OSA's Digital Library

Journal of the Optical Society of America A

Journal of the Optical Society of America A


  • Vol. 15, Iss. 2 — Feb. 1, 1998
  • pp: 389–401

Disk-harmonic coefficients for invariant pattern recognition

Steven C. Verrall and Ramakrishna Kakarala  »View Author Affiliations

JOSA A, Vol. 15, Issue 2, pp. 389-401 (1998)

View Full Text Article

Acrobat PDF (1333 KB)

Browse Journals / Lookup Meetings

Browse by Journal and Year


Lookup Conference Papers

Close Browse Journals / Lookup Meetings

Article Tools



Disk harmonics are defined, and an explanation is given as to why they can be interpreted as the natural generalization of the Fourier basis set onto the unit disk. An existing statistical theory is used to show that a particular set of disk-harmonic coefficients is more suitable for describing images on the unit disk than Zernike or pseudo-Zernike moments are. Zernike moments have been applied to a wide range of problems. However, we concentrate on the problem of invariant pattern recognition and briefly indicate other problems where disk-harmonic coefficients are possibly useful. The effects that different pixel resolutions and discrete white noise have on the three moment or coefficient sets compared in this paper are briefly investigated experimentally.

© 1998 Optical Society of America

OCIS Codes
(070.5010) Fourier optics and signal processing : Pattern recognition
(110.4280) Imaging systems : Noise in imaging systems
(110.6980) Imaging systems : Transforms
(220.1010) Optical design and fabrication : Aberrations (global)

Steven C. Verrall and Ramakrishna Kakarala, "Disk-harmonic coefficients for invariant pattern recognition," J. Opt. Soc. Am. A 15, 389-401 (1998)

Sort:  Author  |  Year  |  Journal  |  Reset


  1. M. Born and E. Wolf, Principles of Optics (Pergamon, New York, 1980).
  2. R. J. Noll, “Zernike polynomials and atmospheric turbulence,” J. Opt. Soc. Am. 66, 207–211 (1976).
  3. M. R. Teague, “Image analysis via the general theory of moments,” J. Opt. Soc. Am. 70, 920–930 (1980).
  4. C.-H. Teh and R. T. Chin, “On image analysis by the methods of moments,” IEEE Trans. Pattern Anal. Mach. Intell. 10, 496–513 (1988).
  5. Y. Sheng and L. Shen, “Orthogonal Fourier–Mellin moments for invariant pattern recognition,” J. Opt. Soc. Am. A 11, 1748–1757 (1994).
  6. A. Papoulis, Systems and Transforms with Applications in Optics (McGraw-Hill, New York, 1968), pp. 40–44.
  7. A. B. Bhatia and E. Wolf, “On the circular polynomials of Zernike and related orthogonal sets,” Proc. Cambridge Philos. Soc. 50, 40–48 (1954).
  8. A. Khotanzad and Y. H. Hong, “Invariant image recognition by Zernike moments,” IEEE Trans. Pattern Anal. Mach. Intell. 12, 489–497 (1990).
  9. A. Khotanzad and J.-H. Lu, “Classification of invariant image representations using a neural network,” IEEE Trans. Acoust. Speech Signal Process. 38, 1028–1038 (1990).
  10. M. R. Azimi-Sadjadi and S. A. Stricker, “Detection and classification of buried dielectric anomalies using neural networks—further results,” IEEE Trans. Instrum. Meas. 43, 34–39 (1994).
  11. J. Wood, “Invariant pattern-recognition—a review,” Pattern Recogn. 29, 1–17 (1996).
  12. M. O. Freeman and B. E. A. Saleh, “Moment invariants in the space and frequency domains,” J. Opt. Soc. Am. A 5, 1073–1084 (1988).
  13. M. I. Heywood and P. D. Noakes, “Fractional central moment method for moment-invariant object classification,” IEE Proc. Vision Image Signal Process. 142, 213–219 (1995).
  14. Y. S. Abu-Mostafa and D. Psaltis, “Recognitive aspects of moment invariants,” IEEE Trans. Pattern Anal. Mach. Intell. PAMI-6, 698–706 (1984).
  15. C.-H. Teh and R. T. Chin, “On digital approximation of moment invariants,” Comput. Vis. Graph. Image Process. 33, 318–326 (1986).
  16. Å. Wallin and O. Kübler, “Complete sets of complex Zernike moment invariants and the role of the pseudoinvariants,” IEEE Trans. Pattern Anal. Mach. Intell. 17, 1106–1110 (1995).
  17. J. Bigün and J. M. H. du Buf, “N-folded symmetries by complex moments in Gabor space and their application to unsupervised texture segmentation,” IEEE Trans. Pattern. Anal. Mach. Intell. 16, 80–87 (1994).
  18. R. Kakarala and J. A. Cadzow, “Estimation of phase for noisy linear-phase signals,” IEEE Trans. Signal Process. 44, 2483–2497 (1996).
  19. R. Courant and D. Hilbert, Methods of Mathematical Physics (Interscience, New York, 1953), Vol. 1, pp. 297–306.
  20. R. M. Bracewell, The Fourier Transform and Its Applications (McGraw-Hill, New York, 1978), pp. 244–249.
  21. J. D. Jackson, Classical Electrodynamics (Wiley, New York, 1975), p. 131.
  22. G. A. Korn, Mathematical Handbook for Scientists and Engineers (McGraw-Hill, New York, 1961), p. 732.
  23. E. R. Kretzmer, “Statistics of television signals,” Bell Syst. Tech. J. 31, 751–763 (1952).
  24. A. Gray and G. B. Mathews, A Treatise on Bessel Functions and Their Applications to Physics (Macmillan, London, 1931).
  25. Y.-N. Hsu, H. H. Arsenault, and G. April, “Rotation-invariant digital pattern recognition using circular harmonic expansion,” Appl. Opt. 21, 4012–4019 (1982).
  26. H. H. Arsenault and Y. Sheng, “Properties of the circular harmonic expansion for rotation-invariant pattern recognition,” Appl. Opt. 25, 3225–3229 (1986).
  27. H. Arsenault, L. Leclerc, G. April, V. Francois, A. Bergeron, and Y. Sheng, “Optical implementation of high-speed pattern recognition,” in Optical Pattern Recognition II, H. J. Caulfield, ed., Proc. SPIE 1134, 186–190 (1989).
  28. G. N. Watson, A Treatise on the Theory of Bessel Functions (Cambridge U. Press, Cambridge, 1966), pp. 596–602.

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.

« Previous Article  |  Next Article »

OSA is a member of CrossRef.

CrossCheck Deposited