Abstract

A new set, to our knowledge, of orthogonal moment functions for describing images is proposed. It is based on the generalized pseudo-Zernike polynomials that are orthogonal on the unit circle. The generalized pseudo-Zernike polynomials are scaled to ensure numerical stability, and some properties are discussed. The performance of the proposed moments is analyzed in terms of image reconstruction capability and invariant character recognition accuracy. Experimental results demonstrate the superiority of generalized pseudo-Zernike moments compared with pseudo-Zernike and Chebyshev–Fourier moments in both noise-free and noisy conditions.

© 2006 Optical Society of America

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  1. R. J. Prokop and A. P. Reeves, 'A survey of moment-based techniques for unoccluded object representation and recognition,' Comput. Vis. Graph. Image Process. 54, 438-460 (1992).
  2. M. K. Hu, 'Visual pattern recognition by moment invariants,' IRE Trans. Inf. Theory IT-8, 179-187 (1962).
  3. S. Dudani, K. Breeding, and R. McGhee, 'Aircraft identification by moment invariants,' IEEE Trans. Comput. 26, 39-45 (1977).
    [CrossRef]
  4. S. O. Belkasim, M. Shridhar, and M. Ahmadi, 'Pattern recognition with moment invariants: a comparative study and new results,' Pattern Recogn. 24, 1117-1138 (1991).
    [CrossRef]
  5. V. Markandey and R. J. P. Figueiredo, 'Robot sensing techniques based on high-dimensional moment invariants and tensor,' IEEE Trans. Rob. Autom. 8, 186-195 (1992).
    [CrossRef]
  6. M. R. Teague, 'Image analysis via the general theory of moments,' J. Opt. Soc. Am. 70, 920-930 (1980).
    [CrossRef]
  7. 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).
    [CrossRef]
  8. Z. L. Ping, R. G. Wu, and Y. L. Sheng, 'Image description with Chebyshev-Fourier moments,' J. Opt. Soc. Am. A 19, 1748-1754 (2002).
    [CrossRef]
  9. Y. L. Sheng and L. X. Shen, 'Orthogonal Fourier-Mellin moments for invariant pattern recognition,' J. Opt. Soc. Am. A 11, 1748-1757 (1994).
    [CrossRef]
  10. R. Mukundan and K. R. Ramakrishnan, Moment Functions in Image Analysis: Theory and Application (World Scientific, 1998).
    [CrossRef]
  11. A. Wünsche, 'Generalized Zernike or disc polynomials,' J. Comput. Appl. Math. 174, 135-163 (2005).
    [CrossRef]
  12. C. F. Dunkl and Y. Xu, Orthogonal Polynomials of Several Variables (Cambridge U. Press, 2001).
    [CrossRef]
  13. C. W. Chong, P. Raveendran, and R. Mukundan, 'The scale invariants of pseudo-Zernike moments,' Pattern Anal. Appl. 6, 176-184 (2003).
    [CrossRef]
  14. C. W. Chong, P. Raveendran, and R. Mukundan, 'A comparative analysis of algorithms for fast computation of Zernike moments,' Pattern Recogn. 36, 731-742 (2003).
    [CrossRef]
  15. S. X. Liao and M. Pawlak, 'On image analysis by moments,' IEEE Trans. Pattern Anal. Mach. Intell. 18, 254-266 (1996).
    [CrossRef]
  16. S. X. Liao and M. Pawlak, 'On the accuracy of Zernike moments for image analysis,' IEEE Trans. Pattern Anal. Mach. Intell. 20, 1358-1364 (1998).
    [CrossRef]
  17. J. Flusser, 'On the independence of rotation moment invariants,' Pattern Recogn. 33, 1405-1410 (2000).
    [CrossRef]
  18. P. T. Yap, P. Raveendran, and S. H. Ong, 'Image analysis by Krawtchouk moments,' IEEE Trans. Image Process. 12, 1367-1377 (2003).
    [CrossRef]

