Abstract

The discrete orthogonal moments are powerful descriptors for image analysis and pattern recognition. However, the computation of these moments is a time consuming procedure. To solve this problem, a new approach that permits the fast computation of Hahn’s discrete orthogonal moments is presented in this paper. The proposed method is based, on the one hand, on the computation of Hahn’s discrete orthogonal polynomials using the recurrence relation with respect to the variable x instead of the order n and the symmetry property of Hahn’s polynomials and, on the other hand, on the application of an innovative image representation where the image is described by a number of homogenous rectangular blocks instead of individual pixels. The paper also proposes a new set of Hahn’s invariant moments under the translation, the scaling, and the rotation of the image. This set of invariant moments is computed as a linear combination of invariant geometric moments from a finite number of image intensity slices. Several experiments are performed to validate the effectiveness of our descriptors in terms of the acceleration of time computation, the reconstruction of the image, the invariability, and the classification. The performance of Hahn’s moment invariants used as pattern features for a pattern classification application is compared with Hu [IRE Trans. Inform. Theory 8, 179 (1962)] and Krawchouk [IEEE Trans. Image Process. 12, 1367 (2003)] moment invariants.

© 2013 Optical Society of America

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  13. P. T. Yap, P. Raveendran, and S. H. Ong, “Image analysis using Hahn moments,” IEEE Trans. Pattern Anal. Mach. Intell. 29, 2057–2062 (2007).
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  14. M. Sayyouri, A. Hmimd, and H. Qjidaa, “A fast computation of Hahn moments for binary and gray-scale images,” in IEEE International Conference on Complex Systems (ICCS’12), Agadir, Morocco, November5–6, 2012.
  15. L. Yang and F. Albregtsen, “Fast and exact computation of Cartesian geometric moments using discrete Green’s theorem,” Pattern Recogn. 29, 1069–1073 (1996).
  16. I. M. Spiliotis and B. G. Mertzios, “Real-time computation of two-dimensional moments on binary images using image block representation,” IEEE Trans. Image Process. 7, 1609–1615 (1998).
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  17. K. M. Hosny, “Exact and fast computation of geometric moments for gray level images,” Appl. Math. Comput. 189, 1214–1222 (2007).
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  18. H. Z. Shu, H. Zhang, B. J. Chen, P. Haigron, and L. M. Luo, “Fast computation of Tchebichef moments for binary and gray-scale images,” IEEE Trans. Image Process. 19, 3171–3180 (2010).
    [CrossRef]
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    [CrossRef]
  21. G. A. Papakostas, E. G. Karakasis, and D. E. Koulouriotis, “Accurate and speedy computation of image Legendre moments for computer vision applications,” Image Vision Comput. 28, 414–423 (2010).
    [CrossRef]
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    [CrossRef]
  23. C.-W. Chong, P. Raveendran, and R. Mukundan, “Translation invariants of Zernike moments,” Pattern Recogn. 36, 1765–1773 (2003).
    [CrossRef]
  24. C.-W. Chong, P. Raveendran, and R. Mukundan, “Translation and scale invariants of Legendre moments,” Pattern Recogn. 37, 119–129 (2004).
    [CrossRef]
  25. H. Zhu, H. Shu, T. Xia, L. Luo, and J. L. Coatrieux, “Translation and scale invariants of Tchebichef moments,” Pattern Recogn. 40, 2530–2542 (2007).
    [CrossRef]
  26. G. A. Papakostas, E. G. Karakasis, and D. E. Koulouriotis, “Novel moment invariants for improved classification performance in computer vision applications,” Pattern Recogn. 43, 58–68 (2010).
    [CrossRef]
  27. H. Zhu, H. Shu, J. Zhou, L. Luo, and J. L. Coatrieux, “Image analysis by discrete orthogonal dual Hahn moments,” Pattern Recogn. Lett. 28, 1688–1704 (2007).
    [CrossRef]
  28. E. G. Karakasis, G. A. Papakostas, D. E. Koulouriotis, and V. D. Tourassis, “Generalized dual Hahn moment invariants,” Pattern Recogn. 46, 1998–2014 (2013).
    [CrossRef]
  29. H. Zhang, H. Z. Shu, G. N. Han, G. Coatrieux, L. M. Luo, and J. L. Coatrieux, “Blurred image recognition by Legendre moment invariants,” IEEE Trans. Image Process. 19, 596–611 (2010).
    [CrossRef]
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    [CrossRef]

2013 (1)

E. G. Karakasis, G. A. Papakostas, D. E. Koulouriotis, and V. D. Tourassis, “Generalized dual Hahn moment invariants,” Pattern Recogn. 46, 1998–2014 (2013).
[CrossRef]

2012 (1)

K. M. Hosny, “Fast computation of accurate Gaussian–Hermite moments for image processing applications,” Digit. Signal Process. 22, 476–485 (2012).
[CrossRef]

2011 (2)

C. Lim, B. Honarvar, K. H. Thung, and R. Paramesran, “Fast computation of exact Zernike moments using cascaded digital filters,” Inf. Sci. 181, 3638–3651 (2011).
[CrossRef]

B. Yang, G. Li, H. Zhang, and M. Dai, “Rotation and translation invariants of Gaussian–Hermite moments,” Pattern Recogn. Lett. 32, 1283–1298 (2011).
[CrossRef]

2010 (5)

H. Zhu, M. Liu, H. Shu, H. Zhang, and L. Luo, “General form for obtaining discrete orthogonal moments,” IET Image Process. 4, 335–352 (2010).
[CrossRef]

H. Z. Shu, H. Zhang, B. J. Chen, P. Haigron, and L. M. Luo, “Fast computation of Tchebichef moments for binary and gray-scale images,” IEEE Trans. Image Process. 19, 3171–3180 (2010).
[CrossRef]

G. A. Papakostas, E. G. Karakasis, and D. E. Koulouriotis, “Accurate and speedy computation of image Legendre moments for computer vision applications,” Image Vision Comput. 28, 414–423 (2010).
[CrossRef]

