Abstract

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

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References

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  1. M. Born, 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).
    [CrossRef]
  3. M. R. Teague, “Image analysis via the general theory of moments,” J. Opt. Soc. Am. 70, 920–930 (1980).
    [CrossRef]
  4. C.-H. Teh, R. T. Chin, “On image analysis by the methods of moments,” IEEE Trans. Pattern Anal. Mach. Intell. 10, 496–513 (1988).
    [CrossRef]
  5. Y. Sheng, L. Shen, “Orthogonal Fourier–Mellin moments for invariant pattern recognition,” J. Opt. Soc. Am. A 11, 1748–1757 (1994).
    [CrossRef]
  6. A. Papoulis, Systems and Transforms with Applications in Optics (McGraw-Hill, New York, 1968), pp. 40–44.
  7. A. B. Bhatia, E. Wolf, “On the circular polynomials of Zernike and related orthogonal sets,” Proc. Cambridge Philos. Soc. 50, 40–48 (1954).
    [CrossRef]
  8. A. Khotanzad, Y. H. Hong, “Invariant image recognition by Zernike moments,” IEEE Trans. Pattern Anal. Mach. Intell. 12, 489–497 (1990).
    [CrossRef]
  9. A. Khotanzad, J.-H. Lu, “Classification of invariant image representations using a neural network,” IEEE Trans. Acoust. Speech Signal Process. 38, 1028–1038 (1990).
    [CrossRef]
  10. M. R. Azimi-Sadjadi, S. A. Stricker, “Detection and classification of buried dielectric anomalies using neural networks—further results,” IEEE Trans. Instrum. Meas. 43, 34–39 (1994).
    [CrossRef]
  11. J. Wood, “Invariant pattern-recognition—a review,” Pattern Recogn. 29, 1–17 (1996).
    [CrossRef]
  12. M. O. Freeman, B. E. A. Saleh, “Moment invariants in the space and frequency domains,” J. Opt. Soc. Am. A 5, 1073–1084 (1988).
    [CrossRef] [PubMed]
  13. M. I. Heywood, 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, D. Psaltis, “Recognitive aspects of moment invariants,” IEEE Trans. Pattern Anal. Mach. Intell. PAMI-6, 698–706 (1984).
    [CrossRef]
  15. C.-H. Teh, R. T. Chin, “On digital approximation of moment invariants,” Comput. Vis. Graph. Image Process. 33, 318–326 (1986).
    [CrossRef]
  16. Å. Wallin, 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).
    [CrossRef]
  17. J. Bigün, 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).
    [CrossRef]
  18. R. Kakarala, J. A. Cadzow, “Estimation of phase for noisy linear-phase signals,” IEEE Trans. Signal Process. 44, 2483–2497 (1996).
    [CrossRef]
  19. R. Courant, 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).
    [CrossRef]
  24. A. Gray, G. B. Mathews, A Treatise on Bessel Functions and Their Applications to Physics (Macmillan, London, 1931).
  25. Y.-N. Hsu, H. H. Arsenault, G. April, “Rotation-invariant digital pattern recognition using circular harmonic expansion,” Appl. Opt. 21, 4012–4019 (1982).
    [CrossRef] [PubMed]
  26. H. H. Arsenault, Y. Sheng, “Properties of the circular harmonic expansion for rotation-invariant pattern recognition,” Appl. Opt. 25, 3225–3229 (1986).
    [CrossRef] [PubMed]
  27. H. Arsenault, L. Leclerc, G. April, V. Francois, A. Bergeron, Y. Sheng, “Optical implementation of high-speed pattern recognition,” in Optical Pattern Recognition II, H. J. Caulfield, ed., Proc. SPIE1134, 186–190 (1989).
    [CrossRef]
  28. G. N. Watson, A Treatise on the Theory of Bessel Functions (Cambridge U. Press, Cambridge, 1966), pp. 596–602.

