Abstract

The Mellin radial harmonic filter and the logarithmic harmonic filter are useful for performing optical scale- and projection-invariant pattern recognition, respectively. To our knowledge, on the basis of the harmonic-function method, no one has been able to obtain more than one invariant property (in addition to the shift invariance) when using the matched-filter approach. A new method of combining the scale-, the projection-, and the shift-invariance properties is proposed, based on two decomposition stages of the input pattern. Computer simulations are presented as well as preliminary experimental results.

© 1995 Optical Society of America

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References

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  1. A. VanderLugt, “Signal detection by complex spatial filtering,” IEEE Trans. Inf. Theory IT-10, 139–145 (1964).
    [Crossref]
  2. Y. N. Hsu, H. H. Arsenault, “Optical pattern recognition using the circular harmonic expansion,” Apppl. Opt. 21, 4016–4019 (1982).
    [Crossref]
  3. D. Mendlovic, E. Marom, N. Konforti, “Shift and scale invariant pattern recognition using Mellen radial harmonics,” Opt. Commun. 67, 172–176 (1988).
    [Crossref]
  4. D. Mendlovic, N. Konforti, E. Marom, “Shift and projection invariant pattern recognition using logarithmic harmonics,” Appl. Opt. 29, 4784–4789 (1990).
    [Crossref] [PubMed]
  5. E. Marom, D. Mendlovic, N. Konforti, “Generalized spatial deformation harmonic filter for distortion invariant pattern recognition,” Opt. Commun. 78, 416–424 (1990).
    [Crossref]
  6. D. Mendlovic, N. Konforti, E. Marom, “Scale and projection invariant pattern recognition,” Appl. Opt. 28, 4982–4986 (1989).
    [Crossref] [PubMed]
  7. D. Casasent, W. T. Chang, “Correlation synthetic discriminant functions,” Appl. Opt. 25, 2343–2350 (1986).
    [Crossref] [PubMed]
  8. A. Moya, E. Tajahuerce, J. Garcia, D. Mendlovic, C. Ferreira, “Method for determining the proper expansion center and order for Mellin radial harmonic filters,” Opt. Commun. 103, 39–45 (1993).
    [Crossref]
  9. D. Mendlovic, Z. Zalevsky, J. Garcia, C. Ferreira, “Logarithmic harmonics proper expansion center and order for efficient projection invariant pattern recognition,” Opt. Commun. 107, 292–299 (1994).
    [Crossref]
  10. A. W. Lohmann, D. P. Paris, “Binary Fraunhofer holograms, generated by computer,” Appl. Opt. 6, 1739–1748 (1967).
    [Crossref] [PubMed]

1994 (1)

D. Mendlovic, Z. Zalevsky, J. Garcia, C. Ferreira, “Logarithmic harmonics proper expansion center and order for efficient projection invariant pattern recognition,” Opt. Commun. 107, 292–299 (1994).
[Crossref]

1993 (1)

A. Moya, E. Tajahuerce, J. Garcia, D. Mendlovic, C. Ferreira, “Method for determining the proper expansion center and order for Mellin radial harmonic filters,” Opt. Commun. 103, 39–45 (1993).
[Crossref]

1990 (2)

D. Mendlovic, N. Konforti, E. Marom, “Shift and projection invariant pattern recognition using logarithmic harmonics,” Appl. Opt. 29, 4784–4789 (1990).
[Crossref] [PubMed]

E. Marom, D. Mendlovic, N. Konforti, “Generalized spatial deformation harmonic filter for distortion invariant pattern recognition,” Opt. Commun. 78, 416–424 (1990).
[Crossref]

1989 (1)

1988 (1)

D. Mendlovic, E. Marom, N. Konforti, “Shift and scale invariant pattern recognition using Mellen radial harmonics,” Opt. Commun. 67, 172–176 (1988).
[Crossref]

1986 (1)

1982 (1)

Y. N. Hsu, H. H. Arsenault, “Optical pattern recognition using the circular harmonic expansion,” Apppl. Opt. 21, 4016–4019 (1982).
[Crossref]

1967 (1)

1964 (1)

A. VanderLugt, “Signal detection by complex spatial filtering,” IEEE Trans. Inf. Theory IT-10, 139–145 (1964).
[Crossref]

Arsenault, H. H.

Y. N. Hsu, H. H. Arsenault, “Optical pattern recognition using the circular harmonic expansion,” Apppl. Opt. 21, 4016–4019 (1982).
[Crossref]

Casasent, D.

Chang, W. T.

Ferreira, C.