H. Zhang, H. Z. Shu, G. N. Han, G. Coatrieux, L. M. Luo, and J. L. Coatrieux, “Blurred image recognition by Legendre moment invariants,” IEEE Trans. Image Process. 19, 596–611 (2010).
[CrossRef]

G. A. Papakostas, E. G. Karakasis, and D. E. Koulouriotis, “Novel moment invariants for improved classification performance in computer vision applications,” Pattern Recogn. 43, 58–68 (2010).
[CrossRef]

2008 (2)

G. A. Papakostas, E. G. Karakasis, and D. E. Koulourisotis, “Efficient and accurate computation of geometric moments on gray-scale images,” Pattern Recogn. 41, 1895–1904 (2008).
[CrossRef]

H. Lin, J. Si, and G. P. Abousleman, “Orthogonal rotation-invariant moments for digital image processing,” IEEE Trans. Image Process. 17, 272–282 (2008).
[CrossRef]

2007 (4)

P. T. Yap, P. Raveendran, and S. H. Ong, “Image analysis using Hahn moments,” IEEE Trans. Pattern Anal. Mach. Intell. 29, 2057–2062 (2007).
[CrossRef]

K. M. Hosny, “Exact and fast computation of geometric moments for gray level images,” Appl. Math. Comput. 189, 1214–1222 (2007).
[CrossRef]

H. Zhu, H. Shu, J. Zhou, L. Luo, and J. L. Coatrieux, “Image analysis by discrete orthogonal dual Hahn moments,” Pattern Recogn. Lett. 28, 1688–1704 (2007).
[CrossRef]

H. Zhu, H. Shu, T. Xia, L. Luo, and J. L. Coatrieux, “Translation and scale invariants of Tchebichef moments,” Pattern Recogn. 40, 2530–2542 (2007).
[CrossRef]

2004 (2)

C.-W. Chong, P. Raveendran, and R. Mukundan, “Translation and scale invariants of Legendre moments,” Pattern Recogn. 37, 119–129 (2004).
[CrossRef]

C. W. Chong, R. Paramesran, and R. Mukundan, “Translation and scale invariants of Legendre moments,” Pattern Recogn. 37, 119–129 (2004).
[CrossRef]

2003 (2)

P. T. Yap, R. Paramesran, and S. H. Ong, “Image analysis by Krawchouk moments,” IEEE Trans. Image Process. 12, 1367–1377 (2003).
[CrossRef]

C.-W. Chong, P. Raveendran, and R. Mukundan, “Translation invariants of Zernike moments,” Pattern Recogn. 36, 1765–1773 (2003).
[CrossRef]

2001 (1)

R. Mukundan, S. H. Ong, and P. A. Lee, “Image analysis by Tchebichef moments,” IEEE Trans. Image Process. 10, 1357–1364 (2001).
[CrossRef]

1998 (1)

I. M. Spiliotis and B. G. Mertzios, “Real-time computation of two-dimensional moments on binary images using image block representation,” IEEE Trans. Image Process. 7, 1609–1615 (1998).
[CrossRef]

1996 (2)

L. Yang and F. Albregtsen, “Fast and exact computation of Cartesian geometric moments using discrete Green’s theorem,” Pattern Recogn. 29, 1069–1073 (1996).

S. X. Liao and M. Pawlak, “On image analysis by moments,” IEEE Trans. Pattern Anal. Mach. Intell. 18, 254–266 (1996).
[CrossRef]

1993 (1)

J. Flusser, “Pattern recognition by affine moment invariants,” Pattern Recogn. 26, 167–174 (1993).
[CrossRef]

1990 (1)

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

1988 (1)

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

1980 (1)

1962 (1)

M. K. Hu, “Visual pattern recognition by moment invariants,” IRE Trans. Inform. Theory 8, 179–187 (1962).

Abousleman, G. P.

H. Lin, J. Si, and G. P. Abousleman, “Orthogonal rotation-invariant moments for digital image processing,” IEEE Trans. Image Process. 17, 272–282 (2008).
[CrossRef]

Albregtsen, F.

L. Yang and F. Albregtsen, “Fast and exact computation of Cartesian geometric moments using discrete Green’s theorem,” Pattern Recogn. 29, 1069–1073 (1996).

Chen, B. J.

H. Z. Shu, H. Zhang, B. J. Chen, P. Haigron, and L. M. Luo, “Fast computation of Tchebichef moments for binary and gray-scale images,” IEEE Trans. Image Process. 19, 3171–3180 (2010).
[CrossRef]

Chin, R. T.

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

Chong, C. W.

C. W. Chong, R. Paramesran, and R. Mukundan, “Translation and scale invariants of Legendre moments,” Pattern Recogn. 37, 119–129 (2004).
[CrossRef]

Chong, C.-W.

C.-W. Chong, P. Raveendran, and R. Mukundan, “Translation and scale invariants of Legendre moments,” Pattern Recogn. 37, 119–129 (2004).
[CrossRef]

C.-W. Chong, P. Raveendran, and R. Mukundan, “Translation invariants of Zernike moments,” Pattern Recogn. 36, 1765–1773 (2003).
[CrossRef]

Coatrieux, G.

H. Zhang, H. Z. Shu, G. N. Han, G. Coatrieux, L. M. Luo, and J. L. Coatrieux, “Blurred image recognition by Legendre moment invariants,” IEEE Trans. Image Process. 19, 596–611 (2010).
[CrossRef]

Coatrieux, J. L.

H. Zhang, H. Z. Shu, G. N. Han, G. Coatrieux, L. M. Luo, and J. L. Coatrieux, “Blurred image recognition by Legendre moment invariants,” IEEE Trans. Image Process. 19, 596–611 (2010).
[CrossRef]

H. Zhu, H. Shu, J. Zhou, L. Luo, and J. L. Coatrieux, “Image analysis by discrete orthogonal dual Hahn moments,” Pattern Recogn. Lett. 28, 1688–1704 (2007).
[CrossRef]

H. Zhu, H. Shu, T. Xia, L. Luo, and J. L. Coatrieux, “Translation and scale invariants of Tchebichef moments,” Pattern Recogn. 40, 2530–2542 (2007).
[CrossRef]

Dai, M.

