Abstract

We propose orthogonal Fourier–Mellin moments, which are more suitable than Zernike moments, for scaleand rotation-invariant pattern recognition. The new orthogonal radial polynomials have more zeros than do the Zernike radial polynomials in the region of small radial distance. The orthogonal Fourier–Mellin moments may be thought of as generalized Zernike moments and orthogonalized complex moments. For small images, the description by the orthogonal Fourier–Mellin moments is better than that by the Zernike moments in terms of image-reconstruction errors and signal-to-noise ratio. Experimental results are shown.

© 1994 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. Y. S. Abu-Mostafa, D. Psaltis, “Recognition aspect of moment invariants,” IEEE Trans. Pattern Anal. Mach. Intell. PAMI-6, 698–706 (1984).
    [CrossRef]
  3. M. O. Freeman, B. E. A. Saleh, “Moment invariants in the space and frequency domains,” J. Opt. Soc. Am. A 5, 1073–1084 (1988).
    [CrossRef] [PubMed]
  4. 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]
  5. M. R. Teague, “Image analysis via the general theory of moments,” J. Opt. Soc. Am. 70, 920–930 (1980).
    [CrossRef]
  6. A. B. Bhatia, E. Wolf, “On the circular polynomials of Zernike and related orthogonal sets,” Proc. Cambridge Philos. Soc. 50, 40–48 (1954).
    [CrossRef]
  7. Y. Sheng, J. Duvernoy, “Circular-Fourier-Radial-Mellin descriptors (FMD’s) for pattern recognition,” J. Opt. Soc. Am. A 3, 885–888 (1986).
    [CrossRef] [PubMed]
  8. D. Jackson, Fourier Series and Orthogonal Polynomials (Mathematical Association of America, New York, 1941), Chaps. VII and XI.
  9. H. H. Arsenault, Y. Sheng, “Properties of the circular harmonic expansion for rotation-invariant pattern recognition,” Appl. Opt. 25, 3225–3229 (1986).
    [CrossRef] [PubMed]
  10. E. R. Kretzmer, “Statistics of television signals,” Bell Syst. Tech. J. 31, 751–763 (1952).
  11. K. Fukunaga, Statistical Pattern Recognition, 2nd ed. (Academic, Boston, 1990), Chap. 9.
  12. A. Khotanzad, Y. H. Hong, “Invariant image recognition by Zernike moments,” IEEE Trans. Pattern Anal. Mach. Intell. 12, 489–497 (1990).
    [CrossRef]

1990 (1)

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

1988 (2)

M. O. Freeman, B. E. A. Saleh, “Moment invariants in the space and frequency domains,” J. Opt. Soc. Am. A 5, 1073–1084 (1988).
[CrossRef] [PubMed]

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]

1986 (2)

1984 (1)

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

1980 (1)

1962 (1)

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

1954 (1)

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

1952 (1)

E. R. Kretzmer, “Statistics of television signals,” Bell Syst. Tech. J. 31, 751–763 (1952).

Abu-Mostafa, Y. S.

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

Arsenault, H. H.

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]

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]

Duvernoy, J.

Freeman, M. O.

Fukunaga, K.

K. Fukunaga, Statistical Pattern Recognition, 2nd ed. (Academic, Boston, 1990), Chap. 9.

Hong, Y. H.

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

Hu, M. K.

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

Jackson, D.

D. Jackson, Fourier Series and Orthogonal Polynomials (Mathematical Association of America, New York, 1941), Chaps. VII and XI.

Khotanzad, A.

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

Kretzmer, E. R.

E. R. Kretzmer, “Statistics of television signals,” Bell Syst. Tech. J. 31, 751–763 (1952).

Psaltis, D.

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

Saleh, B. E. A.

Sheng, Y.

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]

Appl. Opt. (1)

Bell Syst. Tech. J. (1)

E. R. Kretzmer, “Statistics of television signals,” Bell Syst. Tech. J. 31, 751–763 (1952).

IEEE Trans. Inf. Theory (1)

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

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

Y. S. Abu-Mostafa, D. Psaltis, “Recognition aspect 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]

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

J. Opt. Soc. Am. (1)

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

Proc. Cambridge Philos. Soc. (1)

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 (2)

D. Jackson, Fourier Series and Orthogonal Polynomials (Mathematical Association of America, New York, 1941), Chaps. VII and XI.

