mθ).
12π02πUn(rcosθ)cos(mθ)dθ=r{12π02πUn1(rcosθ)cos(|m1|θ)dθ+12π02πUn1(rcosθ)cos((m+1)θ)dθ}12π02πUn2(rcosθ)cos(mθ)dθ.
Rnm(r)=r[Rn1|m1|(r)+Rn1m+1(r)]Rn2m(r).

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