D. Mendlovic, Z. Zalevsky, J. Garcia, C. Ferreira, “Logarithmic harmonics proper expansion center and order for efficient projection invariant pattern recognition,” Opt. Commun. 107, 292–299 (1994).
[Crossref]

A. Moya, E. Tajahuerce, J. Garcia, D. Mendlovic, C. Ferreira, “Method for determining the proper expansion center and order for Mellin radial harmonic filters,” Opt. Commun. 103, 39–45 (1993).
[Crossref]

Garcia, J.

D. Mendlovic, Z. Zalevsky, J. Garcia, C. Ferreira, “Logarithmic harmonics proper expansion center and order for efficient projection invariant pattern recognition,” Opt. Commun. 107, 292–299 (1994).
[Crossref]

A. Moya, E. Tajahuerce, J. Garcia, D. Mendlovic, C. Ferreira, “Method for determining the proper expansion center and order for Mellin radial harmonic filters,” Opt. Commun. 103, 39–45 (1993).
[Crossref]

Hsu, Y. N.

Y. N. Hsu, H. H. Arsenault, “Optical pattern recognition using the circular harmonic expansion,” Apppl. Opt. 21, 4016–4019 (1982).
[Crossref]

Konforti, N.

E. Marom, D. Mendlovic, N. Konforti, “Generalized spatial deformation harmonic filter for distortion invariant pattern recognition,” Opt. Commun. 78, 416–424 (1990).
[Crossref]

D. Mendlovic, N. Konforti, E. Marom, “Shift and projection invariant pattern recognition using logarithmic harmonics,” Appl. Opt. 29, 4784–4789 (1990).
[Crossref] [PubMed]

D. Mendlovic, N. Konforti, E. Marom, “Scale and projection invariant pattern recognition,” Appl. Opt. 28, 4982–4986 (1989).
[Crossref] [PubMed]

D. Mendlovic, E. Marom, N. Konforti, “Shift and scale invariant pattern recognition using Mellen radial harmonics,” Opt. Commun. 67, 172–176 (1988).
[Crossref]

Lohmann, A. W.

Marom, E.

E. Marom, D. Mendlovic, N. Konforti, “Generalized spatial deformation harmonic filter for distortion invariant pattern recognition,” Opt. Commun. 78, 416–424 (1990).
[Crossref]

D. Mendlovic, N. Konforti, E. Marom, “Shift and projection invariant pattern recognition using logarithmic harmonics,” Appl. Opt. 29, 4784–4789 (1990).
[Crossref] [PubMed]

D. Mendlovic, N. Konforti, E. Marom, “Scale and projection invariant pattern recognition,” Appl. Opt. 28, 4982–4986 (1989).
[Crossref] [PubMed]

D. Mendlovic, E. Marom, N. Konforti, “Shift and scale invariant pattern recognition using Mellen radial harmonics,” Opt. Commun. 67, 172–176 (1988).
[Crossref]

Mendlovic, D.

D. Mendlovic, Z. Zalevsky, J. Garcia, C. Ferreira, “Logarithmic harmonics proper expansion center and order for efficient projection invariant pattern recognition,” Opt. Commun. 107, 292–299 (1994).
[Crossref]

A. Moya, E. Tajahuerce, J. Garcia, D. Mendlovic, C. Ferreira, “Method for determining the proper expansion center and order for Mellin radial harmonic filters,” Opt. Commun. 103, 39–45 (1993).
[Crossref]

E. Marom, D. Mendlovic, N. Konforti, “Generalized spatial deformation harmonic filter for distortion invariant pattern recognition,” Opt. Commun. 78, 416–424 (1990).
[Crossref]

D. Mendlovic, N. Konforti, E. Marom, “Shift and projection invariant pattern recognition using logarithmic harmonics,” Appl. Opt. 29, 4784–4789 (1990).
[Crossref] [PubMed]

D. Mendlovic, N. Konforti, E. Marom, “Scale and projection invariant pattern recognition,” Appl. Opt. 28, 4982–4986 (1989).
[Crossref] [PubMed]

D. Mendlovic, E. Marom, N. Konforti, “Shift and scale invariant pattern recognition using Mellen radial harmonics,” Opt. Commun. 67, 172–176 (1988).
[Crossref]

Moya, A.

A. Moya, E. Tajahuerce, J. Garcia, D. Mendlovic, C. Ferreira, “Method for determining the proper expansion center and order for Mellin radial harmonic filters,” Opt. Commun. 103, 39–45 (1993).
[Crossref]

Paris, D. P.

Tajahuerce, E.

A. Moya, E. Tajahuerce, J. Garcia, D. Mendlovic, C. Ferreira, “Method for determining the proper expansion center and order for Mellin radial harmonic filters,” Opt. Commun. 103, 39–45 (1993).
[Crossref]

VanderLugt, A.