B. Yang, G. Li, H. Zhang, and M. Dai, “Rotation and translation invariants of Gaussian–Hermite moments,” Pattern Recogn. Lett. 32, 1283–1298 (2011).
[CrossRef]

Flusser, J.

J. Flusser, “Pattern recognition by affine moment invariants,” Pattern Recogn. 26, 167–174 (1993).
[CrossRef]

Haigron, P.

H. Z. Shu, H. Zhang, B. J. Chen, P. Haigron, and L. M. Luo, “Fast computation of Tchebichef moments for binary and gray-scale images,” IEEE Trans. Image Process. 19, 3171–3180 (2010).
[CrossRef]

Han, G. N.

H. Zhang, H. Z. Shu, G. N. Han, G. Coatrieux, L. M. Luo, and J. L. Coatrieux, “Blurred image recognition by Legendre moment invariants,” IEEE Trans. Image Process. 19, 596–611 (2010).
[CrossRef]

Hmimd, A.

M. Sayyouri, A. Hmimd, and H. Qjidaa, “A fast computation of Charlier moments for binary and gray-scale images,” in The 2nd Edition of the IEEE Colloquium on Information Sciences and Technology (CIST’12), Fez, Morocco, October22–24, 2012.

M. Sayyouri, A. Hmimd, and H. Qjidaa, “A fast computation of Hahn moments for binary and gray-scale images,” in IEEE International Conference on Complex Systems (ICCS’12), Agadir, Morocco, November5–6, 2012.

Honarvar, B.

C. Lim, B. Honarvar, K. H. Thung, and R. Paramesran, “Fast computation of exact Zernike moments using cascaded digital filters,” Inf. Sci. 181, 3638–3651 (2011).
[CrossRef]

Hong, Y. H.

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

Hosny, K. M.

K. M. Hosny, “Fast computation of accurate Gaussian–Hermite moments for image processing applications,” Digit. Signal Process. 22, 476–485 (2012).
[CrossRef]

K. M. Hosny, “Exact and fast computation of geometric moments for gray level images,” Appl. Math. Comput. 189, 1214–1222 (2007).
[CrossRef]

Hu, M. K.

M. K. Hu, “Visual pattern recognition by moment invariants,” IRE Trans. Inform. Theory 8, 179–187 (1962).

Karakasis, E. G.

E. G. Karakasis, G. A. Papakostas, D. E. Koulouriotis, and V. D. Tourassis, “Generalized dual Hahn moment invariants,” Pattern Recogn. 46, 1998–2014 (2013).
[CrossRef]

G. A. Papakostas, E. G. Karakasis, and D. E. Koulouriotis, “Accurate and speedy computation of image Legendre moments for computer vision applications,” Image Vision Comput. 28, 414–423 (2010).
[CrossRef]

G. A. Papakostas, E. G. Karakasis, and D. E. Koulouriotis, “Novel moment invariants for improved classification performance in computer vision applications,” Pattern Recogn. 43, 58–68 (2010).
[CrossRef]

G. A. Papakostas, E. G. Karakasis, and D. E. Koulourisotis, “Efficient and accurate computation of geometric moments on gray-scale images,” Pattern Recogn. 41, 1895–1904 (2008).
[CrossRef]

Khotanzad, A.

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

Koekoek, R.

R. Koekoek, P. A. Lesky, and R. F. Swarttouw, Hypergeometric Orthogonal Polynomials and Their q-Analogues, Springer Monographs in Mathematics, Library of Congress Control Number: 2010923797 (Springer-Verlag, 2010).

Koulouriotis, D. E.

E. G. Karakasis, G. A. Papakostas, D. E. Koulouriotis, and V. D. Tourassis, “Generalized dual Hahn moment invariants,” Pattern Recogn. 46, 1998–2014 (2013).
[CrossRef]

G. A. Papakostas, E. G. Karakasis, and D. E. Koulouriotis, “Novel moment invariants for improved classification performance in computer vision applications,” Pattern Recogn. 43, 58–68 (2010).
[CrossRef]

G. A. Papakostas, E. G. Karakasis, and D. E. Koulouriotis, “Accurate and speedy computation of image Legendre moments for computer vision applications,” Image Vision Comput. 28, 414–423 (2010).
[CrossRef]

Koulourisotis, D. E.

G. A. Papakostas, E. G. Karakasis, and D. E. Koulourisotis, “Efficient and accurate computation of geometric moments on gray-scale images,” Pattern Recogn. 41, 1895–1904 (2008).
[CrossRef]

Lee, P. A.

R. Mukundan, S. H. Ong, and P. A. Lee, “Image analysis by Tchebichef moments,” IEEE Trans. Image Process. 10, 1357–1364 (2001).
[CrossRef]

Lesky, P. A.

R. Koekoek, P. A. Lesky, and R. F. Swarttouw, Hypergeometric Orthogonal Polynomials and Their q-Analogues, Springer Monographs in Mathematics, Library of Congress Control Number: 2010923797 (Springer-Verlag, 2010).

Li, G.

B. Yang, G. Li, H. Zhang, and M. Dai, “Rotation and translation invariants of Gaussian–Hermite moments,” Pattern Recogn. Lett. 32, 1283–1298 (2011).
[CrossRef]

Liao, S. X.

S. X. Liao and M. Pawlak, “On image analysis by moments,” IEEE Trans. Pattern Anal. Mach. Intell. 18, 254–266 (1996).
[CrossRef]

Lim, C.

C. Lim, B. Honarvar, K. H. Thung, and R. Paramesran, “Fast computation of exact Zernike moments using cascaded digital filters,” Inf. Sci. 181, 3638–3651 (2011).
[CrossRef]

Lin, H.

H. Lin, J. Si, and G. P. Abousleman, “Orthogonal rotation-invariant moments for digital image processing,” IEEE Trans. Image Process. 17, 272–282 (2008).
[CrossRef]

Liu, M.

