Abstract

Orthogonal polynomials can be used for representing complex surfaces on a specific domain. In optics, Zernike polynomials have widespread applications in testing optical instruments, measuring wavefront distributions, and aberration theory. This orthogonal set on the unit circle has an appropriate matching with the shape of optical system components, such as entrance and exit pupils. The existence of noise in the process of representation estimation of optical surfaces causes a reduction of precision in the process of estimation. Different strategies are developed to manage unwanted noise effects and to preserve the quality of the estimation. This article studies the modeling of phase wavefront aberrations in third-order optics by using a combination of Zernike and pseudo-Zernike polynomials and shows how this combination may increase the robustness of the estimation process of phase wavefront aberration distribution.

© 2013 Optical Society of America

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References

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  1. B. R. Nijboer, “The diffraction theory of optical aberrations: part II: diffraction pattern in the presence of small aberrations,” Physica 13, 605–620 (1947).
    [CrossRef]
  2. V. F. Zernike, “Beugungstheorie des schneidenver-fahrens und seiner verbesserten form, der phasenkontrastmethode,” Physica 1, 689–704 (1934).
    [CrossRef]
  3. M. Born and E. Wolf, Principles of Optics: Electromagnetic Theory of Propagation, Interference and Diffraction of Light, 6th ed. (Pergamon, 1980).
  4. D. Malacara, Optical Shop Testing, 2nd ed. (Wiley, 1992).
  5. R. J. Noll, “Zernike polynomials and atmospheric turbulence,” J. Opt. Soc. Am. 66, 207–211 (1976).
    [CrossRef]
  6. J. Flusser, B. Zitova, and T. Suk, Moments and Moment Invariants in Pattern Recognition (Wiley, 2009).
  7. R. Mukundan and K. R. Ramakrishnan, Moment Functions in Image Analysis: Theory and Applications (World Scientific, 1998).
  8. K. Rahbar, K. Faez, and E. Attaran Kakhki, “Robust estimation of wave-front aberration distribution function using invariant wavelet transform profilometry,” Opt. Lasers Eng. 51, 246–252 (2013).
    [CrossRef]
  9. K. Rahbar, K. Faez, and E. Attaran Kakhki, “Estimation of phase wave-front aberration distribution function using wavelet transform profilometry,” Appl. Opt. 51, 3380–3386 (2012).
    [CrossRef]
  10. ISO, “Ophthalmic optics and instruments—reporting aberrations of the human eye,” (ISO, 2008).
  11. ANSI, “Ophthalmics—methods of reporting optical aberrations of eyes,” (ANSI, 2010).
  12. A. B. Bhatiaa and E. Wolfa, “On the circle polynomials of Zernike and related orthogonal sets,” Math. Proc. Cambridge Philos. Soc. 50, 40–48 (1954).
    [CrossRef]
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    [CrossRef]
  14. H. Farid and A. C. Popescu, “Blind removal of lens distortion,” J. Opt. Soc. Am. A 18, 2072–2078 (2001).
    [CrossRef]
  15. W. Yu, “Image-based lens geometric distortion correction using minimization of average bicoherence index,” Pattern Recogn. 37, 1175–1187 (2004).
    [CrossRef]
  16. K. Rahbar and K. Faez, “Blind correction of lens aberration using Zernike moments,” in 18th IEEE International Conference on Image Processing (ICIP) (IEEE, 2011), pp. 861–864.
  17. K. Rahbar and K. Faez, “Blind correction of lens aberration using modified Zernike moments,” J. Inform. Commun. Technol. 2, 37–44 (2011).
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    [CrossRef]

2013 (1)

K. Rahbar, K. Faez, and E. Attaran Kakhki, “Robust estimation of wave-front aberration distribution function using invariant wavelet transform profilometry,” Opt. Lasers Eng. 51, 246–252 (2013).
[CrossRef]

2012 (1)

2011 (1)

K. Rahbar and K. Faez, “Blind correction of lens aberration using modified Zernike moments,” J. Inform. Commun. Technol. 2, 37–44 (2011).

