Abstract

We propose radial harmonic Fourier moments, which are shifting, scaling, rotation, and intensity invariant. Compared with Chebyshev–Fourier moments, the new moments have superior performance near the origin and better ability to describe small images in terms of image-reconstruction errors and noise sensitivity. A multidistortion-invariant pattern-recognition experiment was performed with radial harmonic Fourier moments.

© 2003 Optical Society of America

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References

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  1. R. J. Prokop, A. P. Reeves, “A survey of moment-based techniques for unocculuded object representation and recognition,” Graph. Models Image Process. 54, 438–460 (1992).
    [CrossRef]
  2. M. K. Hu, “Visual pattern recognition by moment invariants,” IEEE Trans. Inf. Theory IT-8, 179–187 (1962).
  3. M. R. Teague, “Image analysis via the general theory of moments,” J. Opt. Soc. Am. 70, 920–930 (1980).
    [CrossRef]
  4. Y. L. Sheng, L. X. Shen, “Orthogonal Fourier–Mellin moments for invariant pattern recognition,” J. Opt. Soc. Am. A 11, 1748–1757 (1994).
    [CrossRef]
  5. Z. L. Ping, R. G. Wu, Y. L. Sheng, “Image description with Chebyshev–Fourier moments,” J. Opt. Soc. Am. A 19, 1748–1754 (2002).
    [CrossRef]
  6. 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]
  7. S. X. Liao, M. Pawlak, “On image analysis by moments,” IEEE Trans. Pattern Anal. Mach. Intell. 18, 254–266 (1996).
    [CrossRef]

2002

1996

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

1994

1992

R. J. Prokop, A. P. Reeves, “A survey of moment-based techniques for unocculuded object representation and recognition,” Graph. Models Image Process. 54, 438–460 (1992).
[CrossRef]

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]

1980

1962

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

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]

Hu, M. K.

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

Liao, S. X.

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

Pawlak, M.

S. X. Liao, 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, A. P. Reeves, “A survey of moment-based techniques for unocculuded object representation and recognition,” Graph. Models Image Process. 54, 438–460 (1992).
[CrossRef]

Reeves, A. P.

R. J. Prokop, A. P. Reeves, “A survey of moment-based techniques for unocculuded object representation and recognition,” Graph. Models Image Process. 54, 438–460 (1992).
[CrossRef]

Shen, L. X.

Sheng, Y. L.

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]

Wu, R. G.

Graph. Models Image Process.

R. J. Prokop, A. P. Reeves, “A survey of moment-based techniques for unocculuded object representation and recognition,” Graph. Models Image Process. 54, 438–460 (1992).
[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.

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]

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

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

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Figures (13)

Fig. 1
Fig. 1

(a) Radial polynomials T n ( r ) of RHFMs, (b) radial polynomials R n ( r ) of CHFMs with n = 0 , 1, 2, 9, 10.

Fig. 2
Fig. 2

Reconstructed images of the 26 English alphabet letters with 64 RHFMs, with m and n = 0 , 1, 2, 3, 4, 5, 6, 7.

Fig. 3
Fig. 3

Reconstruction image of “E6” with (a) CHFMs, (b) RHFMs. From top left to bottom right: N = M = 2 , 3, 5, 7, 10, 12, 15, 17, 20.

Fig. 4
Fig. 4

Reconstruction image of “E3” with (a) CHFMs, (b) RHFMs. From top left to bottom right: N = M = 2 , 3, 5, 7, 10, 12, 15, 17, 20.

Fig. 5
Fig. 5

NIRE for a letter E with RHFMs and CHFMs, as a function of the highest degree N of the radial polynomial.

Fig. 6
Fig. 6

Statistical NIRE 2 ¯ as a function of the highest degree N of the radial polynomial.

Fig. 7
Fig. 7

Statistical SNR of the RHFMs and CHFMs with constant α = 3 and m = 0 , 5, 10 as a function of the number of zeros of their radial polynomials.

Fig. 8
Fig. 8

Statistical normalized noisy-image reconstruction error n 2 ¯ with input SNR = 100 as a function of the total number of the RHFMs and CHFMs used in the reconstruction.

Fig. 9
Fig. 9

Normalized noisy-image reconstruction error n 2 for deterministic objects E6 and E3 with additive noise and input SNR = 100 , as a function of the total number of the RHFMs and CHFMs used in the reconstruction.

Fig. 10
Fig. 10

Reconstruction images of noisy images with RHFMs and CHFMs. First row, original noisy images; from left to right, the input SNR is no noise, 100, 10, 1, 0.1. Second row, reconstructed images with RHFMs ( N = M = 7 ) ; third row, reconstructed images with CHFMs ( N = M = 7 ) .

Fig. 11
Fig. 11

Testing image for the letter K. From top left to bottom right: the letter K rotated by 20 and 130 deg, intensity changed by 3 and 6 times, scaled by 2.2 and 3 times, and with zero-mean additive noise of σ = 400 and σ = 1600 .