2005 (1)

A. Wünsche, 'Generalized Zernike or disc polynomials,' J. Comput. Appl. Math. 174, 135-163 (2005).
[CrossRef]

2003 (3)

C. W. Chong, P. Raveendran, and R. Mukundan, 'The scale invariants of pseudo-Zernike moments,' Pattern Anal. Appl. 6, 176-184 (2003).
[CrossRef]

C. W. Chong, P. Raveendran, and R. Mukundan, 'A comparative analysis of algorithms for fast computation of Zernike moments,' Pattern Recogn. 36, 731-742 (2003).
[CrossRef]

P. T. Yap, P. Raveendran, and S. H. Ong, 'Image analysis by Krawtchouk moments,' IEEE Trans. Image Process. 12, 1367-1377 (2003).
[CrossRef]

2002 (1)

2000 (1)

J. Flusser, 'On the independence of rotation moment invariants,' Pattern Recogn. 33, 1405-1410 (2000).
[CrossRef]

1998 (1)

S. X. Liao and M. Pawlak, 'On the accuracy of Zernike moments for image analysis,' IEEE Trans. Pattern Anal. Mach. Intell. 20, 1358-1364 (1998).
[CrossRef]

1996 (1)

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

1994 (1)

1992 (2)

V. Markandey and R. J. P. Figueiredo, 'Robot sensing techniques based on high-dimensional moment invariants and tensor,' IEEE Trans. Rob. Autom. 8, 186-195 (1992).
[CrossRef]

R. J. Prokop and A. P. Reeves, 'A survey of moment-based techniques for unoccluded object representation and recognition,' Comput. Vis. Graph. Image Process. 54, 438-460 (1992).

1991 (1)

S. O. Belkasim, M. Shridhar, and M. Ahmadi, 'Pattern recognition with moment invariants: a comparative study and new results,' Pattern Recogn. 24, 1117-1138 (1991).
[CrossRef]

1988 (1)

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).
[CrossRef]

1980 (1)

1977 (1)

S. Dudani, K. Breeding, and R. McGhee, 'Aircraft identification by moment invariants,' IEEE Trans. Comput. 26, 39-45 (1977).
[CrossRef]

1962 (1)

M. K. Hu, 'Visual pattern recognition by moment invariants,' IRE Trans. Inf. Theory IT-8, 179-187 (1962).

Ahmadi, M.

S. O. Belkasim, M. Shridhar, and M. Ahmadi, 'Pattern recognition with moment invariants: a comparative study and new results,' Pattern Recogn. 24, 1117-1138 (1991).
[CrossRef]

Belkasim, S. O.

S. O. Belkasim, M. Shridhar, and M. Ahmadi, 'Pattern recognition with moment invariants: a comparative study and new results,' Pattern Recogn. 24, 1117-1138 (1991).
[CrossRef]

Breeding, K.

S. Dudani, K. Breeding, and R. McGhee, 'Aircraft identification by moment invariants,' IEEE Trans. Comput. 26, 39-45 (1977).
[CrossRef]

Chin, R. T.

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).
[CrossRef]

Chong, C. W.

C. W. Chong, P. Raveendran, and R. Mukundan, 'The scale invariants of pseudo-Zernike moments,' Pattern Anal. Appl. 6, 176-184 (2003).
[CrossRef]

C. W. Chong, P. Raveendran, and R. Mukundan, 'A comparative analysis of algorithms for fast computation of Zernike moments,' Pattern Recogn. 36, 731-742 (2003).
[CrossRef]

Dudani, S.

S. Dudani, K. Breeding, and R. McGhee, 'Aircraft identification by moment invariants,' IEEE Trans. Comput. 26, 39-45 (1977).
[CrossRef]

Dunkl, C. F.

C. F. Dunkl and Y. Xu, Orthogonal Polynomials of Several Variables (Cambridge U. Press, 2001).
[CrossRef]

Figueiredo, R. J. P.

