OSA's Digital Library

Journal of the Optical Society of America A

Journal of the Optical Society of America A


  • Editor: Franco Gori
  • Vol. 30, Iss. 10 — Oct. 1, 2013
  • pp: 1988–1993

Phase wavefront aberration modeling using Zernike and pseudo-Zernike polynomials

Kambiz Rahbar, Karim Faez, and Ebrahim Attaran Kakhki  »View Author Affiliations

JOSA A, Vol. 30, Issue 10, pp. 1988-1993 (2013)

View Full Text Article

Enhanced HTML    Acrobat PDF (719 KB)

Browse Journals / Lookup Meetings

Browse by Journal and Year


Lookup Conference Papers

Close Browse Journals / Lookup Meetings

Article Tools



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

OCIS Codes
(080.1010) Geometric optics : Aberrations (global)
(080.1005) Geometric optics : Aberration expansions

ToC Category:
Geometric Optics

Original Manuscript: June 18, 2013
Manuscript Accepted: July 30, 2013
Published: September 12, 2013

Kambiz Rahbar, Karim Faez, and Ebrahim Attaran Kakhki, "Phase wavefront aberration modeling using Zernike and pseudo-Zernike polynomials," J. Opt. Soc. Am. A 30, 1988-1993 (2013)

Sort:  Author  |  Year  |  Journal  |  Reset  


  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]
  13. 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]
  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).
  18. 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]
  19. J. H. Bruning, D. R. Herriott, J. E. Gallagher, D. P. Rosenfeld, A. D. White, and D. J. Brangaccio, “Digital wavefront measuring interferometer for testing optical surfaces and lenses,” Appl. Opt. 13, 2693–2703 (1974). [CrossRef]
  20. T. Yatagai, “Fringe scanning Ronchi test for aspherical surfaces,” Appl. Opt. 23, 3676–3679 (1984). [CrossRef]

Cited By

Alert me when this paper is cited

OSA is able to provide readers links to articles that cite this paper by participating in CrossRef's Cited-By Linking service. CrossRef includes content from more than 3000 publishers and societies. In addition to listing OSA journal articles that cite this paper, citing articles from other participating publishers will also be listed.


Fig. 1. Fig. 2. Fig. 3.
Fig. 4.

« Previous Article  |  Next Article »

OSA is a member of CrossRef.

CrossCheck Deposited