1996 (2)

J. Wood, “Invariant pattern-recognition—a review,” Pattern Recogn. 29, 1–17 (1996).
[CrossRef]

R. Kakarala, J. A. Cadzow, “Estimation of phase for noisy linear-phase signals,” IEEE Trans. Signal Process. 44, 2483–2497 (1996).
[CrossRef]

1995 (2)

Å. Wallin, 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).
[CrossRef]

M. I. Heywood, P. D. Noakes, “Fractional central moment method for moment-invariant object classification,” IEE Proc. Vision Image Signal Process. 142, 213–219 (1995).

1994 (3)

J. Bigün, 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).
[CrossRef]

Y. Sheng, L. Shen, “Orthogonal Fourier–Mellin moments for invariant pattern recognition,” J. Opt. Soc. Am. A 11, 1748–1757 (1994).
[CrossRef]

M. R. Azimi-Sadjadi, S. A. Stricker, “Detection and classification of buried dielectric anomalies using neural networks—further results,” IEEE Trans. Instrum. Meas. 43, 34–39 (1994).
[CrossRef]

1990 (2)

A. Khotanzad, Y. H. Hong, “Invariant image recognition by Zernike moments,” IEEE Trans. Pattern Anal. Mach. Intell. 12, 489–497 (1990).
[CrossRef]

A. Khotanzad, J.-H. Lu, “Classification of invariant image representations using a neural network,” IEEE Trans. Acoust. Speech Signal Process. 38, 1028–1038 (1990).
[CrossRef]

1988 (2)

C.-H. Teh, R. T. Chin, “On image analysis by the methods of moments,” IEEE Trans. Pattern Anal. Mach. Intell. 10, 496–513 (1988).
[CrossRef]

M. O. Freeman, B. E. A. Saleh, “Moment invariants in the space and frequency domains,” J. Opt. Soc. Am. A 5, 1073–1084 (1988).
[CrossRef] [PubMed]

1986 (2)

C.-H. Teh, R. T. Chin, “On digital approximation of moment invariants,” Comput. Vis. Graph. Image Process. 33, 318–326 (1986).
[CrossRef]

H. H. Arsenault, Y. Sheng, “Properties of the circular harmonic expansion for rotation-invariant pattern recognition,” Appl. Opt. 25, 3225–3229 (1986).
[CrossRef] [PubMed]

1984 (1)

Y. S. Abu-Mostafa, D. Psaltis, “Recognitive aspects of moment invariants,” IEEE Trans. Pattern Anal. Mach. Intell. PAMI-6, 698–706 (1984).
[CrossRef]

1982 (1)

1980 (1)

1976 (1)

1954 (1)

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

1952 (1)

E. R. Kretzmer, “Statistics of television signals,” Bell Syst. Tech. J. 31, 751–763 (1952).
[CrossRef]

Abu-Mostafa, Y. S.

Y. S. Abu-Mostafa, D. Psaltis, “Recognitive aspects of moment invariants,” IEEE Trans. Pattern Anal. Mach. Intell. PAMI-6, 698–706 (1984).
[CrossRef]

April, G.

Y.-N. Hsu, H. H. Arsenault, G. April, “Rotation-invariant digital pattern recognition using circular harmonic expansion,” Appl. Opt. 21, 4012–4019 (1982).
[CrossRef] [PubMed]

H. Arsenault, L. Leclerc, G. April, V. Francois, A. Bergeron, Y. Sheng, “Optical implementation of high-speed pattern recognition,” in Optical Pattern Recognition II, H. J. Caulfield, ed., Proc. SPIE1134, 186–190 (1989).
[CrossRef]

Arsenault, H.

H. Arsenault, L. Leclerc, G. April, V. Francois, A. Bergeron, Y. Sheng, “Optical implementation of high-speed pattern recognition,” in Optical Pattern Recognition II, H. J. Caulfield, ed., Proc. SPIE1134, 186–190 (1989).
[CrossRef]

Arsenault, H. H.

