Abstract

Chebyshev–Fourier moments for describing images were proposed. After definition of the moments, the multidistortion invariance of the moments was verified. The performance of the moments in describing images was investigated in terms of the normalized image-reconstruction error and the results of the experiments on the noise sensitivity are given.

© 2002 Optical Society of America

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References

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  1. M. K. Hu, “Visual pattern recognition by moment invariants,” IEEE Trans. Inf. Theory IT-8, 179–187 (1962).
  2. M. R. Teague, “Image analysis via the general theory of moments,” J. Opt. Soc. Am. 70, 920–930 (1980).
    [CrossRef]
  3. J. F. Boyce, W. J. Hossack, “Moment invariants for pattern recognition,” Pattern Recogn. Lett. 1, 451–456 (1983).
    [CrossRef]
  4. Y. S. Abu-Mostafa, D. Psaltis, “Recognition aspects of moment invariants,” IEEE Trans. Pattern Anal. Mach. Intell. PAMI-6, 698–706 (1984).
    [CrossRef]
  5. 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]
  6. Y. L. Sheng, L. X. Shen, “Orthogonal Fourier–Mellin moments for invariant pattern recognition,” J. Opt. Soc. Am. A 11, 1748–1757 (1994).
    [CrossRef]
  7. Z. L. Ping, Y. L. Sheng, “Describing image with Chebyshev Moments,” Acta Opt. Sin. (to be published).
  8. A. B. Bhatia, E. Wolf, “On the circular polynomials of Zernike and related orthogonal sets,” Proc. Cambridge Philos. Soc. 50, 40–48 (1954).
    [CrossRef]

1994

1988

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]

1984

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

1983

J. F. Boyce, W. J. Hossack, “Moment invariants for pattern recognition,” Pattern Recogn. Lett. 1, 451–456 (1983).
[CrossRef]

1980

1962

M. K. Hu, “Visual pattern recognition by moment invariants,” IEEE Trans. Inf. Theory IT-8, 179–187 (1962).

1954

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

Abu-Mostafa, Y. S.

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

Boyce, J. F.

J. F. Boyce, W. J. Hossack, “Moment invariants for pattern recognition,” Pattern Recogn. Lett. 1, 451–456 (1983).
[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]

Hossack, W. J.

J. F. Boyce, W. J. Hossack, “Moment invariants for pattern recognition,” Pattern Recogn. Lett. 1, 451–456 (1983).
[CrossRef]

Hu, M. K.

M. K. Hu, “Visual pattern recognition by moment invariants,” IEEE Trans. Inf. Theory IT-8, 179–187 (1962).

Ping, Z. L.

Z. L. Ping, Y. L. Sheng, “Describing image with Chebyshev Moments,” Acta Opt. Sin. (to be published).

Psaltis, D.

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

Shen, L. X.

Sheng, Y. L.

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

Z. L. Ping, Y. L. Sheng, “Describing image with Chebyshev Moments,” Acta Opt. Sin. (to be published).

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]

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]

IEEE Trans. Inf. Theory

M. K. Hu, “Visual pattern recognition by moment invariants,” IEEE Trans. Inf. Theory IT-8, 179–187 (1962).

IEEE Trans. Pattern Anal. Mach. Intell.

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

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]

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

Pattern Recogn. Lett.

J. F. Boyce, W. J. Hossack, “Moment invariants for pattern recognition,” Pattern Recogn. Lett. 1, 451–456 (1983).
[CrossRef]

Proc. Cambridge Philos. Soc.

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

Z. L. Ping, Y. L. Sheng, “Describing image with Chebyshev Moments,” Acta Opt. Sin. (to be published).

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Figures (13)

Fig. 1
Fig. 1

Variations in the value of Rn(r) with r, in the interval 0<r1, n=1, 2, 9, 10.

Fig. 2
Fig. 2

Reconstruction of E with CHFM (in binary). From top left to bottom right: original, 3×3, 5×5, 10×10, 15×15, 20×20.

Fig. 3
Fig. 3

Reconstruction of E with CHFM (in gray). From top left to bottom right: original, 3×3, 5×5, 10×10, 15×15, 20×20.

Fig. 4
Fig. 4

Reconstructed images of the 26 English alphabet letters with 64 CHFM.

Fig. 5
Fig. 5

NIRE ϵ2 for the letter E with CHFM and OFMM as a function of the highest degree N of the radial polynomial.

Fig. 6
Fig. 6

Reconstructed images of E6 with (a) CHFM, (b) OFMM. From top left to bottom right: N=M=2, 3, 5, 7, 10, 12, 15, 17, 20.

Fig. 7
Fig. 7

Reconstructed images of E3 with (a) CHFM, (b) OFMM. From top left to bottom right: N=M=2, 3, 5, 7, 10, 12, 15, 17, 20.

