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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 17, Iss. 26 — Dec. 21, 2009
  • pp: 24269–24281

Direct and inverse discrete Zernike transform

Rafael Navarro, Justo Arines, and Ricardo Rivera  »View Author Affiliations

Optics Express, Vol. 17, Issue 26, pp. 24269-24281 (2009)

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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.

© 2009 OSA

OCIS Codes
(010.7350) Atmospheric and oceanic optics : Wave-front sensing
(220.1010) Optical design and fabrication : Aberrations (global)
(080.1005) Geometric optics : Aberration expansions
(110.7348) Imaging systems : Wavefront encoding

Original Manuscript: July 9, 2009
Revised Manuscript: November 20, 2009
Manuscript Accepted: December 4, 2009
Published: December 18, 2009

Rafael Navarro, Justo Arines, and Ricardo Rivera, "Direct and inverse discrete Zernike transform," Opt. Express 17, 24269-24281 (2009)

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  1. V. N. Mahajan, “Zernike polynomials and wavefront fitting,” in Optical Shop Testing, 3rd ed., D. Malacara, ed. (Wiley, New York, 2007).
  2. R. Navarro and E. Moreno-Barriuso, “Laser ray-tracing method for optical testing,” Opt. Lett. 24(14), 951–953 (1999). [CrossRef] [PubMed]
  3. R. J. Noll, “Phase estimates from slope–type wave–front sensors,” J. Opt. Soc. Am. 68(1), 139–140 (1978). [CrossRef]
  4. R. Cubalchini, “Modal wave-front estimation from phase derivative measurements,” J. Opt. Soc. Am. 69(7), 972–977 (1979). [CrossRef]
  5. J. Y. Wang and D. E. Silva, “Wave-front interpretation with Zernike polynomials,” Appl. Opt. 19(9), 1510–1518 (1980). [CrossRef] [PubMed]
  6. R. K. Tyson, Principles of Adaptive Optics (Academic, Boston, 1991).
  7. J. Alda and G. D. Boreman, “Zernike-based matrix model of deformable mirrors: optimization of aperture size,” Appl. Opt. 32, 2431–2438 (1993). [CrossRef] [PubMed]
  8. G. D. Love, “Wave-front correction and production of Zernike modes with a liquid-crystal spatial light modulator,” Appl. Opt. 36(7), 1517–1520 (1997). [CrossRef] [PubMed]
  9. C.-J. Kim, “Polynomial fit of interferograms,” Appl. Opt. 21(24), 4521–4525 (1982). [CrossRef] [PubMed]
  10. H. van Brug, “Zernike polynomials as a basis for wave-front fitting in lateral shearing interferometry,” Appl. Opt. 36(13), 2788–2790 (1997). [CrossRef] [PubMed]
  11. B. Qi, H. Chen, and N. Dong, “Wavefront fitting of interferograms with Zernike polynomials,” Opt. Eng. 41(7), 1565–1569 (2002). [CrossRef]
  12. J. Nam and J. Rubinstein, “Numerical reconstruction of optical surfaces,” J. Opt. Soc. Am. A 25(7), 1697–1709 (2008). [CrossRef]
  13. J. Schwiegerling, J. Greivenkamp, and J. Miller, “Representation of videokeratoscopic height data with Zernike polynomials,” J. Opt. Soc. Am. A 12(10), 2105–2113 (1995). [CrossRef]
  14. R. J. Noll, “Zernike polynomials and atmospheric turbulence,” J. Opt. Soc. Am. 66(3), 207–211 (1976). [CrossRef]
  15. J. Herrmann, “Cross coupling and aliasing in modal wave-front estimation,” J. Opt. Soc. Am. 71(8), 989–992 (1981). [CrossRef]
  16. J. Herrmann, “Least-squares wave front errors of minimum norm,” J. Opt. Soc. Am. 70(1), 28–35 (1980). [CrossRef]
  17. O. Soloviev and G. Vdovin, “Hartmann-Shack test with random masks for modal wavefront reconstruction,” Opt. Express 13(23), 9570–9584 (2005). [CrossRef] [PubMed]
  18. M. Ares and S. Royo, “Comparison of cubic B-spline and Zernike-fitting techniques in complex wavefront reconstruction,” Appl. Opt. 45(27), 6954–6964 (2006). [CrossRef] [PubMed]
  19. L. Diaz-Santana, G. Walker, and S. X. Bará, “Sampling geometries for ocular aberrometry: A model for evaluation of performance,” Opt. Express 13(22), 8801–8818 (2005). [CrossRef] [PubMed]
  20. W. H. Southwell, “Wave–front estimation from wave–front slope measurements,” J. Opt. Soc. Am. 70(8), 998–1006 (1980). [CrossRef]
  21. E. E. Silbaugh, B. M. Welsh, and M. C. Roggemann, “Characterization of Atmospheric Turbulence Phase Statics Using Wave-Front Slope Measurements,” J. Opt. Soc. Am. A 13(12), 2453–2460 (1996). [CrossRef]
  22. J. Liang, B. Grimm, S. Goelz, and J. F. Bille, “Objective measurement of wave aberrations of the human eye with use of a Hartmann-Shack wave-front sensor,” J. Opt. Soc. Am. A 11, 1949–1957 (1994). [CrossRef]
  23. 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 .
  24. N. U. Mayall and S. Vasilevskis, “Quantitative tests of the Lick Observatory 120-Inch mirror,” Astron. J. 65, 304–317 (1960). [CrossRef]
  25. R. Navarro, “Objective refraction from aberrometry: theory,” J. Biomed. Opt. 14(2), 024021 (2009). [CrossRef] [PubMed]
  26. D. Malacara-Hernandez, “Wavefront fitting with discrete orthogonal polynomials in a unit radius circle,” Opt. Eng. 29(6), 672–675 (1990). [CrossRef]
  27. D. J. Fischer, J. T. O'Bryan, R. Lopez, and H. P. Stahl, “Vector formulation for interferogram surface fitting,” Appl. Opt. 32(25), 4738–4743 (1993). [CrossRef] [PubMed]
  28. 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]

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