Azimi-Sadjadi, M. R.

M. R. Azimi-Sadjadi, S. A. Stricker, “Detection and classification of buried dielectric anomalies using neural networks—further results,” IEEE Trans. Instrum. Meas. 43, 34–39 (1994).
[CrossRef]

Bergeron, A.

H. Arsenault, L. Leclerc, G. April, V. Francois, A. Bergeron, Y. Sheng, “Optical implementation of high-speed pattern recognition,” in Optical Pattern Recognition II, H. J. Caulfield, ed., Proc. SPIE1134, 186–190 (1989).
[CrossRef]

Bhatia, A. B.

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

Bigün, J.

J. Bigün, 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).
[CrossRef]

Born, M.

M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1980).

Bracewell, R. M.

R. M. Bracewell, The Fourier Transform and Its Applications (McGraw-Hill, New York, 1978), pp. 244–249.

Cadzow, J. A.

R. Kakarala, J. A. Cadzow, “Estimation of phase for noisy linear-phase signals,” IEEE Trans. Signal Process. 44, 2483–2497 (1996).
[CrossRef]

Chin, R. T.

C.-H. Teh, R. T. Chin, “On image analysis by the methods of moments,” IEEE Trans. Pattern Anal. Mach. Intell. 10, 496–513 (1988).
[CrossRef]

C.-H. Teh, R. T. Chin, “On digital approximation of moment invariants,” Comput. Vis. Graph. Image Process. 33, 318–326 (1986).
[CrossRef]

Courant, R.

R. Courant, D. Hilbert, Methods of Mathematical Physics (Interscience, New York, 1953), Vol. 1, pp. 297–306.

du Buf, J. M. H.

J. Bigün, 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).
[CrossRef]

Francois, V.

H. Arsenault, L. Leclerc, G. April, V. Francois, A. Bergeron, Y. Sheng, “Optical implementation of high-speed pattern recognition,” in Optical Pattern Recognition II, H. J. Caulfield, ed., Proc. SPIE1134, 186–190 (1989).
[CrossRef]

Freeman, M. O.

Gray, A.

A. Gray, G. B. Mathews, A Treatise on Bessel Functions and Their Applications to Physics (Macmillan, London, 1931).

Heywood, M. I.

M. I. Heywood, P. D. Noakes, “Fractional central moment method for moment-invariant object classification,” IEE Proc. Vision Image Signal Process. 142, 213–219 (1995).

Hilbert, D.

R. Courant, D. Hilbert, Methods of Mathematical Physics (Interscience, New York, 1953), Vol. 1, pp. 297–306.

Hong, Y. H.

A. Khotanzad, Y. H. Hong, “Invariant image recognition by Zernike moments,” IEEE Trans. Pattern Anal. Mach. Intell. 12, 489–497 (1990).
[CrossRef]

Hsu, Y.-N.

Jackson, J. D.

J. D. Jackson, Classical Electrodynamics (Wiley, New York, 1975), p. 131.

Kakarala, R.

R. Kakarala, J. A. Cadzow, “Estimation of phase for noisy linear-phase signals,” IEEE Trans. Signal Process. 44, 2483–2497 (1996).
[CrossRef]

Khotanzad, A.

A. Khotanzad, Y. H. Hong, “Invariant image recognition by Zernike moments,” IEEE Trans. Pattern Anal. Mach. Intell. 12, 489–497 (1990).
[CrossRef]

A. Khotanzad, J.-H. Lu, “Classification of invariant image representations using a neural network,” IEEE Trans. Acoust. Speech Signal Process. 38, 1028–1038 (1990).
[CrossRef]

Korn, G. A.

G. A. Korn, Mathematical Handbook for Scientists and Engineers (McGraw-Hill, New York, 1961), p. 732.

Kretzmer, E. R.

