Abstract

Given an ensemble of images, we consider a method for the extraction of optimal features that are invariant under two-dimensional rotations. Our method consists of first identifying some characteristics of the optimal basis functions and then applying the Karhunen–Loeve algorithm to determine them completely. We perform a few classification experiments with the invariant features, using a neural network as a classifier. The new method is compared with the Zernike moments method.

© 1994 Optical Society of America

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References

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  1. M. K. Hu, “Visual pattern recognition by moment invariants,” IRE Trans. Inf. Theory 8, 179–187 (1962).
    [CrossRef]
  2. T. H. Reiss, “The revised fundamental theorem of moment invariants,” IEEE Pattern Anal. Mach. Intell. 13, 830–834 (1991).
    [CrossRef]
  3. M. Ferraro, T. Caelli, “Relationship between integral transform invariance and Lie group theory,” J. Opt. Soc. Am. A 5, 738–742 (1988).
    [CrossRef]
  4. J. Segman, J. Rubinstein, Y. Y. Zeevi, “The canonical coordinates method for pattern recognition: Theoretical and computational considerations,” IEEE Trans. Pattern Anal. Mach. Intell. 14, 1171–1183 (1992).
    [CrossRef]
  5. K. Kanatani, Group Theoretical Methods in Image Understanding (Springer-Verlag, Berlin, 1989).
  6. R. Lenz, Group Theoretical Methods in Image Processing (Springer-Verlag, Berlin, 1990).
    [CrossRef]
  7. N. Blatt, J. Rubinstein, “The canonical coordinates method II: isomorphism with affine transformations,” Pattern Recognition (to be published).
  8. L. Sirovich, “Turbulence and the dynamics of coherent structures,” Quart. Appl. Math. XLV, 561–590 (1987).
  9. R. Lenz, “Optimal filters for the detection of linear patterns in 2-D and higher dimensional images,” Pattern Recognition 20, 163–172 (1987).
    [CrossRef]
  10. J. Rubinstein, “Invariant low dimensional approximations for distorted patterns,” Technion Tech. Rep. (Technion, Haifa, Israel, 1990).
  11. M. R. Teague, “Image analysis via the general theory of moments,” J. Opt. Soc. Am. 70, 920–930 (1980).
    [CrossRef]
  12. A. Khotanzad, J. H. Lu, “Object recognition using a neural network and invariant Zernike features,” in Proceedings of the IEEE Conference on Vision and Pattern Recognition, (Institute of Electrical and Electronics Engineers, New York, 1989), pp. 200–205.
  13. L. Sirovich, M. Kirbi, “Low dimensional procedure for the characterization of human faces,” J. Opt. Soc. Am. A 4, 519–524 (1987).
    [CrossRef] [PubMed]
  14. N. J. Vilenkin, Special Functions and the Theory of Group Representations (American Mathematical Society, Providence, R.I., 1968).
  15. R. Lippman, “An introduction to computing with neural nets,” IEEE Trans. Acoust. Speech Signal Process. ASSP-4, 179–187 (1987).
  16. D. E. Rumelhart, G. E. Hinton, R. J. Williams, “Learning internal representations by error propagation,” in Parallel Distributed Processing, D. E. Rumelhart, J. L. McClelland, eds. (MIT Press, Cambridge, Mass., 1986), pp. 318–362.
  17. J. L. Lumely, Stochastic Tools in Turbulence (Academic, New York, 1970).
  18. L. Sirovich, H. Park, “Turbulent thermal convection in finite domain: part I,” Phys. Fluids 2, 1649–1665 (1990).
    [CrossRef]
  19. N. Aubry, W. Y. Lian, E. Titi, “Preserving symmetries in the proper orthogonal decomposition,” SIAM J. Sci. Statist. Comp.14, 483–505 (1993).
    [CrossRef]

1992 (1)

J. Segman, J. Rubinstein, Y. Y. Zeevi, “The canonical coordinates method for pattern recognition: Theoretical and computational considerations,” IEEE Trans. Pattern Anal. Mach. Intell. 14, 1171–1183 (1992).
[CrossRef]

1991 (1)

T. H. Reiss, “The revised fundamental theorem of moment invariants,” IEEE Pattern Anal. Mach. Intell. 13, 830–834 (1991).
[CrossRef]

1990 (1)

L. Sirovich, H. Park, “Turbulent thermal convection in finite domain: part I,” Phys. Fluids 2, 1649–1665 (1990).
[CrossRef]

1988 (1)

1987 (4)

L. Sirovich, M. Kirbi, “Low dimensional procedure for the characterization of human faces,” J. Opt. Soc. Am. A 4, 519–524 (1987).
[CrossRef] [PubMed]

R. Lippman, “An introduction to computing with neural nets,” IEEE Trans. Acoust. Speech Signal Process. ASSP-4, 179–187 (1987).