H. Zhu, M. Liu, H. Shu, H. Zhang, and L. Luo, “General form for obtaining discrete orthogonal moments,” IET Image Process. 4, 335–352 (2010).
[CrossRef]

Luo, L.

H. Zhu, M. Liu, H. Shu, H. Zhang, and L. Luo, “General form for obtaining discrete orthogonal moments,” IET Image Process. 4, 335–352 (2010).
[CrossRef]

H. Zhu, H. Shu, T. Xia, L. Luo, and J. L. Coatrieux, “Translation and scale invariants of Tchebichef moments,” Pattern Recogn. 40, 2530–2542 (2007).
[CrossRef]

H. Zhu, H. Shu, J. Zhou, L. Luo, and J. L. Coatrieux, “Image analysis by discrete orthogonal dual Hahn moments,” Pattern Recogn. Lett. 28, 1688–1704 (2007).
[CrossRef]

Luo, L. M.

H. Zhang, H. Z. Shu, G. N. Han, G. Coatrieux, L. M. Luo, and J. L. Coatrieux, “Blurred image recognition by Legendre moment invariants,” IEEE Trans. Image Process. 19, 596–611 (2010).
[CrossRef]

H. Z. Shu, H. Zhang, B. J. Chen, P. Haigron, and L. M. Luo, “Fast computation of Tchebichef moments for binary and gray-scale images,” IEEE Trans. Image Process. 19, 3171–3180 (2010).
[CrossRef]

Mertzios, B. G.

I. M. Spiliotis and B. G. Mertzios, “Real-time computation of two-dimensional moments on binary images using image block representation,” IEEE Trans. Image Process. 7, 1609–1615 (1998).
[CrossRef]

Mukundan, R.

C.-W. Chong, P. Raveendran, and R. Mukundan, “Translation and scale invariants of Legendre moments,” Pattern Recogn. 37, 119–129 (2004).
[CrossRef]

C. W. Chong, R. Paramesran, and R. Mukundan, “Translation and scale invariants of Legendre moments,” Pattern Recogn. 37, 119–129 (2004).
[CrossRef]

C.-W. Chong, P. Raveendran, and R. Mukundan, “Translation invariants of Zernike moments,” Pattern Recogn. 36, 1765–1773 (2003).
[CrossRef]

R. Mukundan, S. H. Ong, and P. A. Lee, “Image analysis by Tchebichef moments,” IEEE Trans. Image Process. 10, 1357–1364 (2001).
[CrossRef]

R. Mukundan and K. R. Ramakrishnan, Moment Functions in Image Analysis (World Scientific, 1998).

Nikiforov, A. F.

A. F. Nikiforov, S. K. Suslov, and V. B. Uvarov, Classical Orthogonal Polynomials of a Discrete Variable (Springer, 1991).

Ong, S. H.

P. T. Yap, P. Raveendran, and S. H. Ong, “Image analysis using Hahn moments,” IEEE Trans. Pattern Anal. Mach. Intell. 29, 2057–2062 (2007).
[CrossRef]

P. T. Yap, R. Paramesran, and S. H. Ong, “Image analysis by Krawchouk moments,” IEEE Trans. Image Process. 12, 1367–1377 (2003).
[CrossRef]

R. Mukundan, S. H. Ong, and P. A. Lee, “Image analysis by Tchebichef moments,” IEEE Trans. Image Process. 10, 1357–1364 (2001).
[CrossRef]

Papakostas, G. A.

E. G. Karakasis, G. A. Papakostas, D. E. Koulouriotis, and V. D. Tourassis, “Generalized dual Hahn moment invariants,” Pattern Recogn. 46, 1998–2014 (2013).
[CrossRef]

G. A. Papakostas, E. G. Karakasis, and D. E. Koulouriotis, “Novel moment invariants for improved classification performance in computer vision applications,” Pattern Recogn. 43, 58–68 (2010).
[CrossRef]

G. A. Papakostas, E. G. Karakasis, and D. E. Koulouriotis, “Accurate and speedy computation of image Legendre moments for computer vision applications,” Image Vision Comput. 28, 414–423 (2010).
[CrossRef]

G. A. Papakostas, E. G. Karakasis, and D. E. Koulourisotis, “Efficient and accurate computation of geometric moments on gray-scale images,” Pattern Recogn. 41, 1895–1904 (2008).
[CrossRef]

Paramesran, R.

C. Lim, B. Honarvar, K. H. Thung, and R. Paramesran, “Fast computation of exact Zernike moments using cascaded digital filters,” Inf. Sci. 181, 3638–3651 (2011).
[CrossRef]

C. W. Chong, R. Paramesran, and R. Mukundan, “Translation and scale invariants of Legendre moments,” Pattern Recogn. 37, 119–129 (2004).
[CrossRef]

P. T. Yap, R. Paramesran, and S. H. Ong, “Image analysis by Krawchouk moments,” IEEE Trans. Image Process. 12, 1367–1377 (2003).
[CrossRef]

Pawlak, M.

S. X. Liao and M. Pawlak, “On image analysis by moments,” IEEE Trans. Pattern Anal. Mach. Intell. 18, 254–266 (1996).
[CrossRef]

Qjidaa, H.

M. Sayyouri, A. Hmimd, and H. Qjidaa, “A fast computation of Hahn moments for binary and gray-scale images,” in IEEE International Conference on Complex Systems (ICCS’12), Agadir, Morocco, November5–6, 2012.

M. Sayyouri, A. Hmimd, and H. Qjidaa, “A fast computation of Charlier moments for binary and gray-scale images,” in The 2nd Edition of the IEEE Colloquium on Information Sciences and Technology (CIST’12), Fez, Morocco, October22–24, 2012.

Ramakrishnan, K. R.

R. Mukundan and K. R. Ramakrishnan, Moment Functions in Image Analysis (World Scientific, 1998).

Raveendran, P.