E. R. Kretzmer, “Statistics of television signals,” Bell Syst. Tech. J. 31, 751–763 (1952).
[CrossRef]

Kübler, O.

Å. Wallin, 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).
[CrossRef]

Leclerc, L.

H. Arsenault, L. Leclerc, G. April, V. Francois, A. Bergeron, Y. Sheng, “Optical implementation of high-speed pattern recognition,” in Optical Pattern Recognition II, H. J. Caulfield, ed., Proc. SPIE1134, 186–190 (1989).
[CrossRef]

Lu, J.-H.

A. Khotanzad, J.-H. Lu, “Classification of invariant image representations using a neural network,” IEEE Trans. Acoust. Speech Signal Process. 38, 1028–1038 (1990).
[CrossRef]

Mathews, G. B.

A. Gray, G. B. Mathews, A Treatise on Bessel Functions and Their Applications to Physics (Macmillan, London, 1931).

Noakes, P. D.

M. I. Heywood, P. D. Noakes, “Fractional central moment method for moment-invariant object classification,” IEE Proc. Vision Image Signal Process. 142, 213–219 (1995).

Noll, R. J.

Papoulis, A.

A. Papoulis, Systems and Transforms with Applications in Optics (McGraw-Hill, New York, 1968), pp. 40–44.

Psaltis, D.

Y. S. Abu-Mostafa, D. Psaltis, “Recognitive aspects of moment invariants,” IEEE Trans. Pattern Anal. Mach. Intell. PAMI-6, 698–706 (1984).
[CrossRef]

Saleh, B. E. A.

Shen, L.

Sheng, Y.

Y. Sheng, L. Shen, “Orthogonal Fourier–Mellin moments for invariant pattern recognition,” J. Opt. Soc. Am. A 11, 1748–1757 (1994).
[CrossRef]

H. H. Arsenault, Y. Sheng, “Properties of the circular harmonic expansion for rotation-invariant pattern recognition,” Appl. Opt. 25, 3225–3229 (1986).
[CrossRef] [PubMed]

H. Arsenault, L. Leclerc, G. April, V. Francois, A. Bergeron, Y. Sheng, “Optical implementation of high-speed pattern recognition,” in Optical Pattern Recognition II, H. J. Caulfield, ed., Proc. SPIE1134, 186–190 (1989).
[CrossRef]

Stricker, S. A.

M. R. Azimi-Sadjadi, S. A. Stricker, “Detection and classification of buried dielectric anomalies using neural networks—further results,” IEEE Trans. Instrum. Meas. 43, 34–39 (1994).
[CrossRef]

Teague, M. R.

Teh, C.-H.

C.-H. Teh, R. T. Chin, “On image analysis by the methods of moments,” IEEE Trans. Pattern Anal. Mach. Intell. 10, 496–513 (1988).
[CrossRef]

C.-H. Teh, R. T. Chin, “On digital approximation of moment invariants,” Comput. Vis. Graph. Image Process. 33, 318–326 (1986).
[CrossRef]

Wallin, Å.

Å. Wallin, 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).
[CrossRef]

Watson, G. N.

G. N. Watson, A Treatise on the Theory of Bessel Functions (Cambridge U. Press, Cambridge, 1966), pp. 596–602.

Wolf, E.

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

M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1980).

Wood, J.

J. Wood, “Invariant pattern-recognition—a review,” Pattern Recogn. 29, 1–17 (1996).
[CrossRef]

Appl. Opt. (2)

Bell Syst. Tech. J. (1)

E. R. Kretzmer, “Statistics of television signals,” Bell Syst. Tech. J. 31, 751–763 (1952).
[CrossRef]

Comput. Vis. Graph. Image Process. (1)

C.-H. Teh, R. T. Chin, “On digital approximation of moment invariants,” Comput. Vis. Graph. Image Process. 33, 318–326 (1986).
[CrossRef]

IEE Proc. Vision Image Signal Process (1)

M. I. Heywood, P. D. Noakes, “Fractional central moment method for moment-invariant object classification,” IEE Proc. Vision Image Signal Process. 142, 213–219 (1995).