A. VanderLugt, “Signal detection by complex spatial filtering,” IEEE Trans. Inf. Theory IT-10, 139–145 (1964).
[Crossref]

Zalevsky, Z.

D. Mendlovic, Z. Zalevsky, J. Garcia, C. Ferreira, “Logarithmic harmonics proper expansion center and order for efficient projection invariant pattern recognition,” Opt. Commun. 107, 292–299 (1994).
[Crossref]

Appl. Opt. (4)

Apppl. Opt. (1)

Y. N. Hsu, H. H. Arsenault, “Optical pattern recognition using the circular harmonic expansion,” Apppl. Opt. 21, 4016–4019 (1982).
[Crossref]

IEEE Trans. Inf. Theory (1)

A. VanderLugt, “Signal detection by complex spatial filtering,” IEEE Trans. Inf. Theory IT-10, 139–145 (1964).
[Crossref]

Opt. Commun. (4)

D. Mendlovic, E. Marom, N. Konforti, “Shift and scale invariant pattern recognition using Mellen radial harmonics,” Opt. Commun. 67, 172–176 (1988).
[Crossref]

E. Marom, D. Mendlovic, N. Konforti, “Generalized spatial deformation harmonic filter for distortion invariant pattern recognition,” Opt. Commun. 78, 416–424 (1990).
[Crossref]

A. Moya, E. Tajahuerce, J. Garcia, D. Mendlovic, C. Ferreira, “Method for determining the proper expansion center and order for Mellin radial harmonic filters,” Opt. Commun. 103, 39–45 (1993).
[Crossref]

D. Mendlovic, Z. Zalevsky, J. Garcia, C. Ferreira, “Logarithmic harmonics proper expansion center and order for efficient projection invariant pattern recognition,” Opt. Commun. 107, 292–299 (1994).
[Crossref]

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Figures (11)

Fig. 1
Fig. 1

F-18 airplane object for computer-simulation tests (outline version).

Fig. 2
Fig. 2

Four shifted, scaled, and projected F-18 airplane input patterns.

Fig. 3
Fig. 3

Correlation at the output plane for the input of Fig. 2.

Fig. 4
Fig. 4

Different input object for computer-simulation tests (out-line version).

Fig. 5
Fig. 5

Correlation at the output plane for the input of Fig. 4.

Fig. 6
Fig. 6

Experimental results for the correlation with the F-18 input, scaled and projected by the factor values of s = 1 and p = 1. The correlation peak is obtained at the negative-first diffraction order, on the left side of the picture).

Fig. 7
Fig. 7

Same as Fig. 6 but with the factor values of s = 0.6 and p = 0.75.

Fig. 8
Fig. 8

Same as Fig. 6 but with the factor values of s = 1 and p = 1.25.

Fig. 9
Fig. 9

Same as Fig. 6 but with the factor values of s = 0.8 and p = 0.75.

Fig. 10
Fig. 10

Same as Fig. 6 but with the factor values of s = 1 and p = 0.75.

Fig. 11
Fig. 11

Same as Fig. 6 but for the Tornado airplane input and with the factor values of s = 1 and p = 1.

Equations (38)