2004 (2)

H. Liu, A. N. Cartwright, and C. Basaran, “Moire interferogram phase extraction: a ridge detection algorithm for continuous wavelet transforms,” Appl. Opt. 43, 850–857 (2004).
[CrossRef]

W. Yu, “Image-based lens geometric distortion correction using minimization of average bicoherence index,” Pattern Recogn. 37, 1175–1187 (2004).
[CrossRef]

2002 (1)

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]

2001 (1)

1984 (1)

1976 (1)

1974 (1)

1954 (1)

A. B. Bhatiaa and E. Wolfa, “On the circle polynomials of Zernike and related orthogonal sets,” Math. Proc. Cambridge Philos. Soc. 50, 40–48 (1954).
[CrossRef]

1947 (1)

B. R. Nijboer, “The diffraction theory of optical aberrations: part II: diffraction pattern in the presence of small aberrations,” Physica 13, 605–620 (1947).
[CrossRef]

1934 (1)

V. F. Zernike, “Beugungstheorie des schneidenver-fahrens und seiner verbesserten form, der phasenkontrastmethode,” Physica 1, 689–704 (1934).
[CrossRef]

Attaran Kakhki, E.

K. Rahbar, K. Faez, and E. Attaran Kakhki, “Robust estimation of wave-front aberration distribution function using invariant wavelet transform profilometry,” Opt. Lasers Eng. 51, 246–252 (2013).
[CrossRef]

K. Rahbar, K. Faez, and E. Attaran Kakhki, “Estimation of phase wave-front aberration distribution function using wavelet transform profilometry,” Appl. Opt. 51, 3380–3386 (2012).
[CrossRef]

Basaran, C.

Bhatiaa, A. B.

A. B. Bhatiaa and E. Wolfa, “On the circle polynomials of Zernike and related orthogonal sets,” Math. Proc. Cambridge Philos. Soc. 50, 40–48 (1954).
[CrossRef]

Born, M.

M. Born and E. Wolf, Principles of Optics: Electromagnetic Theory of Propagation, Interference and Diffraction of Light, 6th ed. (Pergamon, 1980).

Brangaccio, D. J.

Bruning, J. H.

Cartwright, A. N.

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]

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]

Faez, K.

K. Rahbar, K. Faez, and E. Attaran Kakhki, “Robust estimation of wave-front aberration distribution function using invariant wavelet transform profilometry,” Opt. Lasers Eng. 51, 246–252 (2013).
[CrossRef]

K. Rahbar, K. Faez, and E. Attaran Kakhki, “Estimation of phase wave-front aberration distribution function using wavelet transform profilometry,” Appl. Opt. 51, 3380–3386 (2012).
[CrossRef]

K. Rahbar and K. Faez, “Blind correction of lens aberration using modified Zernike moments,” J. Inform. Commun. Technol. 2, 37–44 (2011).

K. Rahbar and K. Faez, “Blind correction of lens aberration using Zernike moments,” in 18th IEEE International Conference on Image Processing (ICIP) (IEEE, 2011), pp. 861–864.

Farid, H.

Flusser, J.

J. Flusser, B. Zitova, and T. Suk, Moments and Moment Invariants in Pattern Recognition (Wiley, 2009).

Gallagher, J. E.

Herriott, D. R.

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]

Liu, H.

Malacara, D.

D. Malacara, Optical Shop Testing, 2nd ed. (Wiley, 1992).

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]

Mukundan, R.

R. Mukundan and K. R. Ramakrishnan, Moment Functions in Image Analysis: Theory and Applications (World Scientific, 1998).

Nijboer, B. R.

B. R. Nijboer, “The diffraction theory of optical aberrations: part II: diffraction pattern in the presence of small aberrations,” Physica 13, 605–620 (1947).
[CrossRef]

Noll, R. J.

Popescu, A. C.

Rahbar, K.