IEEE Trans. Acoust. Speech Signal Process. (1)

A. Khotanzad, J.-H. Lu, “Classification of invariant image representations using a neural network,” IEEE Trans. Acoust. Speech Signal Process. 38, 1028–1038 (1990).
[CrossRef]

IEEE Trans. Instrum. Meas. (1)

M. R. Azimi-Sadjadi, S. A. Stricker, “Detection and classification of buried dielectric anomalies using neural networks—further results,” IEEE Trans. Instrum. Meas. 43, 34–39 (1994).
[CrossRef]

IEEE Trans. Pattern Anal. Mach. Intell. (4)

C.-H. Teh, R. T. Chin, “On image analysis by the methods of moments,” IEEE Trans. Pattern Anal. Mach. Intell. 10, 496–513 (1988).
[CrossRef]

Å. Wallin, 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).
[CrossRef]

Y. S. Abu-Mostafa, D. Psaltis, “Recognitive aspects of moment invariants,” IEEE Trans. Pattern Anal. Mach. Intell. PAMI-6, 698–706 (1984).
[CrossRef]

A. Khotanzad, Y. H. Hong, “Invariant image recognition by Zernike moments,” IEEE Trans. Pattern Anal. Mach. Intell. 12, 489–497 (1990).
[CrossRef]

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

J. Bigün, 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).
[CrossRef]

IEEE Trans. Signal Process. (1)

R. Kakarala, J. A. Cadzow, “Estimation of phase for noisy linear-phase signals,” IEEE Trans. Signal Process. 44, 2483–2497 (1996).
[CrossRef]

J. Opt. Soc. Am. (2)

J. Opt. Soc. Am. A (2)

Pattern Recogn. (1)

J. Wood, “Invariant pattern-recognition—a review,” Pattern Recogn. 29, 1–17 (1996).
[CrossRef]

Proc. Cambridge Philos. Soc. (1)

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

Other (9)

M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1980).

A. Papoulis, Systems and Transforms with Applications in Optics (McGraw-Hill, New York, 1968), pp. 40–44.

R. Courant, D. Hilbert, Methods of Mathematical Physics (Interscience, New York, 1953), Vol. 1, pp. 297–306.

R. M. Bracewell, The Fourier Transform and Its Applications (McGraw-Hill, New York, 1978), pp. 244–249.

J. D. Jackson, Classical Electrodynamics (Wiley, New York, 1975), p. 131.

G. A. Korn, Mathematical Handbook for Scientists and Engineers (McGraw-Hill, New York, 1961), p. 732.

H. Arsenault, L. Leclerc, G. April, V. Francois, A. Bergeron, Y. Sheng, “Optical implementation of high-speed pattern recognition,” in Optical Pattern Recognition II, H. J. Caulfield, ed., Proc. SPIE1134, 186–190 (1989).
[CrossRef]

G. N. Watson, A Treatise on the Theory of Bessel Functions (Cambridge U. Press, Cambridge, 1966), pp. 596–602.

A. Gray, G. B. Mathews, A Treatise on Bessel Functions and Their Applications to Physics (Macmillan, London, 1931).

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Figures (23)

Fig. 1
Fig. 1

Images of the real parts of some FDH’s, where m=0,±1,±2,  ,±7 downward and n=0, 1, 2,  7 across.

Fig. 2
Fig. 2

Same as Fig. 1, but for ZP’s.

Fig. 3
Fig. 3

Same as Fig. 1, but for PZP’s.

Fig. 4
Fig. 4

Images of the real parts of some FDH’s, ordered so that SICnm is decreasing. SICnm decreases across and then downward.

Fig. 5
Fig. 5

Same as Fig. 4, but for ZP’s.

Fig. 6
Fig. 6

Same as Fig. 4, but for PZP’s.

