Abstract

Seidel aberration coefficients can be expressed by Zernike coefficients. The least-squares matrix-inversion method of determining Zernike coefficients from a sampled wave front with measurement noise has been found to be numerically unstable. We present a method of estimating the Seidel aberration coefficients by using a two-dimensional discrete wavelet transform. This method is applied to analyze the wave front of an optical system, and we obtain not only more-accurate Seidel aberration coefficients, but we also speed the computation. Three simulated wave fronts are fitted, and simulation results are shown for spherical aberration, coma, astigmatism, and defocus.

© 2002 Optical Society of America

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References

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  1. M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1975), Section 9.2.
  2. F. Zernike, “Beugungstheorie des Schnridenver-Eahrens und Seiner Verbesserten Form, der Phasenkontrastmethode,” Physica 1, 689 (1934).
    [CrossRef]
  3. J. Y. Wang, D. E. Silva, “Wave-front interpretation with Zernike polynomials,” Appl. Opt. 19, 1510–1518 (1980).
    [CrossRef] [PubMed]
  4. D. Malacara, J. M. Carpio-Valadéz, J. J. Sánchez-Mondragón, “Wave-front fitting with discrete orthogonal polynomials in a unit radius circle,” Opt. Eng. 29, 672–675 (1990).
    [CrossRef]
  5. M. Antonin, M. Barlaud, P. Mathieu, I. Daubechies, “Image coding using vector quantization in the wavelet transform domain,” in Proceedings of the International Conference on Acoustical Speech and Signal Processing (IEEE, New York, 1990), pp. 2297–2300.
    [CrossRef]
  6. D. Philippe, M. Benoit, T. M. Dirk, “Signal adapted multiresolution transform for image coding,” IEEE Trans. Inf. Theory 38, 897–904 (1992).
    [CrossRef]
  7. R. A. Devore, B. Jawerth, P. J. Lucier, “Image compression through wavelet transform coding,” IEEE Trans. Inf. Theory 38, 719–746 (1992).
    [CrossRef]
  8. G. Strang, “Wavelets and dilation equations: a brief introduction,” SIAM (Soc. Ind. Appl. Math.) Rev. 31, 614–627 (1989).
  9. S. G. Mallat, “Multifrequency channel decomposition of images and wavelet models,” IEEE Trans. Acoust. Speech Signal Process. 37, 2091–2110 (1989).
    [CrossRef]
  10. M. Unser, “An improved least squares Laplacian pyramid for image compression,” Signal Process. 27, 187–203 (1992).
    [CrossRef]
  11. D. Malacara, Optical Shop Testing (Wiley, New York, 1992), Chap. 13, p. 465.
  12. S. Mallat, A Wavelet Tour of Signal Processing (Academic, New York, 1998), Chap. 7, pp. 236–240.
  13. S. Mallat, “A theory for multiresolution signal decomposition: The wavelet representation,” IEEE Trans. Pattern Anal. Mach. Intell. 11, 674–693 (1989).
    [CrossRef]
  14. M. Vetterli, J. Kovacevic, Wavelets and Subband Coding (Prentice-Hall, Englewood Cliffs, N.J., 1995).
  15. M. Vetterli, “Multi-dimensional subband coding: some theory and algorithms,” Signal Process. 6, 97–112 (1984).
    [CrossRef]
  16. D. Esteban, C. Galand, “Applications of quadrature mirror filters to split band voice coding schemes,” in Proceedings of the International Conference on Acoustics, Speech, and Signal Processing (IEEE, New York, 1977), pp. 585–589.
  17. Wavelet Toolbox For Use with MATLAB (The Math Works, Natick, Mass., 1997).

1992 (3)

D. Philippe, M. Benoit, T. M. Dirk, “Signal adapted multiresolution transform for image coding,” IEEE Trans. Inf. Theory 38, 897–904 (1992).
[CrossRef]

R. A. Devore, B. Jawerth, P. J. Lucier, “Image compression through wavelet transform coding,” IEEE Trans. Inf. Theory 38, 719–746 (1992).
[CrossRef]

M. Unser, “An improved least squares Laplacian pyramid for image compression,” Signal Process. 27, 187–203 (1992).
[CrossRef]

1990 (1)

D. Malacara, J. M. Carpio-Valadéz, J. J. Sánchez-Mondragón, “Wave-front fitting with discrete orthogonal polynomials in a unit radius circle,” Opt. Eng. 29, 672–675 (1990).
[CrossRef]

1989 (3)

S. Mallat, “A theory for multiresolution signal decomposition: The wavelet representation,” IEEE Trans. Pattern Anal. Mach. Intell. 11, 674–693 (1989).
[CrossRef]

G. Strang, “Wavelets and dilation equations: a brief introduction,” SIAM (Soc. Ind. Appl. Math.) Rev. 31, 614–627 (1989).