P. T. Yap, P. Raveendran, and S. H. Ong, “Image analysis using Hahn moments,” IEEE Trans. Pattern Anal. Mach. Intell. 29, 2057–2062 (2007).
[CrossRef]

C.-W. Chong, P. Raveendran, and R. Mukundan, “Translation and scale invariants of Legendre moments,” Pattern Recogn. 37, 119–129 (2004).
[CrossRef]

C.-W. Chong, P. Raveendran, and R. Mukundan, “Translation invariants of Zernike moments,” Pattern Recogn. 36, 1765–1773 (2003).
[CrossRef]

Sayyouri, M.

M. Sayyouri, A. Hmimd, and H. Qjidaa, “A fast computation of Charlier moments for binary and gray-scale images,” in The 2nd Edition of the IEEE Colloquium on Information Sciences and Technology (CIST’12), Fez, Morocco, October22–24, 2012.

M. Sayyouri, A. Hmimd, and H. Qjidaa, “A fast computation of Hahn moments for binary and gray-scale images,” in IEEE International Conference on Complex Systems (ICCS’12), Agadir, Morocco, November5–6, 2012.

Shu, H.

H. Zhu, M. Liu, H. Shu, H. Zhang, and L. Luo, “General form for obtaining discrete orthogonal moments,” IET Image Process. 4, 335–352 (2010).
[CrossRef]

H. Zhu, H. Shu, T. Xia, L. Luo, and J. L. Coatrieux, “Translation and scale invariants of Tchebichef moments,” Pattern Recogn. 40, 2530–2542 (2007).
[CrossRef]

H. Zhu, H. Shu, J. Zhou, L. Luo, and J. L. Coatrieux, “Image analysis by discrete orthogonal dual Hahn moments,” Pattern Recogn. Lett. 28, 1688–1704 (2007).
[CrossRef]

Shu, H. Z.

H. Zhang, H. Z. Shu, G. N. Han, G. Coatrieux, L. M. Luo, and J. L. Coatrieux, “Blurred image recognition by Legendre moment invariants,” IEEE Trans. Image Process. 19, 596–611 (2010).
[CrossRef]

H. Z. Shu, H. Zhang, B. J. Chen, P. Haigron, and L. M. Luo, “Fast computation of Tchebichef moments for binary and gray-scale images,” IEEE Trans. Image Process. 19, 3171–3180 (2010).
[CrossRef]

Si, J.

H. Lin, J. Si, and G. P. Abousleman, “Orthogonal rotation-invariant moments for digital image processing,” IEEE Trans. Image Process. 17, 272–282 (2008).
[CrossRef]

Spiliotis, I. M.

I. M. Spiliotis and B. G. Mertzios, “Real-time computation of two-dimensional moments on binary images using image block representation,” IEEE Trans. Image Process. 7, 1609–1615 (1998).
[CrossRef]

Suslov, S. K.

A. F. Nikiforov, S. K. Suslov, and V. B. Uvarov, Classical Orthogonal Polynomials of a Discrete Variable (Springer, 1991).

Swarttouw, R. F.

R. Koekoek, P. A. Lesky, and R. F. Swarttouw, Hypergeometric Orthogonal Polynomials and Their q-Analogues, Springer Monographs in Mathematics, Library of Congress Control Number: 2010923797 (Springer-Verlag, 2010).

Teague, M. R.

Teh, C. H.

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

Thung, K. H.

C. Lim, B. Honarvar, K. H. Thung, and R. Paramesran, “Fast computation of exact Zernike moments using cascaded digital filters,” Inf. Sci. 181, 3638–3651 (2011).
[CrossRef]

Tourassis, V. D.

E. G. Karakasis, G. A. Papakostas, D. E. Koulouriotis, and V. D. Tourassis, “Generalized dual Hahn moment invariants,” Pattern Recogn. 46, 1998–2014 (2013).
[CrossRef]

Uvarov, V. B.

A. F. Nikiforov, S. K. Suslov, and V. B. Uvarov, Classical Orthogonal Polynomials of a Discrete Variable (Springer, 1991).

Xia, T.

H. Zhu, H. Shu, T. Xia, L. Luo, and J. L. Coatrieux, “Translation and scale invariants of Tchebichef moments,” Pattern Recogn. 40, 2530–2542 (2007).
[CrossRef]

Yang, B.

B. Yang, G. Li, H. Zhang, and M. Dai, “Rotation and translation invariants of Gaussian–Hermite moments,” Pattern Recogn. Lett. 32, 1283–1298 (2011).
[CrossRef]

Yang, L.

L. Yang and F. Albregtsen, “Fast and exact computation of Cartesian geometric moments using discrete Green’s theorem,” Pattern Recogn. 29, 1069–1073 (1996).

Yap, P. T.

P. T. Yap, P. Raveendran, and S. H. Ong, “Image analysis using Hahn moments,” IEEE Trans. Pattern Anal. Mach. Intell. 29, 2057–2062 (2007).
[CrossRef]

P. T. Yap, R. Paramesran, and S. H. Ong, “Image analysis by Krawchouk moments,” IEEE Trans. Image Process. 12, 1367–1377 (2003).
[CrossRef]

Zhang, H.

B. Yang, G. Li, H. Zhang, and M. Dai, “Rotation and translation invariants of Gaussian–Hermite moments,” Pattern Recogn. Lett. 32, 1283–1298 (2011).
[CrossRef]

H. Zhu, M. Liu, H. Shu, H. Zhang, and L. Luo, “General form for obtaining discrete orthogonal moments,” IET Image Process. 4, 335–352 (2010).
[CrossRef]

H. Z. Shu, H. Zhang, B. J. Chen, P. Haigron, and L. M. Luo, “Fast computation of Tchebichef moments for binary and gray-scale images,” IEEE Trans. Image Process. 19, 3171–3180 (2010).
[CrossRef]

H. Zhang, H. Z. Shu, G. N. Han, G. Coatrieux, L. M. Luo, and J. L. Coatrieux, “Blurred image recognition by Legendre moment invariants,” IEEE Trans. Image Process. 19, 596–611 (2010).
[CrossRef]

Zhou, J.