Fig. 7
Fig. 7

Statistical NIRE’s for FDHC’s (solid curve) and ZM’s (dashed curve).

Fig. 8
Fig. 8

Percent reduction in statistical NIRE when FDHC’s are used in place of ZM’s.

Fig. 9
Fig. 9

NIRE plots for the 64×64-pixel-resolution character A. Three sets of moments or coefficients are compared: FDHC’s (solid curve), PZM’s (dashed curve), and ZM’s (dotted curve).

Fig. 10
Fig. 10

Same as Fig. 9, but for 32×32-pixel resolution.

Fig. 11
Fig. 11

Same as Fig. 9, but for 16×16-pixel resolution.

Fig. 12
Fig. 12

Reconstructions of the 64×64-pixel-resolution character A. From left to right are the original image and reconstructions using 50, 100, 150, and 200 moments or coefficients, respectively. FDHC’s, PZM’s, and ZM’s are used in the top, middle, and bottom rows, respectively.

Fig. 13
Fig. 13

Same as Fig. 12, but for 32×32-pixel resolution.

Fig. 14
Fig. 14

Reconstructions of the 16×16-pixel-resolution character A. From left to right are the original image and reconstructions using 50, 100, and 150 moments or coefficients, respectively. FDHC’s, PZM’s, and ZM’s are used in the top, middle, and bottom rows, respectively.

Fig. 15
Fig. 15

Expected NIRE plots for the 64×64-pixel-resolution noisy character A. Three sets of moments or coefficients are compared: FDHC’s (solid curve), PZM’s (dashed curve), and ZM’s (dotted curve).

Fig. 16
Fig. 16

Same as Fig. 15, but for 32×32-pixel resolution.

Fig. 17
Fig. 17

Same as Fig. 15, but for 16×16-pixel resolution.

Fig. 18
Fig. 18

Reconstructions of the 64×64-pixel-resolution noisy character A. From left to right are the original image and reconstructions using 50, 100, 150, and 200 moments or coefficients, respectively. FDHC’s, PZM’s, and ZM’s are used in the top, middle, and bottom rows, respectively.

Fig. 19
Fig. 19

Same as Fig. 18, but for 32×32-pixel resolution.

Fig. 20
Fig. 20

Reconstructions of the 16×16-pixel-resolution noisy character A. From left to right are the original image and reconstructions using 50, 100, and 150 moments or coefficients, respectively. FDHC’s, PZM’s, and ZM’s are used in the top, middle, and bottom rows, respectively.

Fig. 21
Fig. 21

Expected NIRE plots for the discrete white noise at a resolution of 64×64 pixels. Three sets of moments or coefficients are compared: FDHC’s (solid curve), PZM’s (dashed curve), and ZM’s (dotted curve).

Fig. 22
Fig. 22

Same as Fig. 21, but for a resolution of 32×32 pixels.

Fig. 23
Fig. 23

Same as Fig. 21, but for a resolution of 16×16 pixels.

Tables (2)

Tables Icon

Table 1 Roots of Bessel Functions and Their First Derivatives

Tables Icon

Table 2 First 19 FDHC’s, Ordered So That SICnm Is Decreasing

Equations (45)