V. Markandey and R. J. P. Figueiredo, 'Robot sensing techniques based on high-dimensional moment invariants and tensor,' IEEE Trans. Rob. Autom. 8, 186-195 (1992).
[CrossRef]

Flusser, J.

J. Flusser, 'On the independence of rotation moment invariants,' Pattern Recogn. 33, 1405-1410 (2000).
[CrossRef]

Hu, M. K.

M. K. Hu, 'Visual pattern recognition by moment invariants,' IRE Trans. Inf. Theory IT-8, 179-187 (1962).

Liao, S. X.

S. X. Liao and M. Pawlak, 'On the accuracy of Zernike moments for image analysis,' IEEE Trans. Pattern Anal. Mach. Intell. 20, 1358-1364 (1998).
[CrossRef]

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

Markandey, V.

V. Markandey and R. J. P. Figueiredo, 'Robot sensing techniques based on high-dimensional moment invariants and tensor,' IEEE Trans. Rob. Autom. 8, 186-195 (1992).
[CrossRef]

McGhee, R.

S. Dudani, K. Breeding, and R. McGhee, 'Aircraft identification by moment invariants,' IEEE Trans. Comput. 26, 39-45 (1977).
[CrossRef]

Mukundan, R.

C. W. Chong, P. Raveendran, and R. Mukundan, 'The scale invariants of pseudo-Zernike moments,' Pattern Anal. Appl. 6, 176-184 (2003).
[CrossRef]

C. W. Chong, P. Raveendran, and R. Mukundan, 'A comparative analysis of algorithms for fast computation of Zernike moments,' Pattern Recogn. 36, 731-742 (2003).
[CrossRef]

R. Mukundan and K. R. Ramakrishnan, Moment Functions in Image Analysis: Theory and Application (World Scientific, 1998).
[CrossRef]

Ong, S. H.

P. T. Yap, P. Raveendran, and S. H. Ong, 'Image analysis by Krawtchouk moments,' IEEE Trans. Image Process. 12, 1367-1377 (2003).
[CrossRef]

Pawlak, M.

S. X. Liao and M. Pawlak, 'On the accuracy of Zernike moments for image analysis,' IEEE Trans. Pattern Anal. Mach. Intell. 20, 1358-1364 (1998).
[CrossRef]

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

Ping, Z. L.

Prokop, R. J.

R. J. Prokop and A. P. Reeves, 'A survey of moment-based techniques for unoccluded object representation and recognition,' Comput. Vis. Graph. Image Process. 54, 438-460 (1992).

Ramakrishnan, K. R.

R. Mukundan and K. R. Ramakrishnan, Moment Functions in Image Analysis: Theory and Application (World Scientific, 1998).
[CrossRef]

Raveendran, P.

C. W. Chong, P. Raveendran, and R. Mukundan, 'The scale invariants of pseudo-Zernike moments,' Pattern Anal. Appl. 6, 176-184 (2003).
[CrossRef]

P. T. Yap, P. Raveendran, and S. H. Ong, 'Image analysis by Krawtchouk moments,' IEEE Trans. Image Process. 12, 1367-1377 (2003).
[CrossRef]

C. W. Chong, P. Raveendran, and R. Mukundan, 'A comparative analysis of algorithms for fast computation of Zernike moments,' Pattern Recogn. 36, 731-742 (2003).
[CrossRef]

Reeves, A. P.

R. J. Prokop and A. P. Reeves, 'A survey of moment-based techniques for unoccluded object representation and recognition,' Comput. Vis. Graph. Image Process. 54, 438-460 (1992).

Shen, L. X.

Sheng, Y. L.

Shridhar, M.

S. O. Belkasim, M. Shridhar, and M. Ahmadi, 'Pattern recognition with moment invariants: a comparative study and new results,' Pattern Recogn. 24, 1117-1138 (1991).
[CrossRef]

Teague, M. R.