H. Zhu, H. Shu, J. Zhou, L. Luo, and J. L. Coatrieux, “Image analysis by discrete orthogonal dual Hahn moments,” Pattern Recogn. Lett. 28, 1688–1704 (2007).
[CrossRef]

Zhu, H.

H. Zhu, M. Liu, H. Shu, H. Zhang, and L. Luo, “General form for obtaining discrete orthogonal moments,” IET Image Process. 4, 335–352 (2010).
[CrossRef]

H. Zhu, H. Shu, T. Xia, L. Luo, and J. L. Coatrieux, “Translation and scale invariants of Tchebichef moments,” Pattern Recogn. 40, 2530–2542 (2007).
[CrossRef]

H. Zhu, H. Shu, J. Zhou, L. Luo, and J. L. Coatrieux, “Image analysis by discrete orthogonal dual Hahn moments,” Pattern Recogn. Lett. 28, 1688–1704 (2007).
[CrossRef]

Appl. Math. Comput. (1)

K. M. Hosny, “Exact and fast computation of geometric moments for gray level images,” Appl. Math. Comput. 189, 1214–1222 (2007).
[CrossRef]

Digit. Signal Process. (1)

K. M. Hosny, “Fast computation of accurate Gaussian–Hermite moments for image processing applications,” Digit. Signal Process. 22, 476–485 (2012).
[CrossRef]

IEEE Trans. Image Process. (6)

H. Zhang, H. Z. Shu, G. N. Han, G. Coatrieux, L. M. Luo, and J. L. Coatrieux, “Blurred image recognition by Legendre moment invariants,” IEEE Trans. Image Process. 19, 596–611 (2010).
[CrossRef]

H. Z. Shu, H. Zhang, B. J. Chen, P. Haigron, and L. M. Luo, “Fast computation of Tchebichef moments for binary and gray-scale images,” IEEE Trans. Image Process. 19, 3171–3180 (2010).
[CrossRef]

H. Lin, J. Si, and G. P. Abousleman, “Orthogonal rotation-invariant moments for digital image processing,” IEEE Trans. Image Process. 17, 272–282 (2008).
[CrossRef]

R. Mukundan, S. H. Ong, and P. A. Lee, “Image analysis by Tchebichef moments,” IEEE Trans. Image Process. 10, 1357–1364 (2001).
[CrossRef]

P. T. Yap, R. Paramesran, and S. H. Ong, “Image analysis by Krawchouk moments,” IEEE Trans. Image Process. 12, 1367–1377 (2003).
[CrossRef]

I. M. Spiliotis and B. G. Mertzios, “Real-time computation of two-dimensional moments on binary images using image block representation,” IEEE Trans. Image Process. 7, 1609–1615 (1998).
[CrossRef]

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

P. T. Yap, P. Raveendran, and S. H. Ong, “Image analysis using Hahn moments,” IEEE Trans. Pattern Anal. Mach. Intell. 29, 2057–2062 (2007).
[CrossRef]

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

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

S. X. Liao and M. Pawlak, “On image analysis by moments,” IEEE Trans. Pattern Anal. Mach. Intell. 18, 254–266 (1996).
[CrossRef]

IET Image Process. (1)

H. Zhu, M. Liu, H. Shu, H. Zhang, and L. Luo, “General form for obtaining discrete orthogonal moments,” IET Image Process. 4, 335–352 (2010).
[CrossRef]

Image Vision Comput. (1)

G. A. Papakostas, E. G. Karakasis, and D. E. Koulouriotis, “Accurate and speedy computation of image Legendre moments for computer vision applications,” Image Vision Comput. 28, 414–423 (2010).
[CrossRef]

Inf. Sci. (1)

C. Lim, B. Honarvar, K. H. Thung, and R. Paramesran, “Fast computation of exact Zernike moments using cascaded digital filters,” Inf. Sci. 181, 3638–3651 (2011).
[CrossRef]

IRE Trans. Inform. Theory (1)

M. K. Hu, “Visual pattern recognition by moment invariants,” IRE Trans. Inform. Theory 8, 179–187 (1962).

J. Opt. Soc. Am. (1)

Pattern Recogn. (9)

C. W. Chong, R. Paramesran, and R. Mukundan, “Translation and scale invariants of Legendre moments,” Pattern Recogn. 37, 119–129 (2004).
[CrossRef]

L. Yang and F. Albregtsen, “Fast and exact computation of Cartesian geometric moments using discrete Green’s theorem,” Pattern Recogn. 29, 1069–1073 (1996).

E. G. Karakasis, G. A. Papakostas, D. E. Koulouriotis, and V. D. Tourassis, “Generalized dual Hahn moment invariants,” Pattern Recogn. 46, 1998–2014 (2013).
[CrossRef]

G. A. Papakostas, E. G. Karakasis, and D. E. Koulourisotis, “Efficient and accurate computation of geometric moments on gray-scale images,” Pattern Recogn. 41, 1895–1904 (2008).
[CrossRef]

J. Flusser, “Pattern recognition by affine moment invariants,” Pattern Recogn. 26, 167–174 (1993).
[CrossRef]

C.-W. Chong, P. Raveendran, and R. Mukundan, “Translation invariants of Zernike moments,” Pattern Recogn. 36, 1765–1773 (2003).
[CrossRef]

C.-W. Chong, P. Raveendran, and R. Mukundan, “Translation and scale invariants of Legendre moments,” Pattern Recogn. 37, 119–129 (2004).
[CrossRef]

H. Zhu, H. Shu, T. Xia, L. Luo, and J. L. Coatrieux, “Translation and scale invariants of Tchebichef moments,” Pattern Recogn. 40, 2530–2542 (2007).
[CrossRef]

G. A. Papakostas, E. G. Karakasis, and D. E. Koulouriotis, “Novel moment invariants for improved classification performance in computer vision applications,” Pattern Recogn. 43, 58–68 (2010).
[CrossRef]

Pattern Recogn. Lett. (2)

H. Zhu, H. Shu, J. Zhou, L. Luo, and J. L. Coatrieux, “Image analysis by discrete orthogonal dual Hahn moments,” Pattern Recogn. Lett. 28, 1688–1704 (2007).
[CrossRef]

B. Yang, G. Li, H. Zhang, and M. Dai, “Rotation and translation invariants of Gaussian–Hermite moments,” Pattern Recogn. Lett. 32, 1283–1298 (2011).
[CrossRef]

Other (7)

M. Sayyouri, A. Hmimd, and H. Qjidaa, “A fast computation of Hahn moments for binary and gray-scale images,” in IEEE International Conference on Complex Systems (ICCS’12), Agadir, Morocco, November5–6, 2012.

