Abstract

The computing method for orthogonal Fourier–Mellin moments in a polar coordinate system is presented in detail. The image expressed in a Cartesian system has to be transformed into a polar coordinate system first when we calculate the orthogonal Fourier–Mellin moments of the image in a polar coordinate system, which will increase both computational complexity and error. To solve the problem, a new direct computing method for orthogonal Fourier–Mellin moments in a Cartesian coordinate system is proposed, which can avoid the image transformation between two coordinate systems and eliminate the rounding error in coordinate transformation and decrease the computational complexity.

© 2009 Optical Society of America

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References

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2000

Z. L. Ping and Y. L. Sheng, “Fourier-Mellin descriptor and interpolated feature space trajectories for three-dimensional object recognition,” Opt. Eng. 39, 1260-1266 (2000).
[CrossRef]

1996

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

1994

1988

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]

1986

1980

1976

1926

M. K. Hu, “Visual pattern recognition by moment invariants,” IRE Trans. Inf. Theory 8, 179-187 (1926).

Arsennault, H. H.

Bhatia, A. B.

Casasent, D.

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]

Hu, M. K.

M. K. Hu, “Visual pattern recognition by moment invariants,” IRE Trans. Inf. Theory 8, 179-187 (1926).

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]

Pawlak, M.

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.

Z. L. Ping and Y. L. Sheng, “Fourier-Mellin descriptor and interpolated feature space trajectories for three-dimensional object recognition,” Opt. Eng. 39, 1260-1266 (2000).
[CrossRef]

Psaltis, D.

Shen, L. X.

Sheng, Y.

Sheng, Y. L.

Z. L. Ping and Y. L. Sheng, “Fourier-Mellin descriptor and interpolated feature space trajectories for three-dimensional object recognition,” Opt. Eng. 39, 1260-1266 (2000).
[CrossRef]

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]

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]

Wolf, E.

Appl. Opt.

IEEE Trans. Pattern Anal. Mach. Intell.

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]

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

IRE Trans. Inf. Theory

M. K. Hu, “Visual pattern recognition by moment invariants,” IRE Trans. Inf. Theory 8, 179-187 (1926).

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

Opt. Eng.

Z. L. Ping and Y. L. Sheng, “Fourier-Mellin descriptor and interpolated feature space trajectories for three-dimensional object recognition,” Opt. Eng. 39, 1260-1266 (2000).
[CrossRef]

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Figures (4)

Fig. 1
Fig. 1

Illustration of conversion of a Cartesian coordinate system to a polar coordinate system.

Fig. 2
Fig. 2

Effects of image conversion in the two coordinate systems. Left, original gray image with a 256 × 256 matrix; right, reconstruction image with a 256 × 256 matrix. The origin of the polar coordinate system is at the center of the right image, so the range of the angular coordinate of the reconstruction image is (0,255), and the range of radial coordinates is (0,128). The right-hand image shows that the center of the image is fuzzy and that some pixels are missing at the edge.

Fig. 3
Fig. 3

Integral region due to the definition of OFMMs in a polar coordinate system in Eq. (1).

Fig. 4
Fig. 4

Letter E at the top, original image with 64 × 64   pixels . Second row, experimental results obtained by calculating the OFMMs and reconstructing the images in a polar coordinate system. Third row, images reconstructed in the Cartesian coordinate system with the OFMMs that have been calculated in a polar coordinate system. Fourth row, experimental results in a Cartesian coordinate system. K = L = 6 , 7, and 8 in columns 1, 2, and 3, respectively.

Equations (24)

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V ( r cos θ , r sin θ ) = R n ( r ) exp ( j m θ ) .
ϕ m n = 1 2 π α n 0 2 π 0 1 f ( r , θ ) Q n ( r ) exp ( j m θ ) r d r d θ ,
0 1 Q n ( r ) Q k ( r ) r d r = α n δ n k ,
Q n ( r ) = s = 0 n α n s r s ,
α n s = ( 1 ) n + s ( n + s + 1 ) ! ( n s ) ! s ! ( s + 1 ) ! ,
ϕ m n = 1 2 π α n k = 0 K l = 0 L f ( k r , l θ ) Q n ( k r ) exp ( j m l θ ) k r r θ .
Δ R = 1 M .
i = r Δ R .
Δ θ = 2 π N ,
θ = arctan ( y x ) + k π ,
j = θ Δ θ ,
f ̂ ( r , θ ) n = 0 L m = K K ϕ n m Q n ( r ) exp ( j m θ ) ,
x = r cos ( θ ) ,
y = sin ( θ ) ,
r = i Δ r ,
θ = j Δ θ ,
r s e j m θ = ( x + j y ) ( s m ) 2 ( x j y ) ( s + m ) 2 ,
Q n ( r ) = s = 0 n α n s r s ,
Q n ( r ) exp ( j m θ ) = s = 0 n α n , s r s exp ( j m θ ) = s = 0 n α n , s ( x + j y ) p ( x j y ) q ,
ϕ m n = 1 2 π α n 0 2 π 0 1 f ( r , θ ) Q n ( r ) exp ( j m θ ) r d r d θ = 1 2 π α n s 1 f ( r , θ ) Q n ( r ) exp ( j m θ ) d s = 1 2 π α n s 2 f ( r , θ ) Q n ( r ) exp ( j m θ ) d s + 1 2 π α n s 1 s 2 f ( r , θ ) Q n ( r ) exp ( j m θ ) d s .
ϕ m n = 1 2 π α n s 2 f ( r , θ ) Q n ( r ) exp ( j m θ ) d s .
ϕ m n = 1 2 π α n s 2 f ( x , y ) s = 0 n α n , s ( x + j y ) p ( x j y ) q d x d y ,
ϕ m n = 1 2 π α n s 2 f ( x , y ) Q n ( r ) exp ( j m θ ) d x d y ,
f ̂ ( x , y ) n = 0 L m = K K ϕ n m Q n ( r ) exp ( j m θ ) ,

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