L. Sirovich, “Turbulence and the dynamics of coherent structures,” Quart. Appl. Math. XLV, 561–590 (1987).

R. Lenz, “Optimal filters for the detection of linear patterns in 2-D and higher dimensional images,” Pattern Recognition 20, 163–172 (1987).
[CrossRef]

1980 (1)

1962 (1)

M. K. Hu, “Visual pattern recognition by moment invariants,” IRE Trans. Inf. Theory 8, 179–187 (1962).
[CrossRef]

Aubry, N.

N. Aubry, W. Y. Lian, E. Titi, “Preserving symmetries in the proper orthogonal decomposition,” SIAM J. Sci. Statist. Comp.14, 483–505 (1993).
[CrossRef]

Blatt, N.

N. Blatt, J. Rubinstein, “The canonical coordinates method II: isomorphism with affine transformations,” Pattern Recognition (to be published).

Caelli, T.

Ferraro, M.

Hinton, G. E.

D. E. Rumelhart, G. E. Hinton, R. J. Williams, “Learning internal representations by error propagation,” in Parallel Distributed Processing, D. E. Rumelhart, J. L. McClelland, eds. (MIT Press, Cambridge, Mass., 1986), pp. 318–362.

Hu, M. K.

M. K. Hu, “Visual pattern recognition by moment invariants,” IRE Trans. Inf. Theory 8, 179–187 (1962).
[CrossRef]

Kanatani, K.

K. Kanatani, Group Theoretical Methods in Image Understanding (Springer-Verlag, Berlin, 1989).

Khotanzad, A.

A. Khotanzad, J. H. Lu, “Object recognition using a neural network and invariant Zernike features,” in Proceedings of the IEEE Conference on Vision and Pattern Recognition, (Institute of Electrical and Electronics Engineers, New York, 1989), pp. 200–205.

Kirbi, M.

Lenz, R.

R. Lenz, “Optimal filters for the detection of linear patterns in 2-D and higher dimensional images,” Pattern Recognition 20, 163–172 (1987).
[CrossRef]

R. Lenz, Group Theoretical Methods in Image Processing (Springer-Verlag, Berlin, 1990).
[CrossRef]

Lian, W. Y.

N. Aubry, W. Y. Lian, E. Titi, “Preserving symmetries in the proper orthogonal decomposition,” SIAM J. Sci. Statist. Comp.14, 483–505 (1993).
[CrossRef]

Lippman, R.

R. Lippman, “An introduction to computing with neural nets,” IEEE Trans. Acoust. Speech Signal Process. ASSP-4, 179–187 (1987).

Lu, J. H.

A. Khotanzad, J. H. Lu, “Object recognition using a neural network and invariant Zernike features,” in Proceedings of the IEEE Conference on Vision and Pattern Recognition, (Institute of Electrical and Electronics Engineers, New York, 1989), pp. 200–205.

Lumely, J. L.

J. L. Lumely, Stochastic Tools in Turbulence (Academic, New York, 1970).

Park, H.

L. Sirovich, H. Park, “Turbulent thermal convection in finite domain: part I,” Phys. Fluids 2, 1649–1665 (1990).
[CrossRef]

Reiss, T. H.

T. H. Reiss, “The revised fundamental theorem of moment invariants,” IEEE Pattern Anal. Mach. Intell. 13, 830–834 (1991).
[CrossRef]

Rubinstein, J.

J. Segman, J. Rubinstein, Y. Y. Zeevi, “The canonical coordinates method for pattern recognition: Theoretical and computational considerations,” IEEE Trans. Pattern Anal. Mach. Intell. 14, 1171–1183 (1992).
[CrossRef]

J. Rubinstein, “Invariant low dimensional approximations for distorted patterns,” Technion Tech. Rep. (Technion, Haifa, Israel, 1990).

N. Blatt, J. Rubinstein, “The canonical coordinates method II: isomorphism with affine transformations,” Pattern Recognition (to be published).

Rumelhart, D. E.

D. E. Rumelhart, G. E. Hinton, R. J. Williams, “Learning internal representations by error propagation,” in Parallel Distributed Processing, D. E. Rumelhart, J. L. McClelland, eds. (MIT Press, Cambridge, Mass., 1986), pp. 318–362.