M. Sayyouri, A. Hmimd, and H. Qjidaa, “A fast computation of Charlier moments for binary and gray-scale images,” in The 2nd Edition of the IEEE Colloquium on Information Sciences and Technology (CIST’12), Fez, Morocco, October22–24, 2012.

R. Koekoek, P. A. Lesky, and R. F. Swarttouw, Hypergeometric Orthogonal Polynomials and Their q-Analogues, Springer Monographs in Mathematics, Library of Congress Control Number: 2010923797 (Springer-Verlag, 2010).

A. F. Nikiforov, S. K. Suslov, and V. B. Uvarov, Classical Orthogonal Polynomials of a Discrete Variable (Springer, 1991).

http://www.dabi.temple.edu/~shape/MPEG7/dataset.html .

R. Mukundan and K. R. Ramakrishnan, Moment Functions in Image Analysis (World Scientific, 1998).

http://www.cs.columbia.edu/CAVE/software/softlib/coil-20.php .

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Figures (12)

Fig. 1.
Fig. 1.

Flowchart of the two methods for the computation of Hahn’s moments. (a) The proposed fast method and (b) the direct method.

Fig. 2.
Fig. 2.

Set of test binary images. Spoon, Jar, Horse, Car, and Device.

Fig. 3.
Fig. 3.

Average computation time for binary image using the direct method and the proposed method.

Fig. 4.
Fig. 4.

Set of test gray-scale images. House, Woman, Pepper, Mandrill, and Lake.

Fig. 5.
Fig. 5.

Average computation time for gray-scale images using the direct method and the proposed method.

Fig. 6.
Fig. 6.

(a) Woman gray-scale image, (b) noisy image by salt and pepper, and (c) noisy image by white Gaussian.

Fig. 7.
Fig. 7.

(a) MSE for the binary image of Horse by two methods, (b) MSE for gray-scale image of Woman by two methods, and (c) MSE for gray-scale image of Woman with salt and pepper and white Gaussian noise.

Fig. 8.
Fig. 8.

Set of the transformed binary image character “R” by translation, scaling, and rotation.

Fig. 9.
Fig. 9.

Binary images as a training set for invariant character recognition in the experiment.

Fig. 10.
Fig. 10.

Part of the images of the testing set in the experiment.

Fig. 11.
Fig. 11.

Collection of the COIL-20 objects.

Fig. 12.
Fig. 12.

Set of test binary images (a) Cup, (b) Duck, (c) Box, (d) Cat, and (e) Object.

Tables (9)

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Table 1. Hahn’s Moment Invariants (a=10,b=10), Translation of Character “R”

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Table 2. Hahn’s Moment Invariants (a=10,b=10), Scaling of Character “R”

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Table 3. Hahn’s Moment Invariants (a=10,b=10), Rotation of Character “R”

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Table 4. Hahn’s Moment Invariants (a=10,b=10), Mixed Transformation of Character “R”

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Table 5. Classification Results of the Set of Binary English Characters and Numbers by Using d1 Distance

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Table 6. Classification Results of the Set of Binary English Characters and Numbers by Using d2 Distance

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Table 7. Classification Results of COILL-20 Objects Database by Using d1 Distance

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Table 8. Classification Results of COILL-20 Objects Database by Using d2 Distance

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Table 9. Average Times and Reduction Percentage of Hahn’s Invariant Moments for Binary and Gray-Scale Images

Equations (50)