Teh, C. H.

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).
[CrossRef]

Wu, R. G.

Wünsche, A.

A. Wünsche, 'Generalized Zernike or disc polynomials,' J. Comput. Appl. Math. 174, 135-163 (2005).
[CrossRef]

Xu, Y.

C. F. Dunkl and Y. Xu, Orthogonal Polynomials of Several Variables (Cambridge U. Press, 2001).
[CrossRef]

Yap, P. T.

P. T. Yap, P. Raveendran, and S. H. Ong, 'Image analysis by Krawtchouk moments,' IEEE Trans. Image Process. 12, 1367-1377 (2003).
[CrossRef]

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

R. J. Prokop and A. P. Reeves, 'A survey of moment-based techniques for unoccluded object representation and recognition,' Comput. Vis. Graph. Image Process. 54, 438-460 (1992).

IEEE Trans. Comput. (1)

S. Dudani, K. Breeding, and R. McGhee, 'Aircraft identification by moment invariants,' IEEE Trans. Comput. 26, 39-45 (1977).
[CrossRef]

IEEE Trans. Image Process. (1)

P. T. Yap, P. Raveendran, and S. H. Ong, 'Image analysis by Krawtchouk moments,' IEEE Trans. Image Process. 12, 1367-1377 (2003).
[CrossRef]

IEEE Trans. Pattern Anal. Mach. Intell. (3)

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

S. X. Liao and M. Pawlak, 'On the accuracy of Zernike moments for image analysis,' IEEE Trans. Pattern Anal. Mach. Intell. 20, 1358-1364 (1998).
[CrossRef]

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).
[CrossRef]

IEEE Trans. Rob. Autom. (1)

V. Markandey and R. J. P. Figueiredo, 'Robot sensing techniques based on high-dimensional moment invariants and tensor,' IEEE Trans. Rob. Autom. 8, 186-195 (1992).
[CrossRef]

IRE Trans. Inf. Theory (1)

M. K. Hu, 'Visual pattern recognition by moment invariants,' IRE Trans. Inf. Theory IT-8, 179-187 (1962).

J. Comput. Appl. Math. (1)

A. Wünsche, 'Generalized Zernike or disc polynomials,' J. Comput. Appl. Math. 174, 135-163 (2005).
[CrossRef]

J. Opt. Soc. Am. (1)

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

Pattern Anal. Appl. (1)

C. W. Chong, P. Raveendran, and R. Mukundan, 'The scale invariants of pseudo-Zernike moments,' Pattern Anal. Appl. 6, 176-184 (2003).
[CrossRef]

Pattern Recogn. (3)

C. W. Chong, P. Raveendran, and R. Mukundan, 'A comparative analysis of algorithms for fast computation of Zernike moments,' Pattern Recogn. 36, 731-742 (2003).
[CrossRef]

J. Flusser, 'On the independence of rotation moment invariants,' Pattern Recogn. 33, 1405-1410 (2000).
[CrossRef]

S. O. Belkasim, M. Shridhar, and M. Ahmadi, 'Pattern recognition with moment invariants: a comparative study and new results,' Pattern Recogn. 24, 1117-1138 (1991).
[CrossRef]

Other (2)

R. Mukundan and K. R. Ramakrishnan, Moment Functions in Image Analysis: Theory and Application (World Scientific, 1998).
[CrossRef]

C. F. Dunkl and Y. Xu, Orthogonal Polynomials of Several Variables (Cambridge U. Press, 2001).
[CrossRef]

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Figures (14)

Fig. 1
Fig. 1

Plots of normalized radial polynomials R ̃ p q α ( r ) : (a) α = 0 ; (b) α = 1 ; (c) α = 2 .

Fig. 2
Fig. 2

Plots of weighted radial polynomials R ¯ p q α ( r ) and their zero distributions with different values of α: (a) α = 0 ; (b) α = 10 ; (c) α = 20 ; (d) α = 30 ; (e) α = 40 .