Segman, J.

J. Segman, J. Rubinstein, Y. Y. Zeevi, “The canonical coordinates method for pattern recognition: Theoretical and computational considerations,” IEEE Trans. Pattern Anal. Mach. Intell. 14, 1171–1183 (1992).
[CrossRef]

Sirovich, L.

L. Sirovich, H. Park, “Turbulent thermal convection in finite domain: part I,” Phys. Fluids 2, 1649–1665 (1990).
[CrossRef]

L. Sirovich, M. Kirbi, “Low dimensional procedure for the characterization of human faces,” J. Opt. Soc. Am. A 4, 519–524 (1987).
[CrossRef] [PubMed]

L. Sirovich, “Turbulence and the dynamics of coherent structures,” Quart. Appl. Math. XLV, 561–590 (1987).

Teague, M. R.

Titi, E.

N. Aubry, W. Y. Lian, E. Titi, “Preserving symmetries in the proper orthogonal decomposition,” SIAM J. Sci. Statist. Comp.14, 483–505 (1993).
[CrossRef]

Vilenkin, N. J.

N. J. Vilenkin, Special Functions and the Theory of Group Representations (American Mathematical Society, Providence, R.I., 1968).

Williams, R. J.

D. E. Rumelhart, G. E. Hinton, R. J. Williams, “Learning internal representations by error propagation,” in Parallel Distributed Processing, D. E. Rumelhart, J. L. McClelland, eds. (MIT Press, Cambridge, Mass., 1986), pp. 318–362.

Zeevi, Y. Y.

J. Segman, J. Rubinstein, Y. Y. Zeevi, “The canonical coordinates method for pattern recognition: Theoretical and computational considerations,” IEEE Trans. Pattern Anal. Mach. Intell. 14, 1171–1183 (1992).
[CrossRef]

IEEE Pattern Anal. Mach. Intell. (1)

T. H. Reiss, “The revised fundamental theorem of moment invariants,” IEEE Pattern Anal. Mach. Intell. 13, 830–834 (1991).
[CrossRef]

IEEE Trans. Acoust. Speech Signal Process. (1)

R. Lippman, “An introduction to computing with neural nets,” IEEE Trans. Acoust. Speech Signal Process. ASSP-4, 179–187 (1987).

IEEE Trans. Pattern Anal. Mach. Intell. (1)

J. Segman, J. Rubinstein, Y. Y. Zeevi, “The canonical coordinates method for pattern recognition: Theoretical and computational considerations,” IEEE Trans. Pattern Anal. Mach. Intell. 14, 1171–1183 (1992).
[CrossRef]

IRE Trans. Inf. Theory (1)

M. K. Hu, “Visual pattern recognition by moment invariants,” IRE Trans. Inf. Theory 8, 179–187 (1962).
[CrossRef]

J. Opt. Soc. Am. (1)

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

Pattern Recognition (1)

R. Lenz, “Optimal filters for the detection of linear patterns in 2-D and higher dimensional images,” Pattern Recognition 20, 163–172 (1987).
[CrossRef]

Phys. Fluids (1)

L. Sirovich, H. Park, “Turbulent thermal convection in finite domain: part I,” Phys. Fluids 2, 1649–1665 (1990).
[CrossRef]

Quart. Appl. Math. (1)

L. Sirovich, “Turbulence and the dynamics of coherent structures,” Quart. Appl. Math. XLV, 561–590 (1987).

Other (9)

K. Kanatani, Group Theoretical Methods in Image Understanding (Springer-Verlag, Berlin, 1989).

R. Lenz, Group Theoretical Methods in Image Processing (Springer-Verlag, Berlin, 1990).
[CrossRef]

N. Blatt, J. Rubinstein, “The canonical coordinates method II: isomorphism with affine transformations,” Pattern Recognition (to be published).

J. Rubinstein, “Invariant low dimensional approximations for distorted patterns,” Technion Tech. Rep. (Technion, Haifa, Israel, 1990).

A. Khotanzad, J. H. Lu, “Object recognition using a neural network and invariant Zernike features,” in Proceedings of the IEEE Conference on Vision and Pattern Recognition, (Institute of Electrical and Electronics Engineers, New York, 1989), pp. 200–205.

N. J. Vilenkin, Special Functions and the Theory of Group Representations (American Mathematical Society, Providence, R.I., 1968).

D. E. Rumelhart, G. E. Hinton, R. J. Williams, “Learning internal representations by error propagation,” in Parallel Distributed Processing, D. E. Rumelhart, J. L. McClelland, eds. (MIT Press, Cambridge, Mass., 1986), pp. 318–362.

