Abstract

An invertible discrete Zernike transform, DZT is proposed and implemented. Three types of non-redundant samplings, random, hybrid (perturbed deterministic) and deterministic (spiral) are shown to provide completeness of the resulting sampled Zernike polynomial expansion. When completeness is guaranteed, then we can obtain an orthonormal basis, and hence the inversion only requires transposition of the matrix formed by the basis vectors (modes). The discrete Zernike modes are given for different sampling patterns and number of samples. The DZT has been implemented showing better performance, numerical stability and robustness than the standard Zernike expansion in numerical simulations. Non-redundant (critical) sampling along with an invertible transformation can be useful in a wide variety of applications.

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References

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2009

R. Navarro, “Objective refraction from aberrometry: theory,” J. Biomed. Opt. 14(2), 024021 (2009).
[CrossRef] [PubMed]

J. Arines, E. Pailos, P. Prado, and S. Bará, “The contribution of the fixational eye movements to the variability of the measured ocular aberration,” Ophthalmic Physiol. Opt. 29(3), 281–287 (2009).
[CrossRef] [PubMed]

2008

2006

2005

2002

B. Qi, H. Chen, and N. Dong, “Wavefront fitting of interferograms with Zernike polynomials,” Opt. Eng. 41(7), 1565–1569 (2002).
[CrossRef]

1999

1997

1996

1995

1994

1993

1990

D. Malacara-Hernandez, “Wavefront fitting with discrete orthogonal polynomials in a unit radius circle,” Opt. Eng. 29(6), 672–675 (1990).
[CrossRef]

1982

1981

1980

1979

1978

1976

1960

N. U. Mayall and S. Vasilevskis, “Quantitative tests of the Lick Observatory 120-Inch mirror,” Astron. J. 65, 304–317 (1960).
[CrossRef]

Alda, J.

Ares, M.

Arines, J.

J. Arines, E. Pailos, P. Prado, and S. Bará, “The contribution of the fixational eye movements to the variability of the measured ocular aberration,” Ophthalmic Physiol. Opt. 29(3), 281–287 (2009).
[CrossRef] [PubMed]

Bará, S.

J. Arines, E. Pailos, P. Prado, and S. Bará, “The contribution of the fixational eye movements to the variability of the measured ocular aberration,” Ophthalmic Physiol. Opt. 29(3), 281–287 (2009).
[CrossRef] [PubMed]

Bará, S. X.

Bille, J. F.

Boreman, G. D.

Chen, H.

B. Qi, H. Chen, and N. Dong, “Wavefront fitting of interferograms with Zernike polynomials,” Opt. Eng. 41(7), 1565–1569 (2002).
[CrossRef]

Cubalchini, R.

Diaz-Santana, L.

Dong, N.

B. Qi, H. Chen, and N. Dong, “Wavefront fitting of interferograms with Zernike polynomials,” Opt. Eng. 41(7), 1565–1569 (2002).
[CrossRef]

Fischer, D. J.

Goelz, S.

Greivenkamp, J.

Grimm, B.

Herrmann, J.

Kim, C.-J.

Liang, J.

Lopez, R.

Love, G. D.

Malacara-Hernandez, D.

D. Malacara-Hernandez, “Wavefront fitting with discrete orthogonal polynomials in a unit radius circle,” Opt. Eng. 29(6), 672–675 (1990).
[CrossRef]

Mayall, N. U.

N. U. Mayall and S. Vasilevskis, “Quantitative tests of the Lick Observatory 120-Inch mirror,” Astron. J. 65, 304–317 (1960).
[CrossRef]

Miller, J.

Moreno-Barriuso, E.

Nam, J.

Navarro, R.

Noll, R. J.

O'Bryan, J. T.

Pailos, E.

J. Arines, E. Pailos, P. Prado, and S. Bará, “The contribution of the fixational eye movements to the variability of the measured ocular aberration,” Ophthalmic Physiol. Opt. 29(3), 281–287 (2009).
[CrossRef] [PubMed]

Prado, P.

J. Arines, E. Pailos, P. Prado, and S. Bará, “The contribution of the fixational eye movements to the variability of the measured ocular aberration,” Ophthalmic Physiol. Opt. 29(3), 281–287 (2009).
[CrossRef] [PubMed]

Qi, B.

B. Qi, H. Chen, and N. Dong, “Wavefront fitting of interferograms with Zernike polynomials,” Opt. Eng. 41(7), 1565–1569 (2002).
[CrossRef]

Roggemann, M. C.