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

2=2x2+2y2.
2g=-λg,λ0,
guv(x, y)=exp[iπ(ux+vy)],
dnm(r, θ)=Jm(2πlnmr)exp(imθ),
2π01Jm(2πlnmr)Jm(2πlνmr)r dr=anmδnν,
Dnm(ρ, ϕ)=02π0dnm(r, θ)×exp[-i2πρr cos(θ-ϕ)]r drdθ.
2πi-mJm(r)=02π exp(-ir cos ψ+imψ)dψ
Dnm(ρ, ϕ)=2π exp[im(ϕ-π/2)]×0Jm(2πlnmr)Jm(2πρr)r dr.
|Dnm(ρ, ϕ)|=2π0Jm(2πlnmr)Jm(2πρr)r dr.
|Dnm(ρ, ϕ)|=δ(ρ-lnm)/2πρ,ρ, lnm00,ρ=0andlnm0δ(u, v),lnm=0,
Cff(x, y, u, v)=Cff(0, 0)×exp{-α[(x-u)2+(y-v)2]1/2},
Cff(0, 0)=E{[f(x, y)]2}
Cff(ξ, η)=Cff(0, 0)exp[-α(ξ2+μ2)1/2],
Cff(R)=Cff(0, 0)exp(-αR),
Sff(ρ)=2παCff(0, 0)(4π2ρ2+α2)-3/2.
2π01Jm(2πlnmr)Jm(2πlνmr)r dr=lνmJm(2πlnm)Jm(2πlνm)-lnmJm(2πlνm)Jm(2πlnm)lnm2-lνm2,νnπ[Jm(2πlnm)]2+1-m2πlnm2[Jm(2πlnm)]2,ν=n0,π,ν=n=0
[lνmJm(2πlnm)Jm(2πlνm)-lnmJm(2πlνm)Jm(2πlnm)]/(lnm2-lνm2)=0,
anm=π{[Jm(2πlnm)]2+[1-(m/2πlnm)2]×[Jm(2πlnm)]2},n0,
a00=π.
dnm(r, θ)=Jm(2πlnmr)exp(imθ),0r1,
f(r, θ)=m=-n=0BnmJm(2πlnmr)exp(imθ),0r1.
Bnm=1anm02π01f(r, θ)Jm(2πlnmr)exp(-imθ)r drdθ,
anm=π[1-(m/2πlnm)2][Jm(2πlnm)]2
Bnm=12πimanm02πF(lnm, ϕ)exp(-imϕ)dϕ,lnm0F(0, 0)/πlnm=0,
Vnm(r, θ)=Rnm(r)exp(imθ),
Rnm=s=0(n-|m|)/2 (-1)s(n-s)!rn-2ss!n+|m|2-s!n-|m|2-s!,
n-|m|=even,|m|n.
Rnm=s=0n-|m| (-1)s(2n+1-s)!rn-ss!(n-|m|-s)!(n+|m|+1-s)!,
SICnm=anm Var{Bnm}/Cff(0, 0),
fˆ(r, θ)=λ=1NBλdλ(r, θ),
f(r, θ)=λ=1Bλdλ(r, θ).
2¯(N)=E02π01[f(r, θ)-fˆ(r, θ)]2r drdθE02π01[f(r, θ)]2r drdθ.
2¯(N)=1-λ=1Naλ Var{Bλ}πCff(0, 0)
=1-1πλ=1NSICλ,
SICλ=aλCff(0, 0)02π0102π01Cff(r, θ, ρ, ϕ)×Jm(2πlλr)Jm(2πlλρ)×cos[m(θ-ϕ)]r drdθ ρ dρdϕ
=aλ01Jm(2πlλρ)×02π01exp[-α(r2+ρ2-2rρ cos ψ)1/2]×Jm(2πlλr)cos(mψ)r drdψ ρ dρ,
SNR=1πσ202π01[f(r, θ)]2r drdθ,
f(r, θ)=12πm=-am(r)exp(jmθ),
am(r)=02πf(r, θ)exp(-jmθ)dθ.
am(r)=2πn=1BnmJm(2πlnmr),0r1.
amN(r)=2πn=1NBnmJm(2πlnmr)
amN(r)=01xam(x)n=1N 2Jm(2πlnmx)Jm(2πlnmr)[1-(m/2πlnm)2][Jm(2πlnm)]2dx.
amN(r)=01xam(x)SN(r, x; 0)dx,
SN(r, x; 0)=n=1N 2Jm(2πlnmx)Jm(2πlnmr)[1-(m/2πlnm)2][Jm(2πlnm)]2.
xSN(r, x; 0)δ(r-x),N,

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