K. Rahbar, K. Faez, and E. Attaran Kakhki, “Robust estimation of wave-front aberration distribution function using invariant wavelet transform profilometry,” Opt. Lasers Eng. 51, 246–252 (2013).
[CrossRef]

K. Rahbar, K. Faez, and E. Attaran Kakhki, “Estimation of phase wave-front aberration distribution function using wavelet transform profilometry,” Appl. Opt. 51, 3380–3386 (2012).
[CrossRef]

K. Rahbar and K. Faez, “Blind correction of lens aberration using modified Zernike moments,” J. Inform. Commun. Technol. 2, 37–44 (2011).

K. Rahbar and K. Faez, “Blind correction of lens aberration using Zernike moments,” in 18th IEEE International Conference on Image Processing (ICIP) (IEEE, 2011), pp. 861–864.

Ramakrishnan, K. R.

R. Mukundan and K. R. Ramakrishnan, Moment Functions in Image Analysis: Theory and Applications (World Scientific, 1998).

Rosenfeld, D. P.

Suk, T.

J. Flusser, B. Zitova, and T. Suk, Moments and Moment Invariants in Pattern Recognition (Wiley, 2009).

White, A. D.

Wolf, E.

M. Born and E. Wolf, Principles of Optics: Electromagnetic Theory of Propagation, Interference and Diffraction of Light, 6th ed. (Pergamon, 1980).

Wolfa, E.

A. B. Bhatiaa and E. Wolfa, “On the circle polynomials of Zernike and related orthogonal sets,” Math. Proc. Cambridge Philos. Soc. 50, 40–48 (1954).
[CrossRef]

Yatagai, T.

Yu, W.

W. Yu, “Image-based lens geometric distortion correction using minimization of average bicoherence index,” Pattern Recogn. 37, 1175–1187 (2004).
[CrossRef]

Zernike, V. F.

V. F. Zernike, “Beugungstheorie des schneidenver-fahrens und seiner verbesserten form, der phasenkontrastmethode,” Physica 1, 689–704 (1934).
[CrossRef]

Zitova, B.

J. Flusser, B. Zitova, and T. Suk, Moments and Moment Invariants in Pattern Recognition (Wiley, 2009).

Appl. Opt. (4)

IEEE Trans. Biomed. Eng. (1)

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]

J. Inform. Commun. Technol. (1)

K. Rahbar and K. Faez, “Blind correction of lens aberration using modified Zernike moments,” J. Inform. Commun. Technol. 2, 37–44 (2011).

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (1)

Math. Proc. Cambridge Philos. Soc. (1)

A. B. Bhatiaa and E. Wolfa, “On the circle polynomials of Zernike and related orthogonal sets,” Math. Proc. Cambridge Philos. Soc. 50, 40–48 (1954).
[CrossRef]

Opt. Lasers Eng. (1)

K. Rahbar, K. Faez, and E. Attaran Kakhki, “Robust estimation of wave-front aberration distribution function using invariant wavelet transform profilometry,” Opt. Lasers Eng. 51, 246–252 (2013).
[CrossRef]

Pattern Recogn. (1)

W. Yu, “Image-based lens geometric distortion correction using minimization of average bicoherence index,” Pattern Recogn. 37, 1175–1187 (2004).
[CrossRef]

Physica (2)

B. R. Nijboer, “The diffraction theory of optical aberrations: part II: diffraction pattern in the presence of small aberrations,” Physica 13, 605–620 (1947).
[CrossRef]

V. F. Zernike, “Beugungstheorie des schneidenver-fahrens und seiner verbesserten form, der phasenkontrastmethode,” Physica 1, 689–704 (1934).
[CrossRef]

Other (7)

M. Born and E. Wolf, Principles of Optics: Electromagnetic Theory of Propagation, Interference and Diffraction of Light, 6th ed. (Pergamon, 1980).

D. Malacara, Optical Shop Testing, 2nd ed. (Wiley, 1992).

J. Flusser, B. Zitova, and T. Suk, Moments and Moment Invariants in Pattern Recognition (Wiley, 2009).

R. Mukundan and K. R. Ramakrishnan, Moment Functions in Image Analysis: Theory and Applications (World Scientific, 1998).

ISO, “Ophthalmic optics and instruments—reporting aberrations of the human eye,” (ISO, 2008).

ANSI, “Ophthalmics—methods of reporting optical aberrations of eyes,” (ANSI, 2010).