S. G. Mallat, “Multifrequency channel decomposition of images and wavelet models,” IEEE Trans. Acoust. Speech Signal Process. 37, 2091–2110 (1989).
[CrossRef]

1984 (1)

M. Vetterli, “Multi-dimensional subband coding: some theory and algorithms,” Signal Process. 6, 97–112 (1984).
[CrossRef]

1980 (1)

1934 (1)

F. Zernike, “Beugungstheorie des Schnridenver-Eahrens und Seiner Verbesserten Form, der Phasenkontrastmethode,” Physica 1, 689 (1934).
[CrossRef]

Antonin, M.

M. Antonin, M. Barlaud, P. Mathieu, I. Daubechies, “Image coding using vector quantization in the wavelet transform domain,” in Proceedings of the International Conference on Acoustical Speech and Signal Processing (IEEE, New York, 1990), pp. 2297–2300.
[CrossRef]

Barlaud, M.

M. Antonin, M. Barlaud, P. Mathieu, I. Daubechies, “Image coding using vector quantization in the wavelet transform domain,” in Proceedings of the International Conference on Acoustical Speech and Signal Processing (IEEE, New York, 1990), pp. 2297–2300.
[CrossRef]

Benoit, M.

D. Philippe, M. Benoit, T. M. Dirk, “Signal adapted multiresolution transform for image coding,” IEEE Trans. Inf. Theory 38, 897–904 (1992).
[CrossRef]

Born, M.

M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1975), Section 9.2.

Carpio-Valadéz, J. M.

D. Malacara, J. M. Carpio-Valadéz, J. J. Sánchez-Mondragón, “Wave-front fitting with discrete orthogonal polynomials in a unit radius circle,” Opt. Eng. 29, 672–675 (1990).
[CrossRef]

Daubechies, I.

M. Antonin, M. Barlaud, P. Mathieu, I. Daubechies, “Image coding using vector quantization in the wavelet transform domain,” in Proceedings of the International Conference on Acoustical Speech and Signal Processing (IEEE, New York, 1990), pp. 2297–2300.
[CrossRef]

Devore, R. A.

R. A. Devore, B. Jawerth, P. J. Lucier, “Image compression through wavelet transform coding,” IEEE Trans. Inf. Theory 38, 719–746 (1992).
[CrossRef]

Dirk, T. M.

D. Philippe, M. Benoit, T. M. Dirk, “Signal adapted multiresolution transform for image coding,” IEEE Trans. Inf. Theory 38, 897–904 (1992).
[CrossRef]

Esteban, D.

D. Esteban, C. Galand, “Applications of quadrature mirror filters to split band voice coding schemes,” in Proceedings of the International Conference on Acoustics, Speech, and Signal Processing (IEEE, New York, 1977), pp. 585–589.

Galand, C.

D. Esteban, C. Galand, “Applications of quadrature mirror filters to split band voice coding schemes,” in Proceedings of the International Conference on Acoustics, Speech, and Signal Processing (IEEE, New York, 1977), pp. 585–589.

Jawerth, B.

R. A. Devore, B. Jawerth, P. J. Lucier, “Image compression through wavelet transform coding,” IEEE Trans. Inf. Theory 38, 719–746 (1992).
[CrossRef]

Kovacevic, J.

M. Vetterli, J. Kovacevic, Wavelets and Subband Coding (Prentice-Hall, Englewood Cliffs, N.J., 1995).

Lucier, P. J.

R. A. Devore, B. Jawerth, P. J. Lucier, “Image compression through wavelet transform coding,” IEEE Trans. Inf. Theory 38, 719–746 (1992).
[CrossRef]

Malacara, D.

D. Malacara, J. M. Carpio-Valadéz, J. J. Sánchez-Mondragón, “Wave-front fitting with discrete orthogonal polynomials in a unit radius circle,” Opt. Eng. 29, 672–675 (1990).
[CrossRef]

D. Malacara, Optical Shop Testing (Wiley, New York, 1992), Chap. 13, p. 465.

Mallat, S.