K. Fukunaga, Statistical Pattern Recognition, 2nd ed. (Academic, Boston, 1990), Chap. 9.

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Figures (16)

Fig. 1
Fig. 1

(a) Radial polynomials Qn(r) of the OFMM’s with n = 0,1, 2, …,9; (b) Zernike radial polynomials Rn|m| (r) with m = 10 and n = 10,12, …, 28.

Fig. 2
Fig. 2

Normalized image reconstruction error ε2 for a letter E with OFMM’s and ZM’s, as a function of the highest degree N of the radial polynomials.

Fig. 3
Fig. 3

(a) NIRE ε2 for a letter E6 and a half-size letter E3 with OFMM’s and ZM’s, as a function of the total number of moments used in the reconstruction; (b) reconstructed binary and gray-level images with OFMM’s of up to and including n = 20. Left, original images; right, reconstructed images.

Fig. 4
Fig. 4

Reconstructed images for E3 with OFMM’s. From top left to bottom right: original image and reconstructed images with OFMM’s of N = M = 0,1, 2, …,7, with N = M = 7 corresponding to a total of 64 independent OFMM’s. The reconstructed images were not thresholded.

Fig. 5
Fig. 5

Reconstructed images for E3 with ZM’s. From top left to bottom right: original image and reconstructed images with ZM’s of N = 0, 2,4, …, 14, with N = M = 7 corresponding to a total of 64 independent ZM’s. The reconstructed images were not thresholded.

Fig. 6
Fig. 6

Autocorrelation Cff(x, y; u, υ) as a function of R = [(xu)2 + (yυ)2]½ with different constants α The dashed curve is the autocorrelation of a uniform unit circle.

Fig. 7
Fig. 7

Statistical NIRE ε2 as a function of the total number of the OFMM’s and ZM’s used in the reconstruction.

Fig. 8
Fig. 8

Statistical SNR of the OFMM and the ZM with constant α = 3, image size k = 1, and order m = 0,5,10 as a function of the number of zeros of their radial polynomials.

Fig. 9
Fig. 9

Statistical SNR of OFMM’s with m = 5, n = 0, 5 and of ZM’s with m = 5, n = 5, 15, with the number of zeros equal to 0 and 5 respectively, as a function of image size k.

Fig. 10
Fig. 10

Statistical normalized noisy-image reconstruction error ɛ n 2 ¯ with input SNR = 100 as a function of the total number of the OFMM’s and ZM’s used in the reconstruction.

Fig. 11
Fig. 11

Normalized noisy-image reconstruction error εn2 for deterministic objects E6 and E3 with additive noise and input SNR = 100, as a function of the total number of the OFMM’s and ZM’s used in the reconstruction.

Fig. 12
Fig. 12

Reconstructed images with the OFMM’s of the E3 corrupted by an additive Gaussian noise of zero mean and σ = 185. From top left to bottom right: original image and reconstructed images with OFMM’s of AT = 0, 1, 2, …, 7, with N = 7 corresponding to a total of 64 independent OFMM’s. The reconstructed images were not thresholded.

Fig. 13
Fig. 13

Reconstructed images with the ZM’s of the E3 corrupted by additive Gaussian noise of zero mean and σ = 185. From top left to bottom right: original image and reconstructed images with ZM’s of N = 0, 2, 4,…, 14, with N = 14 corresponding to a total of 64 ZM’s. The reconstructed images were not thresholded.

Fig. 14
Fig. 14

Reconstructed images of the 26 English alphabet letters with 25 OFMM’s of n and m = 0,1,2,3,4.

Fig. 15
Fig. 15

Testing images for the letter A. From top left to bottom right: the letter A rotated by 20 and 130 deg, scaled by k = 0.555 and 0.455, and with zero-mean additive noise of σ = 25.5 and σ = 255.