K. Rahbar and K. Faez, “Blind correction of lens aberration using Zernike moments,” in 18th IEEE International Conference on Image Processing (ICIP) (IEEE, 2011), pp. 861–864.

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Figures (4)

Fig. 1.
Fig. 1.

Basic elements of a symmetrical optical system.

Fig. 2.
Fig. 2.

Geometry of imaginative scheme to extract the wavefront aberration function.

Fig. 3.
Fig. 3.

Phase demodulation of aberrational fringe image. (a) Predefined phase aberration function, (b) image of fringe pattern on the exit pupil of the visual system in ξ direction, (c) image of fringe pattern on the exit pupil of the visual system in δ direction, (d) unwrapped phase of the fringe pattern in ξ direction, (e) unwrapped phase of the fringe pattern in δ direction, and (f) recovered total aberration.

Fig. 4.
Fig. 4.

Comparison graph of the stability of the proposed model and the original one with respect to the variation of the Gaussian noise standard deviation.

Tables (2)

Tables Icon

Table 1. Comparison of Pseudo-Zernike and Zernike Polynomials with Seidel Aberration Distribution

Tables Icon

Table 2. Zernike Polynomial Coefficients Corresponding to the Seidel Aberration

Equations (22)

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Z n m ( ρ , θ ) = N m n R m n ( ρ ) Θ ( θ ) .
N n m = 2 ( n + 1 ) 1 + δ m 0 ,
R n m ( ρ ) = s = 0 ( n | m | ) / 2 ( 1 ) s ( n s ) ! ρ n 2 s s ! [ 0.5 ( n + | m | ) s ] ! [ 0.5 ( n | m | ) s ] ! .
Θ ( θ ) = { cos ( m θ ) ; for m 0 sin ( m θ ) ; for m < 0 .
Z ˜ n m ( ρ , θ ) = N m n R ˜ m n ( ρ ) Θ ( θ ) .
R ˜ n m ( ρ ) = s = 0 n | m | ( 1 ) s · ( 2 n + 1 s ) ! ρ n s s ! [ n | m | s ] ! [ n + | m | + 1 s ] ! .
( X , Y ) = ( h x , h y ) + ( ϵ x , ϵ y ) .
W ( x , y ) = m , n w m , n Ƶ n m ( x , y ) .
W ( x , y ) y = ϵ y ( x , y ) f , W ( x , y ) x = ϵ x ( x , y ) f ,
ϵ y ( x , y ) f = j w m , n Ƶ n m ( x , y ) y , ϵ x ( x , y ) f = j w m , n Ƶ n m ( x , y ) x .
W ( x , y ) = m = 0 n = 0 , 2 , 4 w n , m Z n m ( x , y ) + m = ± 1 n = 1 w n , m Z n m ( x , y ) + m = ± 2 n = 2 w n , m Z n m ( x , y ) + m = ± 1 n = 2 w n , m Z ˜ n m ( x , y ) .
W ( ξ , δ ) ξ = x f , W ( ξ , δ ) δ = y f .
f ( x , δ ) = a ( x , δ ) + γ ( x , δ ) cos [ 2 π d ( x + η ) ] ,
f ( x , δ ) = a ( x , δ ) + γ ( x , δ ) cos [ 2 π d ( f W ( ξ , δ ) ξ + η ) ] .
2 π d ( f W ( ξ , δ ) ξ ) = 2 π m , m = 0 , ± 1 , .
W ( ξ , δ ) ξ = d f m .
W ( s , b ) = 1 s ψ * ( s b s ) g ( ξ ) d ξ ,
ϕ ( s , b ) = t g 1 [ I W ( s , b ) R W ( s , b ) ] .
2 π d ( f W ( ξ , δ ) ξ ) = t g 1 [ I W ( s , b ) R W ( s , b ) ] .
W ( ξ ) = W ( ξ , δ ) ξ d ξ .
W ( ξ , δ ) = j ω j Ƶ j ( ξ , δ ) ,
W ( ξ , δ ) = j ω j Ƶ j ( ξ , δ ) + α j Φ n ( μ , σ 2 ) .

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