Fig. 3
Fig. 3

Image reconstruction of the letter E of size 31 × 31 without noise.

Fig. 4
Fig. 4

Image reconstruction of a Chinese character of size 63 × 63 without noise.

Fig. 5
Fig. 5

Plot of reconstruction error for E without noise.

Fig. 6
Fig. 6

Plot of reconstruction error for the Chinese character without noise.

Fig. 7
Fig. 7

E with 5% salt-and-pepper noises added.

Fig. 8
Fig. 8

E with 10% salt-and-pepper noises added.

Fig. 9
Fig. 9

Reconstruction error for E with 5% salt-and-pepper noises.

Fig. 10
Fig. 10

Reconstruction error for E with 10% salt-and-pepper noises.

Fig. 11
Fig. 11

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

Fig. 12
Fig. 12

Part of the images of the testing set with 15% salt-and-pepper noises in the first experiment.

Fig. 13
Fig. 13

Grayscale images of the training set used in the second experiment.

Fig. 14
Fig. 14

Part of the images of the testing set with σ 2 = 0.10 Gaussian white noises in the second experiment.

Tables (3)

Tables Icon

Table 1 Comparison of Positions of the Radial Real-Valued GPZP Zeros with Different α

Tables Icon

Table 2 Classification Results of the First Experiment

Tables Icon

Table 3 Classification Results of the Second Experiment

Equations (35)