Fig. 16
Fig. 16

Misclassification rate of the OFMM’s and the ZM’s with input noise σ = 25.5 and σ = 255 as a function of the ratio M10/M00 that was used to calculate the scale ratio.

Tables (1)

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Table 1 Positions of the First Zeros of Qn(r) and Rn|10|(r)

Equations (55)

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M s , m = 0 2 π 0 r s f ( r , θ ) exp ( j m θ ) r d r d θ ,
Φ n m = 1 2 π a 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 = a n δ n k ,
1 , r , r 2 , r n .
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 ) ! ,
Q 0 ( r ) = 1 , Q 1 ( r ) = ( 2 3 r ) , Q 2 ( r ) = 3 12 r + 10 r 2 , Q 3 ( r ) = ( 4 30 r + 60 r 2 35 r 3 ) , Q 4 ( r ) = 5 60 r + 210 r 2 280 r 3 + 126 r 4 , Q 5 ( r ) = ( 6 105 r + 560 r 2 260 r 3 + 1260 r 4 462 r 5 ) ,
Φ nm = n + 1 π s = 0 n α ns M sm .
V ( r cos θ , r sin θ ) = R n ( r ) exp ( j m θ ) ,
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 ,
p = s m 2 , q = s + m 2 .
Φ n m = n + 1 π s = 0 n α n s C ( s m / 2 ) , ( s + m ) / 2 ,
C p q = f ( x , y ) ( x + j y ) p ( x + j y ) q d x d y .
f ̂ ( r , θ ) n = 0 N m = M M Φ n m Q n ( r ) exp ( j m θ ) .
ɛ 2 = [ f ( x , y ) f ̂ ( x , y ) ] 2 d x d y f ( x , y ) d x d y ,
E = { f ( x , y ) } = 0 ,
ɛ 2 ¯ = E { 1 1 [ f ( x , y ) f ̂ ( x , y ) ] 2 d x d y } E { 1 1 [ f ( x , y ) ] 2 d x d y } .
ɛ 2 ¯ ( N , M ) = 1 1 π k 2 n = 0 N m = M M n + 1 π × 0 2 π 0 k 0 2 π 0 k C ff ( x , y , u , υ ) C ff ( 0 , 0 ) × Q n ( r ) Q n ( ρ ) cos [ m ( θ ϕ ) ] r d r d θ ρ d ρ d ϕ ,
C ff ( x , y , u , υ ) = C ff ( 0 , 0 ) exp ( α 1 | x u | α 2 | y υ | ) ,
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
C ff ( x , y , u , υ ) = C ff ( 0 , 0 ) exp { α [ ( x u ) 2 + ( y υ ) 2 ] 1 / 2 } ,
C ff ( x , y , u , υ ) = { C ff ( 0 , 0 ) exp [ α k R ] R 2 k 0 R > 2 k ,
C nn ( x , y , u , υ ) = σ 2 δ ( x u , y υ ) ,
SNR n m = var { ( Φ n m ) f } var { ( Φ n m ) noise } = 1 σ 2 var { ( Φ n m ) f } ,
var { ( Φ n m ) f } = 0 2 π 0 k 0 2 π 0 k C ff ( x , y , u , υ ) Q n ( r ) Q n ( ρ ) × cos [ m ( θ ϕ ) ] r d r d θ ρ d ρ d ϕ ,
ɛ 2 ¯ ( N , M ) = E { 1 1 [ f ( x , y ) f ̂ ( x , y ) n ̂ ( x , y ) ] 2 d x d y } E { 1 1 [ f ( x , y ) ] 2 d x d y } = ɛ 2 ¯ ( N , M ) E { 0 2 π 0 1 [ n ̂ ( r , θ ) ] 2 r d r d θ } E { 0 2 π 0 1 [ f ( r , θ ) ] 2 r d r d θ } = ɛ 2 ¯ ( N , M ) + N total π SNR input ,