Fig. 8
Fig. 8

Statistical SNRs of the CHFM and the OFMM with constant α=3 and m=0, 5, 10 as a function of the number of zeros of their radial polynomials.

Fig. 9
Fig. 9

Statistical NIRE ϵ¯n2 with the input SNR=100 as a function of the total number of CHFM and OFMM used in the reconstruction.

Fig. 10
Fig. 10

NIRE ϵn2 for deterministic images with additive noise and input SNR=100, as a function of the total number of the CHFM and OFMM used in the reconstruction.

Fig. 11
Fig. 11

Reconstructed images of E6 corrupted by additive zero-mean and input SNR=100 white noise with (a) CHFM and (b) OFMM. From top left to bottom right: N=M=2, 3, 5, 7, 10, 12, 15, 17, 20.

Fig. 12
Fig. 12

Reconstructed images of E3 corrupted by additive zero-mean and input SNR=100 white noise with (a) CHFM and (b) OFMM. From top left to bottom right: N=M=2, 3, 5, 7, 10, 12, 15, 17, 20.

Fig. 13
Fig. 13

Reconstructed images of noisy images with CHFM and OFMM. The first row is the original noisy images: from left to right, the input SNR is no noise, 100, 10, 1, 0.1. The second row is the reconstructed images with CHFM (N=M=7). The third row is reconstructed images with OFMM (N=M=7).

Equations (31)

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V(r cos θ, r sin θ)=Rn(r)exp(jmθ),
Un*(r)=Un(2r-1),
Un(cos θ)=sin(n+1)θsin θ,
sin(n+1)θ=sin θ(2 cos θ)n-n-11(2 cos θ)n-2+n-22(2 cos θ)n-4-.
Un*(r)=Un(cos θ)=k=0(n+2)/2(-1)k (n-k)!k!(n-2k)!×[2(2r-1)]n-2k.
w(r)=(r-r2)1/2
01Un*(r)Uk*(r)W(r)dr=π8δnk.
Rn(r)=8πUn*(r)W1/2(r)r-1/2=8π1-rr1/4k=0(n+2)/2(-1)k (n-k)!k!(n-2k)!×[2(2r-1)]n-2k.
01Rn(r)Rk(r)rdr=δnk.
Pnm(r, θ)=Rn(r)exp(jmθ).
02π01Pnm(r, θ)Pkl(r, θ)rdrdθ=δnkml.
f(r, θ)=n=0m=-+ΨnmRn(r)exp(jmθ).
HereΨnm=02π01f(r, θ)Rn(r)exp(-jmθ)rdrdθ.
f(r, θ)n=0Nm=-MMΨnmRn(r)exp(jmθ).
Msm=02π01gf(r/k, θ)rs exp(-jmθ)rdrdθ=gks+2Msm,
k=M10M00M10M00,
g=M10M00M10M002 M00M00.
Ψnm=02π0kgf(r/k, θ)Rn(r/k)exp(-jmθ)rdrdθ,
Ψnm=gk202π01f(ρ, θ)Rn(ρ)exp(-jmθ)ρdρdθ=gk2Ψnm,
Ψnm=Ψnm/gk2.
Φnm=12πan 02π01f(r, θ)Qn(r)exp(jmθ)rdrdθ,
Qn(r)=s=0nansrs,
ans=(-1)n+s (n+s+1)!(n-s)!s!(s+1)!.
ϵ2=-[f(x, y)-fˆ(x, y)]2dxdy-f2(x, y)dxdy,
Cnn(x, y, u, v)=σ2δ(x-u, y-v),
SNRnm=var[(Ψnm)f]var[(Ψnm)noise]=1σ2 var[(Ψnm)f],
var{(Ψnm)f}=02π0k02π0kCff(x, y, u, v)Rn(r)Rm(ρ)×cos[m(θ-ϕ)]rdrdθρdρdϕ,
Cff(x, y, u, v)=Cff(0, 0)exp-αk[(x-u)2+(y-v)2],
Cff(0, 0)=E{[f(r, θ)]2}=1πk2 02π0k[f(r, θ)]2rdrdθ,
ϵ¯n2(N, m)=E-11-11[f(x, y)-fˆ(x, y)-nˆ(x, y)]2dxdyE-11-11[f(x, y)]2dxdy=ϵ¯2(N, M)+E02π01nˆ[(r, θ)]2rdrdθE02π01[f(r, θ)]rdrdθ=ϵ¯2(N, M)+NtotalπSNRinput,
ϵ¯2(N, M)=E-11-11[f(x, y)-fˆ(x, y)]2dxdyE-11-11[f(x, y)]dxdy

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