J. L. Lumely, Stochastic Tools in Turbulence (Academic, New York, 1970).

N. Aubry, W. Y. Lian, E. Titi, “Preserving symmetries in the proper orthogonal decomposition,” SIAM J. Sci. Statist. Comp.14, 483–505 (1993).
[CrossRef]

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Figures (11)

Fig. 1
Fig. 1

Set of 24 test library images. The resolution is 100 × 100 pixels.

Fig. 2
Fig. 2

Test images in the polar plane.

Fig. 3
Fig. 3

Transformation of some of the library images from the polar plane back to the Cartesian plane.

Fig. 4
Fig. 4

Real parts of the optimal basis functions ϕl, l = 1, 2,…, 25.

Fig. 5
Fig. 5

Absolute values of the optimal basis functions ϕl, l = 1, 2,…, 25.

Fig. 6
Fig. 6

Phases of the optimal basis functions ϕl, l = 1, 2,…, 25.

Fig. 7
Fig. 7

Reconstruction of the 24 library images with the use of 50 IKL coefficients.

Fig. 8
Fig. 8

Reconstruction of the 24 library images with the use of 200 IKL coefficients.

Fig. 9
Fig. 9

Mean square error in the reconstruction of the library images with the use of 50 IKL coefficients.

Fig. 10
Fig. 10

Reconstruction of the 24 library images with the use of 50 ZM’s.

Fig. 11
Fig. 11

Reconstruction of the 24 library images with the use of 200 ZM’s.

Tables (3)

Tables Icon

Table 1 Performance Table of the Recognition Experiments with the Multilayer Perceptron for the IKL and ZM Methods

Tables Icon

Table 2 Statistics of the Pairwise Difference between the Normalized Feature Vectors for the IKL and the ZM Methods

Tables Icon

Table 3 Statistics of the Scattering of Feature Vectors for Each Image and Four Rotations of it

Equations (23)

Equations on this page are rendered with MathJax. Learn more.

1 N j = 1 N f j f j 2
f = l = 1 L f , ϕ l ϕ l .
I [ ϕ l , f i ] = I [ ϕ l , T f i ] ,
j = 1 L 1 N j = 1 N | α l j | 2 ,
max 1 N j = 1 N | α 1 j | 2 , ϕ 1 = 1 .
max 1 N j = 1 N | α 2 j | 2 , ϕ 2 = 1 , ϕ 2 , ϕ 1 = 0 ,
ϕ l ( r , θ ) = exp ( i n l θ ) h l ( r ) ,
f j ( r , θ ) = k b k j ( r ) exp ( i k θ )
max 1 N j = 1 N | f j , ϕ 1 | 2 = max 0 0 K n 1 ( r , r ) h 1 ( r ) h 1 ( r ) r d r r d r ,
K k ( r , r ) = 1 N j = 1 N b k j ( r ) b ¯ k j ( r ) .
max 1 N j = 1 N | f j , ϕ 1 | 2 = λ n 1 1 .
b k j ( r ) = 0 2 π f j ( r , θ ) exp ( i k θ ) d θ , k = 0 , 1 , 2 , .
ϕ l ( r , θ ) = exp ( i n l θ ) h l ( r ) , l = 1 , 2 , , L .
f j f j 1 N k = 1 N f k .
f a , t = f I ( a , t ) , J ( a , t ) ,
I ( a , t ) = ( a r max 2 / R ) 1 / 2 cos t π T , J ( a , t ) = ( a r max 2 / R ) 1 / 2 sin t π T .
F a , τ j = 1 2 T t f a , t j exp ( i τ t π / T ) , j = 1 , 2 , , N , a = 1 , 2 , , R , τ = 0 , 1 , , T 1.
K a , a τ = 1 N j = 1 N F a , τ j F a , τ j , τ = 0 , 1 , , T 1 , a , a = 1 , 2 , , R .
α l = a = 1 R t = 1 2 T f a , t exp ( i n l t π T ) υ ¯ a l , l = 1 , 2 , , L ,
MSE ( f ) = 1 10 4 I J ( f I , J f I , J ) 2 .
V n m ( r , θ ) = exp ( i m θ ) R n m ( r ) ,
R n m ( r ) = n + 1 × s = 0 ( n | m | ) / 2 ( 1 ) s ( n s ) ! s ! ( n + | m | 2 s ) ! ( n | m | 2 s ) ! r n 2 s , 0 r 1 .
J i = 1 1 + exp ( j W j i Y j ) ,

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