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

σ(x)Δhn(a,b)(x;N)+τ(x)hn(a,b)(x;N)+λnhn(a,b)(x;N)=0
σ(x)=x(N+ax),τ(x)=(b+1)(N1)(a+b+2)x,andλn=n(a+b+n+1).
hn(a,b)(x;N)=(1)nn!w(x)n[wn(x)]
wn(x)=Γ(N+ax)Γ(n+b+1+x)Γ(Nnx)Γ(x+1),
nf(x)=k=0n(nk)(1)kf(xk).
hn(a,b)(x;N)=(1)n(b+1)n(Nn)nn!F23(n,x,n+1+a+b;b+1,1N;1)=k=0nαk,n(a,b)xk.
F23(a1,a2,a3;b1,b1;x)=k=0(a1)k(a2)k(a3)k(b1)k(b2)kxkk!.
(x)0=1and(x)k=x(x+1)(x+k1);k1.
hn(a,b)(x;N)=k=0n(1)k(nk)!k!(b+n)!(b+k)!(a+b+k+n)!(a+b+n)!(x)k.
x=0N1hn(a,b)(x;N)hm(a,b)(x;N)w(x)=ρ(n)δnm,
ρ(n)=Γ(a+n+1)Γ(b+n+1)(a+b+n+1)N(a+b+2n+1)n!(Nn1)!,
h˜n(a,b)(x,N)=hn(a,b)(x,N)w(x)ρ(n).
w(x+1)=(N1x)(b+1+x)(N+ax1)w(x),withw(0)=Γ(N+a)Γ(b+1)Γ(N).
ρ(n+1)=(a+n+1)(b+n+1)(a+b+n+N+1)(Nn1)(a+b+2n+1)(n+1)(a+b+n+1)(a+b+2n+3)ρ(n),withρ(0)=Γ(a+1)Γ(b+1)Γ(a+b+N+1)(a+b+1)(N1)!Γ(a+b+1).
x=0N1h˜n(a,b)(x;N)h˜m(a,b)(x;N)=δnm.
h˜n(a,b)(x;N)=B×DAh˜n1(a,b)(x;N)C×EAh˜n2(a,b)(x;N).
A=n(a+b+n)(a+b+2n1)(a+b+2n),B=xab+2N24(b2a2)(a+b+2N)4(a+b+2n2)(a+b+2n),C=(a+n1)(b+n1)(a+b+2n2)×(a+b+N+n1)(Nn+1)(a+b+2n1),D=n(a+b+n)(a+b+2n+1)(Nn)(a+n)(b+n)(a+b+2n1)(a+b+n+N),E=n(n1)(a+b+n)(a+n)(a+n1)(b+n)(b+n1)(Nn+1)(Nn)×(a+b+n1)(a+b+2n+1)(a+b+2n3)(a+b+n+N)(a+b+n+N1).
h˜0(a,b)(x,N)=w(x)ρ(0),h˜1(a,b)(x,N)=[(b+1)(N1)+(a+b+2)x]w(x)ρ(1).
ΔPn(x)=Pn(x+1)2Pn(x)+Pn(x1).
h˜n(a,b)(x,N)=w(x)σ(x1)+τ(x1)×[2σ(x1)+τ(x1)λnw(x1)h˜n(a,b)(x1,N)σ(x1)+τ(x1)w(x2)h˜n(a,b)(x2,N)].
h˜n(a,b)(0,N)=(1N)n(n+bn)w(0)ρ(n),h˜n(a,b)(1,N)=(n+b+1)(Nn1)n(N+a1)(b+1)(N1)w(1)w(0)h˜n(a,b)(0,N).
h˜n(a,b)(N1x,N)=(1)nh˜n(b,a)(x,N).
h˜n(a,b)(x,N)=(1)nh˜n(b,a)(N1x,N).
Hnm=x=0M1y=0N1h˜n(a,b)(x,M)h˜m(a,b)(y,N)f(x,y),
f(x,y)={bi,i=0,1,.,K1},
Hnm=i=0k1x=x1,bix2,biy=y1,biy2,bih˜n(a,b)(x;M)h˜n(a,b)(y;N)=i=0k1Hnmbi,
Hnmbi=x=x1,bix2,biy=y1,biy1,bih˜n(a,b)(x;M)h˜n(a,b)(y;N)=x=x1,bix2,bih˜n(a,b)(x;M)y=y1,biy2,bih˜n(a,b)(y;N)=Sn(x1,bi,x2,bi)Sm(y1,bi,y2,bi),
Sn(x1,bi,x2,bi)=x=x1,bix2,bih˜n(a,b)(x;M)andSm(y1,bi,y2,bi)=y=y1,biy2,bih˜n(a,b)(y;N).
f(x,y)=i=1Lfi(x,y),
f(x,y)={fi(x,y),i=1,2,.,L},withfi(x,y)={bij,j=1,2,.,Ki1},
Hnm=x=0M1y=ON1h˜n(a,b)(x;M)h˜n(a,b)(y;N)i=1Lfi(x,y)=i=1Lx=0M1y=ON1h˜n(a,b)(x;M)h˜n(a,b)(y;N)fi(x,y)=i=1LfiHnm(i),
Hnm=x=0M1y=0N1f(x,y)h˜n(a,b)(x;M)h˜m(a,b)(y;N).
f(x,y)=x=0M1y=0N1Hnmh˜n(a,b)(x;M)h˜m(a,b)(y;N).
f^(x,y)=n=0maxm=0lHnm,mh˜nm(a,b)(x;M)h˜m(a,b)(y;N).
MSE=1MNi=1Mj=1N(f(xi,yj)f^(xi,yj))2.
GMnm=x=0N1y=0M1xnymf(x,y).
GMInm=GM00γx=0N1y=0N1[(xx¯)cosθ+(yy¯)sinθ]n×[(yy¯)cosθ(xx¯)sinθ]mf(x,y),
γ=n+m2+1,x¯=GM10GM00,y¯=GM01GM00,andθ=12tan1(2μ11μ20μ02).
μnm=x=0N1y=0M1(xx¯)n(yy¯)mf(x,y).
HMInm=[ρ(n)ρ(m)]1/2i=0nj=0mαi,n(a,b)αj,n(a,b)Vi,j,
Vnm=q=0np=0m(np)(mq)(N×M2)((p+q)/2)+1×(N2)np×(M2)mp×GMIpq.
GMInm=GM00γi=0nj=0m(ni)(mj)(cosθ)i+j(sinθ)n+mij×(1)mjμm+ij,n+ji=i=0nj=0m(ni)(mj)(cosθ)i+j(sinθ)n+mij×(1)mjηm+ij,n+ji,
ηnm=μnmGM00γ.
ηnm=μnmGM00γ=1GM00γx=0N1y=0M1(xx¯)n(yy¯)mf(x,y)=1GM00γx=0N1y=0M1(xx¯)n(yy¯)m(k=1sfk(x,y))=1GM00γk=1Sfk×xk=0N1yk=0M1(xx¯)n(yy¯)m=1GM00γk=1Sfk×j=0k[(xk=x1,bjx2,bj(xx¯)n)(yk=y1,bjy2,bj(yy¯)m)]=1GM00γk=1Sfk×ηnmk,
ηnmk=j=0k[(xk=x1,bjx2,bj(xx¯)n)(yk=y1,bjy2,bj(yy¯)m)],
GMInm=i=0nj=0m(ni)(mj)(cosθ)i+j(sinθ)n+mij×(1)mjηm+ij,n+ji=1GM00γi=0nj=0m(ni)(mj)(cosθ)i+j(sinθ)n+mij×(1)mjk=1Sfk×ηm+ij,n+jik=1GM00γk=1Sfki=0nj=0m(ni)(mj)(cosθ)i+j(sinθ)n+mij×(1)mj×ηm+ij,n+jik.
V=[HMI00,HMI01,HMI10,HMI11,HMI02,HMI20,HMI21,HMI12,HMI22]
d1(x,y)=i=1n(xiyi)2,
d2(x,y)=i=1nxiyii=1n(xi)2i=1n(yi)2.
η=Number of correctly classified imagesThe total of images used in the test×100%.

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