Royo, S.

Rubinstein, J.

Schwiegerling, J.

Silbaugh, E. E.

Silva, D. E.

Soloviev, O.

Southwell, W. H.

Stahl, H. P.

van Brug, H.

Vasilevskis, S.

N. U. Mayall and S. Vasilevskis, “Quantitative tests of the Lick Observatory 120-Inch mirror,” Astron. J. 65, 304–317 (1960).
[CrossRef]

Vdovin, G.

Walker, G.

Wang, J. Y.

Welsh, B. M.

Appl. Opt.

Astron. J.

N. U. Mayall and S. Vasilevskis, “Quantitative tests of the Lick Observatory 120-Inch mirror,” Astron. J. 65, 304–317 (1960).
[CrossRef]

J. Biomed. Opt.

R. Navarro, “Objective refraction from aberrometry: theory,” J. Biomed. Opt. 14(2), 024021 (2009).
[CrossRef] [PubMed]

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

Ophthalmic Physiol. Opt.

J. Arines, E. Pailos, P. Prado, and S. Bará, “The contribution of the fixational eye movements to the variability of the measured ocular aberration,” Ophthalmic Physiol. Opt. 29(3), 281–287 (2009).
[CrossRef] [PubMed]

Opt. Eng.

D. Malacara-Hernandez, “Wavefront fitting with discrete orthogonal polynomials in a unit radius circle,” Opt. Eng. 29(6), 672–675 (1990).
[CrossRef]

B. Qi, H. Chen, and N. Dong, “Wavefront fitting of interferograms with Zernike polynomials,” Opt. Eng. 41(7), 1565–1569 (2002).
[CrossRef]

Opt. Express

Opt. Lett.

Other

E. Anderson, Z. Bai, C. Bischof, S. Blackford, J. Demmel, J. Dongarra, J. Du Croz, A. Greenbaum, S. Hammarling, A. McKenney, and D. Sorensen, LAPACK User's Guide, 3rd Ed., (SIAM, Philadelphia, 1999), http://www.netlib.org/lapack/lug/lapack_lug.html .

V. N. Mahajan, “Zernike polynomials and wavefront fitting,” in Optical Shop Testing, 3rd ed., D. Malacara, ed. (Wiley, New York, 2007).

R. K. Tyson, Principles of Adaptive Optics (Academic, Boston, 1991).

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Figures (4)

Fig. 1
Fig. 1

Examples of sampling patterns with 91 points providing singular (hexagonal and hexapolar, upper panels) and invertible (random and spiral, lower panels) (Z).

Fig. 2
Fig. 2

Inverse of condition number of matrix (Z) .versus standard deviation of random perturbations of the coordinates of the sampling grid, for different sampling schemes with I = 36: Square, hexagonal, hexapolar and spiral.

Fig. 4
Fig. 4

Modes of the DZT (continued).

Fig. 3
Fig. 3

Modes of the DZT, Q n m (only m ≥ 0 are shown), for different sampling schemes: random (R), perturbed hexagonal (H) and spiral (S). The three upper rows correspond to I = 36 samples and the three lower rows to I = 91. Bottom row represents the continuous (I = ∞) Zernike modes.

Tables (2)

Tables Icon

Table 1 Rank of matrix Z for different sampling schemes (rows) and number of samples (columns). Square (Sq), Hexagonal (H)

Tables Icon

Table 2 RMS errors obtained with standard (Z) and discrete (Q) representations in coefficients (C) and in wavefront (W) for hexagonal, random and spiral sampling patterns. All values are in micrometers.

Equations (11)

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W ( ρ , θ ) = n , m c n m Z n m ( ρ , θ )
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 }
R n | m | ( ρ ) = s = 0 ( n | m | ) / 2 ( 1 ) s ( n s ) ! s ! [ 0.5 ( n + | m | ) s ] ! [ 0.5 ( n | m | ) s ] ! ρ n 2 s ,
N n m = 2 ( n + 1 ) 1 + δ m 0
i w i 2 / I n , m c n m 2 .
C = ( Z T Z ) 1 Z T W .
C = ( D Z T D Z ) 1 D Z M
c n m = 0 1 0 2 π W ( ρ , θ ) Z n m ( ρ , θ ) ρ d θ d ρ .
Z = Q R
D Z T ( c j ) = W i = j c j Q i j
D Z T 1 ( W i ) = c j = i W i Q i j .

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