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Z p q = p + 1 π 0 2 π 0 1 V p q * ( r , θ ) f ( r , θ ) r d r d θ , p = 0 , 1 , 2 , , ; 0 q p ,
V p q ( r , θ ) = R p q ( r ) exp ( j q θ ) .
R p q ( r ) = s = 0 p q ( 1 ) k ( 2 p + 1 s ) ! s ! ( p q s ) ! ( p + q + 1 s ) ! r p s .
0 2 π 0 1 V p q ( r , θ ) V l k * ( r , θ ) r d r d θ = π ( p + 1 ) δ p l δ q k ,
V p q α ( z , z * ) z ( q + q ) 2 ( z * ) ( q q ) 2 P p q ( α , 2 q + 1 ) ( 2 ( z z * ) 1 2 1 ) = z ( p + q ) 2 ( z * ) ( p q ) 2 ( α + 1 ) p q ( p q ) ! F 1 2 ( p + q , p q 1 ; α + 1 ; 1 1 ( z z * ) 1 2 ) ,
F 1 2 ( a , b ; c ; x ) = k = 0 ( a ) k ( b ) k ( c ) k x k k ! .
( a ) k = a ( a + 1 ) ( a + 2 ) ( a + k 1 ) , with ( a ) 0 = 1 .
V p q α ( z , z * ) = ( p + q + 1 ) ! ( α + 1 ) p + q + 1 s = 0 p q ( 1 ) s ( α + 1 ) 2 p + 1 s s ! ( p q s ) ! ( p + q + 1 s ) ! z ( p + q s ) 2 ( z * ) ( p q s ) 2 .
V p q α ( r , θ ) V p q α ( r exp ( j θ ) , r exp ( j θ ) ) = R p q α ( r ) exp ( j q θ ) ,
R p q α ( r ) = ( p + q + 1 ) ! ( α + 1 ) p + q + 1 s = 0 p q ( 1 ) s ( α + 1 ) 2 p + 1 s s ! ( p q s ) ! ( p + q + 1 s ) ! r p s .
R p q ( r ) = R p q 0 ( r ) = r q P p q ( 0 , 2 q + 1 ) ( 2 r 1 ) .
R p q α ( r ) = ( M 1 r + M 2 ) R p 1 , q α ( r ) + M 3 R p 2 , q α ( r ) , for p q 2 ,
M 1 = ( 2 p + 1 + α ) ( 2 p + α ) ( p + q + 1 + α ) ( p q ) ,
M 2 = ( p + q + 1 ) ( α + 2 p ) p + q + α + 1 + M 1 ( p + q ) ( p q 1 ) ( 2 p 1 + α ) ,
M 3 = ( p + q ) ( p + q + 1 ) ( 2 p 2 + α ) ( 2 p 1 + α ) 2 ( p + q + α + 1 ) ( p + q + α ) + M 2 ( p + q ) ( 2 p 2 + α ) p + q + α M 1 ( p + q ) ( p + q 1 ) ( p q 2 ) 2 ( p + q + α ) .
R q q α ( r ) = r q ,
R q + 1 , q α ( r ) = ( α + 3 + 2 q ) r q + 1 2 ( q + 1 ) r q .
0 1 R p q α ( r ) R l q α ( r ) ( 1 r ) α r d r = ( p q + 1 ) 2 q + 1 ( 2 p + α + 2 ) ( α + 1 + p q ) 2 q + 1 δ p l .
0 1 0 2 π V p q α ( r , θ ) [ V m n α ( r , θ ) ] * ( 1 r ) α r d r d θ = 2 π ( p q + 1 ) 2 q + 1 ( 2 p + α + 2 ) ( α + 1 + p q ) 2 q + 1 δ p m δ q n .
R ̃ p q α ( r ) = R p q α ( r ) ( 2 p + α + 2 ) ( α + 1 + p q ) 2 q + 1 2 π ( p q + 1 ) 2 q + 1 .
R ¯ p q α ( r ) = R p q α ( r ) ( 2 p + α + 2 ) ( α + 1 + p q ) 2 q + 1 2 π ( p q + 1 ) 2 q + 1 ( 1 r ) α 2 .
V ¯ p q α ( r , θ ) = R ¯ p q α ( r ) exp ( j q θ ) ;
0 2 π 0 1 V ¯ p q α ( r , θ ) [ V ¯ n m α ( r , θ ) ] * r d r d θ = δ p n δ q m .
Z ¯ p q α = 0 2 π 0 1 [ V ¯ p q α ( r , θ ) ] * f ( r , θ ) r d r d θ .
f ( r , θ ) = p = 0 q Z ¯ p q α V ¯ p q α ( r , θ ) .
f ̃ ( r , θ ) = p = 0 M { Z ¯ p 0 α ( c ) R ¯ p 0 α ( r ) + 2 q > 0 [ Z ¯ p q α ( c ) cos ( q θ ) + q > 0 Z ¯ p q α ( s ) sin ( q θ ) ] R ¯ p q α ( r ) } ,
Z ¯ p q α ( c ) = 0 2 π 0 1 R ¯ p q α ( r ) f ( r , θ ) cos ( q θ ) r d r d θ ,
Z ¯ p q α ( s ) = 0 2 π 0 1 R ¯ p q α ( r ) f ( r , θ ) sin ( q θ ) r d r d θ , q 0 .
Z ¯ p q α = 2 ( N 1 ) 2 s = 0 N 1 t = 0 N 1 R ¯ p q α ( r s t ) exp ( j q θ s t ) f ( s , t ) ,
r s t = ( c 1 s + c 2 ) 2 + ( c 1 t + c 2 ) 2 , θ s t = tan 1 ( c 1 t + c 2 c 1 s + c 2 ) , c 1 = 2 N 1 , c 2 = 1 2 .
ϵ = x = 0 N 1 y = 0 N 1 f ( x , y ) T ( f ̂ ( x , y ) ) ,
T ( u ) = { 1 u 0.5 0 u < 0.5 .
V = [ Z ¯ 20 α , Z ¯ 21 α , Z ¯ 22 α , Z ¯ 30 α , Z ¯ 31 α , Z ¯ 32 α , Z ¯ 33 α ] ,
d ( V s , V t ( k ) ) = j = 1 T ( ν s j ν t j ( k ) ) 2 ,
η = Number of correctly classified images The total number of images used in the test 100 % .

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