Fig. 12
Fig. 12

Slice of the 25-dimensional feature space, namely, ( x 1 = 0 ,   x 2 = 0 ,   x 3 ,   x 4 , ,   x 25 = 0 ) T .

Fig. 13
Fig. 13

Weighted distance d i ( M ,   N ) as a function of the highest circular and radical order M and N.

Equations (37)

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P nm ( r ,   θ ) = T n ( r ) exp ( jm θ ) ,
T n ( r ) = 1 r if n = 0 2 r   sin [ ( n + 1 ) π r ] if n is odd 2 r cos ( n π r ) if n is even .
0 2 π 0 1 P nm ( r ,   θ ) P kl ( r ,   θ ) r d r d θ = δ nmkl ,
f ( r ,   θ ) = n = 0 m = - + ϕ nm T n ( r ) exp ( jm θ ) ,
ϕ nm = 0 2 π 0 1 f ( r ,   θ ) T n ( r ) exp ( - jm θ ) r d r d θ ,
M sm = 0 2 π 0 1 gf ( r / k ,   θ ) r s exp ( - jm θ ) r d r d θ
= gk s + 2 0 2 π 0 1 f ( r ,   θ ) r s exp ( - jm θ ) r d r d θ = gk s + 2 M sm ,
k = M 10 M 00 / M 10 M 00 ,
g = M 10 M 00 / M 10 M 00 2 M 00 M 00 .
ϕ nm = 0 2 π 0 k gf ( r / k ,   θ ) T n ( r / k ) exp ( - jm θ ) r d r d θ = gk 2 0 2 π 0 1 f ( ρ ,   θ ) T n ( ρ ) exp ( - jm θ ) ρ d ρ d θ = gk 2 ϕ nm .
Φ nm = ϕ nm / gk 2 .
R n ( r ) = 8 π 1 - r r 1 / 4 × k = 0 ( n + 2 ) / 2 ( - 1 ) k ( n - k ) ! k ! ( n - 2 k ) !   [ 2 ( 2 r - 1 ) ] n - 2 k .
lim r 0   T n ( r ) = 0 when n is odd when n is even ,
lim r 0   R n ( r ) = .
f ˆ ( r ,   θ ) n = 0 N m = - M M Φ nm T n ( r ) exp ( jm θ ) ,
2 = - + [ f ( x ,   y ) - f ˆ ( x ,   y ) ] 2 d x d y - + f 2 ( x ,   y ) d x d y .
2 ¯ = E - 1 1 [ f ( x ,   y ) - f ˆ ( x ,   y ) ] 2 d x d y E - 1 1 [ f ( x ,   y ) ] 2 d x d y .
SNR nm = var [ ( Φ nm ) f ] var [ ( Φ nm ) noise ] = 1 σ 2 var [ ( Φ nm ) f ] ,
var [ ( Φ nm ) f ]
= 0 2 π 0 k 0 2 π 0 k C ff ( x ,   y ,   u ,   v ) T n ( r ) T m ( ρ )
× cos [ m ( θ - ϕ ) ] r d r d θ ρ d ρ d ϕ ,
C ff ( x ,   y ,   u ,   v )
= C ff ( 0 ,   0 ) exp { - α [ ( x - u ) 2 + ( y - v ) 2 ] 1 / 2 } ,
C ff   ( 0 ,   0 ) = E { [ f ( x ,   y ) ] 2 } = 1 π k 2   0 2 π 0 k [ f ( r ,   θ ) ] 2 r d r d θ ,
n 2 ¯ ( N ,   M )
= E - 1 1 [ f ( x ,   y ) - f ˆ ( x ,   y ) - n ˆ ( x ,   y ) ] 2 d x d y E - 1 1 [ f ( x ,   y ) ] 2 d x d y = 2 ¯ ( N ,   M ) + E 0 2 π 0 1 [ n ˆ ( r ,   θ ) ] 2 r d r d θ E { f ( r ,   θ ) ] 2 r d r d θ }
= 2 ¯ ( N ,   M ) + N π SNR input ,
n 2 ( N ,   M ) = 2 ( N ,   M ) + N total π SNR input .
d i ( M ,   N ) = n , m = 0 M , N [ | Φ mn | - ( | Φ mn | ) i ] 2 ( σ mn ) i 2 1 / 2
φ n ( r ) = 1 n = 0 sin [ ( n + 1 ) π r ] n = odd cos ( n π r ) n = even .
ϕ n ( r ) = φ n ( r ) / r .
0 1 ϕ n ( r ) ϕ m ( r ) r d r = 0 1 φ n ( r ) φ m ( r ) d r .
( f ,   g ) 1 = 0 1 fgr d r ,
r f ( r ) = n = 0 ( f r ,   φ n ) φ n ,
( f r ,   φ n ) = 0 1 f r φ n d r = 0 1 f   φ n r   r d r
= 0 1 f ϕ n r d r = ( f ,   ϕ n ) 1 .
f ( r ) = n = 0 ( f ,   ϕ n ) 1 ϕ n .

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