S. Mallat, “A theory for multiresolution signal decomposition: The wavelet representation,” IEEE Trans. Pattern Anal. Mach. Intell. 11, 674–693 (1989).
[CrossRef]

S. Mallat, A Wavelet Tour of Signal Processing (Academic, New York, 1998), Chap. 7, pp. 236–240.

Mallat, S. G.

S. G. Mallat, “Multifrequency channel decomposition of images and wavelet models,” IEEE Trans. Acoust. Speech Signal Process. 37, 2091–2110 (1989).
[CrossRef]

Mathieu, P.

M. Antonin, M. Barlaud, P. Mathieu, I. Daubechies, “Image coding using vector quantization in the wavelet transform domain,” in Proceedings of the International Conference on Acoustical Speech and Signal Processing (IEEE, New York, 1990), pp. 2297–2300.
[CrossRef]

Philippe, D.

D. Philippe, M. Benoit, T. M. Dirk, “Signal adapted multiresolution transform for image coding,” IEEE Trans. Inf. Theory 38, 897–904 (1992).
[CrossRef]

Sánchez-Mondragón, J. J.

D. Malacara, J. M. Carpio-Valadéz, J. J. Sánchez-Mondragón, “Wave-front fitting with discrete orthogonal polynomials in a unit radius circle,” Opt. Eng. 29, 672–675 (1990).
[CrossRef]

Silva, D. E.

Strang, G.

G. Strang, “Wavelets and dilation equations: a brief introduction,” SIAM (Soc. Ind. Appl. Math.) Rev. 31, 614–627 (1989).

Unser, M.

M. Unser, “An improved least squares Laplacian pyramid for image compression,” Signal Process. 27, 187–203 (1992).
[CrossRef]

Vetterli, M.

M. Vetterli, “Multi-dimensional subband coding: some theory and algorithms,” Signal Process. 6, 97–112 (1984).
[CrossRef]

M. Vetterli, J. Kovacevic, Wavelets and Subband Coding (Prentice-Hall, Englewood Cliffs, N.J., 1995).

Wang, J. Y.

Wolf, E.

M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1975), Section 9.2.

Zernike, F.

F. Zernike, “Beugungstheorie des Schnridenver-Eahrens und Seiner Verbesserten Form, der Phasenkontrastmethode,” Physica 1, 689 (1934).
[CrossRef]

Appl. Opt. (1)

IEEE Trans. Acoust. Speech Signal Process (1)

S. G. Mallat, “Multifrequency channel decomposition of images and wavelet models,” IEEE Trans. Acoust. Speech Signal Process. 37, 2091–2110 (1989).
[CrossRef]

IEEE Trans. Inf. Theory (2)

D. Philippe, M. Benoit, T. M. Dirk, “Signal adapted multiresolution transform for image coding,” IEEE Trans. Inf. Theory 38, 897–904 (1992).
[CrossRef]

R. A. Devore, B. Jawerth, P. J. Lucier, “Image compression through wavelet transform coding,” IEEE Trans. Inf. Theory 38, 719–746 (1992).
[CrossRef]

IEEE Trans. Pattern Anal. Mach. Intell. (1)

S. Mallat, “A theory for multiresolution signal decomposition: The wavelet representation,” IEEE Trans. Pattern Anal. Mach. Intell. 11, 674–693 (1989).
[CrossRef]

Opt. Eng. (1)

D. Malacara, J. M. Carpio-Valadéz, J. J. Sánchez-Mondragón, “Wave-front fitting with discrete orthogonal polynomials in a unit radius circle,” Opt. Eng. 29, 672–675 (1990).
[CrossRef]

Physica (1)

F. Zernike, “Beugungstheorie des Schnridenver-Eahrens und Seiner Verbesserten Form, der Phasenkontrastmethode,” Physica 1, 689 (1934).
[CrossRef]

SIAM (Soc. Ind. Appl. Math.) Rev. (1)

G. Strang, “Wavelets and dilation equations: a brief introduction,” SIAM (Soc. Ind. Appl. Math.) Rev. 31, 614–627 (1989).

Signal Process (2)

M. Vetterli, “Multi-dimensional subband coding: some theory and algorithms,” Signal Process. 6, 97–112 (1984).
[CrossRef]

M. Unser, “An improved least squares Laplacian pyramid for image compression,” Signal Process. 27, 187–203 (1992).
[CrossRef]

Other (7)

D. Malacara, Optical Shop Testing (Wiley, New York, 1992), Chap. 13, p. 465.

S. Mallat, A Wavelet Tour of Signal Processing (Academic, New York, 1998), Chap. 7, pp. 236–240.

M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1975), Section 9.2.