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f ( r , θ ; x 0 , y 0 ) = N = - N = f N ( θ ; x 0 , y 0 ) r i 2 π N - 1 ,
f N ( θ ; x 0 , y 0 ) = 1 L r 0 R f ( r , θ ; x 0 , y 0 ) r - i 2 π N - 1 r d r ;
f ( x , y ) = 1 2 L N = - N = f N ( y ) x i 2 π N - 1 / 2 ,
f N ( y ) = - X x 0 f ( x , y ) ( - x ) - i 2 π N - 1 / 2 d x + x 0 X f ( x , y ) x - i 2 π N - 1 / 2 d x ,
f r ( r , θ ) = f M 1 ( θ ) r i 2 π M 1 - 1
f M 1 ( θ ) = r 0 R f ( r , θ ) r - i 2 π M i - 1 r d r ,
H M ( y ) = exp ( - L ) 1 f r ( r , θ ) x - i 2 π M - 1 / 2 d x .
C ( x , y ) = - - f ( x , y ; x , y ) H M * ( y ) x - i 2 π M - 1 / 2 d x d y ,
f ( x , y ; x , y ) = f ( x + x , y + y ) .
f [ ( x + a ) + x , ( y + b ) + y ] = f [ x + ( x + a ) , y + ( y + b ) ] .
g ( x , y ; x , y ) = f ( α x , α y ; x , y ) ,
C ( α ) ( x , y ) = - - g ( x , y ; x , y ) H M * ( y ) x - i 2 π M - 1 / 2 d x d y .
C ( x , y ) = A C ( α ) ( x , y ) ,
f ( x , y ) = 1 2 L N f N ( y ) x i 2 π N - 1 / 2 ,
f N ( y ) = exp ( - L ) 1 f ( x , y ) x - i 2 π N - 1 / 2 d x .
g ( x , y ) = f ( α x , α y ) = 1 2 L N g N ( y ) x i 2 π N - 1 / 2 ,
g N ( y ) = exp ( - L ) / α 1 / α f ( α x , α y ) x - i 2 π N - 1 / 2 d x .
g N ( y ) = ( 1 α ) - i 2 π N + 1 / 2 exp ( - L ) 1 f ( z , α y ) z - i 2 π N - 1 / 2 d z .
g N ( y α ) = ( 1 α ) - i 2 π N + 1 / 2 exp ( - L ) 1 f ( z , y ) z - i 2 π N - 1 / 2 d z .
g N ( y α ) = ( 1 α ) - i 2 π N + 1 / 2 exp ( - L ) 1 f ( x , y ) x - i 2 π N - 1 / 2 d x ,
g N ( y ) = ( 1 / α ) - i 2 π N + 1 / 2 f N ( α y ) .
C ( α ) ( x , y ) = 1 2 L N - exp ( - L ) 1 α i 2 π N - 1 / 2 f N ( α y ; x , y ) × x i 2 π N - 1 / 2 H M * ( y ) x - i 2 π M - 1 / 2 d x d y = 1 2 L N - exp ( - L ) 1 α i 2 π N - 3 / 2 f N ( y ; x , y ) × x i 2 π N - 1 / 2 x - i 2 π M - 1 / 2 H M * ( y α ) d x d y ,
exp ( - L ) 1 x i 2 π N - 1 / 2 x - i 2 π M - 1 / 2 d x = { 0 for N = M 0 otherwise ,
C ( α ) ( x , y ) = A - f M ( y ; x , y ) H M * ( y α ) d y .
C ( x , y ) = - f M ( y ; x , y ) H M * ( y ) d y ;
H M * ( y ) = B H M * ( y / α ) ,
C ( β ) ( x , y ) = - - f ( β x , y ; x , y ) H M * ( y ) x - i 2 π - 1 / 2 d x d y .
f ( β x , y ) = 2 2 L N f N ( β ) ( y ) x i 2 π N - 1 / 2
f N ( β ) ( y ) = exp ( - L ) / β 1 / β f ( β x , y ) x - i 2 π N - 1 / 2 d x .
f N ( β ) ( y ) = ( 1 β ) - i 2 π N + 1 / 2 exp ( - L ) 1 f ( z , y ) z - i 2 π N - 1 / 2 d z = ( 1 β ) - i 2 π N + 1 / 2 f N ( y ) .
f ( β x , y ) = 1 2 L N ( 1 β ) - i 2 π N + 1 / 2 f N ( y ) x i 2 π N - 1 / 2 .
C ( β ) ( x , y ) = 1 2 L - N β i 2 π N - 1 / 2 f N ( y ) H M * ( y ) × exp ( - L ) 1 x i 2 π N - 1 / 2 x - i 2 π M - 1 / 2 d x d y .
C ( β ) ( x , y ) = ( 1 β ) - i 2 π M + 1 / 2 - f M ( y ) H M * ( y ) d y .
H M ( y ) = exp ( - L ) 1 f r ( r , θ ) x - i 2 π M - 1 / 2 d x = exp ( - L ) 1 f r ( x , y ) x - i 2 π M - 1 / 2 d x = 0 f r ( x , y ) x - 2 π M - 1 / 2 d x ,
f r ( r / β , θ ) = f r ( x / β , y / β ) = f M 1 ( θ ) ( 1 / β ) i 2 π M 1 - 1 r i 2 π M 1 - 1 = ( 1 / β ) i 2 π M 1 - 1 f r ( x , y ) ;
H M ( y / β ) = 0 f r ( x , y / β ) x - i 2 π M - 1 / 2 d x = 0 f r ( x / β , y / β ) ( x / β ) - i 2 π M - 1 / 2 d x β .
H M ( y / β ) = K 0 f r ( x , y ) x - i 2 π M - 1 / 2 d x = K exp ( - L ) 1 f r ( x , y ) x - i 2 π M - 1 / 2 d x = K H M ( y ) ,
K = ( β ) - i 2 π ( M 1 - M ) - 3 / 2 .

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