SNR input = k 2 C ff ( 0 , 0 ) σ 2 .
M s m = 0 2 π 0 1 g f ( r / k , θ ) r s r d r d θ = g k s + 2 M s m ,
K = ( M 10 M 00 ) / ( M 10 M 00 ) ,
g = [ ( M 10 M 00 ) / ( M 10 M 00 ) ] 2 ( M 00 M 10 ) .
d i = { n , m = 0 when m = 0 , n 0 , 1 4 [ | Φ n m | ( | Φ n m | ) i ] 2 ( σ n m ) i 2 } 1 / 2 ,
1 , r , r 2 , r n
w ( r ) = r q 1 ( 1 r ) p q ,
0 1 w ( r ) G n ( p , q , r ) G n ( p , q , r ) d r = b n ( p , q ) δ n n ,
b n ( p , q ) = n ! [ ( q 1 ) ! ] 2 ( p q + n ) ! ( q + n 1 ) ! ( p + n 1 ) ! ( p + 2 n ) .
G n ( p , q , r ) = n ! ( q 1 ) ! ( p + n 1 ) ! × s = 0 n ( 1 ) s ( p + n + s 1 ) ! ( n s ) ! s ! ( q + s 1 ) ! r s .
0 1 Q n ( r ) Q k ( r ) r d r = a n δ n k ,
0 1 G n ( 2 , 2 , r ) G k ( 2 , 2 , r ) r d r = 1 2 ( n + 1 ) 3 δ n k .
Q n ( r ) = a n 1 / 2 [ 2 ( n + 1 ) 3 ] 1 / 2 G n ( 2 , 2 , r ) .
{ z 1 + [ 1 2 z ( 1 2 t ) + z 2 ] 1 / 2 } m ( 2 z t ) m [ 1 2 z ( 1 2 t ) + z 2 ] 1 / 2 = s = 0 ( m + s s ) G s ( m + 1 , m + 1 , t ) z s .
G n ( 2 , 2 , 1 ) = ( 1 ) n ( n + 1 ) .
( a n ) 1 / 2 = ( 1 ) n ( 2 n + 2 ) 1 / 2 .
Q n ( r ) = ( 1 ) n ( n + 1 n ) G n ( 2 , 2 , r ) .
G n ( 2 , 2 , r ) = 1 n + 1 s = 0 ( 1 ) s ( n + s + 1 ) ! ( n s ) ! s ! ( s + 1 ) ! r s .
Q n ( 2 , 2 , r ) = ( 1 ) n s = 0 n ( 1 ) s ( n + s + 1 ) ! ( n s ) ! s ! ( s + 1 ) ! r s .
{ z 1 + [ 1 2 z ( 1 2 t ) + z 2 ] 1 / 2 } m ( 2 z t ) m [ 1 2 z ( 1 2 t ) + z 2 ] 1 / 2 = s = 0 ( m + s s ) G s ( m + 1 , m + 1 , t ) z s .
Q n ( r ) = ( 1 ) n ( n + 1 n ) G n ( 2 , 2 , r ) .
z + 1 [ 1 + 2 z ( 1 2 r ) + z 2 ] 1 / 2 2 z r [ 1 + 2 z ( 1 2 r ) + z 2 ] 1 / 2 = s = 0 Q s ( r ) z s .
R m + 2 s | m | ( r ) = ( 1 ) n ( m + s s ) r m G s ( m + 1 , m + 1 , r 2 ) .
{ z + 1 [ 1 + 2 z ( 1 2 r 2 ) + z 2 ] 1 / 2 } m ( 2 z r ) m [ 1 + 2 z ( 1 2 r 2 ) + z 2 ] 1 / 2 = s = 0 R m + 2 s | m | ( r ) z s .
P m + s | m | ( r ) = ( 1 ) s ( 2 m + s + 1 s ) r m G s ( 2 m + 2 , 2 m + 2 , r ) .
{ z + 1 [ 1 + 2 z ( 1 2 r ) + z 2 ] 1 / 2 } 2 m + 1 ( 2 z ) 2 m + 1 r m + 1 [ 1 + 2 ( 1 2 r ) + z 2 ] 1 / 2 = s = 0 P m + s | m | ( r ) z s .
r Q n ( r 2 ) = R 2 n + 1 1 ( r ) ,
Q n ( r ) = P n 0 ( r ) .
r P n | m | ( r 2 ) = R 2 n + 1 2 | m | + 1 ( r ) .

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