M. Antonin, M. Barlaud, P. Mathieu, I. Daubechies, “Image coding using vector quantization in the wavelet transform domain,” in Proceedings of the International Conference on Acoustical Speech and Signal Processing (IEEE, New York, 1990), pp. 2297–2300.
[CrossRef]

D. Esteban, C. Galand, “Applications of quadrature mirror filters to split band voice coding schemes,” in Proceedings of the International Conference on Acoustics, Speech, and Signal Processing (IEEE, New York, 1977), pp. 585–589.

Wavelet Toolbox For Use with MATLAB (The Math Works, Natick, Mass., 1997).

M. Vetterli, J. Kovacevic, Wavelets and Subband Coding (Prentice-Hall, Englewood Cliffs, N.J., 1995).

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Figures (4)

Fig. 1
Fig. 1

Contour of the test wave front W 1 estimated (a) without noise, (b) with noise by the DWT method, and (c) with noise by the LSMI method.

Fig. 2
Fig. 2

Contour of the test wave front W 2 estimated (a) without noise, (b) with noise by the DWT method, and (c) with noise by the LSMI method.

Fig. 3
Fig. 3

Contour of the test wave front W 3 estimated (a) without noise, (b) with noise by the DWT method, and (c) with noise by the LSMI method.

Fig. 4
Fig. 4

Simulated cure (dotted curves) and cure fitting with added noise by the DWT method (solid curves) and by the LSMI method (dashed curves) of the test wave fronts: (a) W 1(x, 0), (b) W 1(0, y), (c) W 2(x, 0), (d) W 2(0, y), (e) W 3(x, 0), (f) W 3(0, y).

Tables (4)

Tables Icon

Table 1 Zernike Polynomials Un,m up to Fourth Degree

Tables Icon

Table 2 Results of Computer Simulation

Tables Icon

Table 3 Results of Computer Simulation

Tables Icon

Table 4 Results of Computer Simulation

Equations (30)

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

Wx, y=s=05 wsUsx, y=w0x+w1y+w2x2+y2+w3x2+3y2+w4yx2+y2+w5x2+y22,
Wxi, yi=j=1N ajZjxi, yi, i=1, 2,, M,
W=Za.
Δ=i=1M |ajZjxi, yi-Wi|2.
ZTZa=ZTW,
a=ZTZ-1ZTW.
tilt inx=w0=a2-2a8,
tilt iny=w1=a3-2a9,
defocus=w2=2a5-6a13-a42+a621/2,
astigmatism=w3=4a42+4a921/2,
coma=w4=9a82+9a921/2,
spherical=w5=6a13,
Vj=Vj-1+Wj-1
Vj=Wj-1  Wj-2 W1  V0.
Aj-1m, n=k,lZ h˜kh˜lAj2m-k, 2n-l,
Dj-11m, n=k,lZ h˜kg˜lAj2m-k, 2n-l,
Dj-12m, n=k,lZ g˜kh˜lAj2m-k, 2n-l,
Dj-13m, n=k,lZ g˜kg˜lAj2m-k, 2n-l.
Ajm, n=4 k,lZ hkhlAj-1m-k/2, n-l/2+4 k,lZ hkglDj-11m-k/2, n-l/2+4 k,lZ gkhlDj-12m-k/2, n-l/2+4 k,lZ gkglDj-13m-k/2, n-l/2,
h˜l=h-l, g˜l=g-l, gl=-11-lh1-l.
Ajm, nWm, n.
Aj=k,lZ Wx, y, ϕj,k,lx, y=s=05 wsk,lZ Usx, y, ϕj,k,lx, y=s=05 wsAjs,
Aj=Ajsw.
w=AjsTAjs-1AjsTAj.
Ŵx, y=s=05 ŵsUsx, y.
ei=Ŵxi, yi-Wxi, yi,
E=iei2.
W1x, y=-3.2x2+y2+0.9x2+3y2+1.1yx2+y2+3.1x2+y22,
W2x, y=-4.05x2+y2+1.35x2+3y2+3.25x2+y22,
W3x, y=0.15y-1.55x2+y2+3.55yx2+y2+2.05x2+y22.

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