Abstract

The new techniques of amplitude compensated matched filtering using circular harmonic expansion and a Mellin transform are presented. These techniques yield much better discrimination and sharper autocorrelation peaks compared with the classical matched spatial filtering. Aside from these advantages, the amplitude compensated circular harmonic expansion matched filtering can also yield rotation invariance, and the amplitude compensated Mellin transform matched filtering can yield scale invariance. By computer simulation, we have verified the unique advantages of the new methods. We also provide 3-D graphs of autocorrelation and cross correlation.

© 1990 Optical Society of America

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References

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  1. A. B. VanderLugt, “Signal Detection By Complex Spatial Filtering,” IEEE Trans. Inf. Theory IT-10, 139–145 (1964).
    [CrossRef]
  2. A. V. Oppenheim, J. S. Lim, “The Importance of Phase in Signals,” Proc. IEEE 69, 529–541 (1981).
    [CrossRef]
  3. J. L. Horner, P. D. Gianino, “Phase-Only Matched Filtering,” Appl. Opt. 23, 812–816 (1984).
    [CrossRef] [PubMed]
  4. G. G. Mu, X.-M. Wang, Z.-Q. Wang, “Amplitude-Compensated Matched Filtering,” Appl. Opt. 27, 3461–3463 (1988).
    [CrossRef] [PubMed]
  5. Y-N. Hsu, H. H. Arsenault, G. April, “Rotation-Invariant Digital Pattern Recognition Using Circular Harmonic Expansion,” Appl. Opt. 21, 4012–4015 (1982).
    [CrossRef] [PubMed]
  6. Y-N. Hsu, H. H. Arsenault, “Optical Pattern Recognition Using Circular Harmonic Expansion,” Appl. Opt. 21, 4016–4019 (1982).
    [CrossRef] [PubMed]
  7. D. Casasent, D. Psaltis, “Scale Invariant Optical Correlation Using Mellin Transforms,” Opt. Commun. 17, 59–63 (1976).
    [CrossRef]
  8. J. Rosen, J. Shamir, “Circular Harmonic Phase Filters for Efficient Rotation-Invariant Pattern Recognition,” Appl. Opt. 27, 2895–2899 (1988).
    [CrossRef] [PubMed]
  9. D. L. Flannery, J. S. Loomis, M. E. Milkovich, “Transform-Ratio Ternary Phase-Amplitude Filter Formulation for Improved Correlation Discrimination,” Appl. Opt. 27, 4079–4083 (1988).
    [CrossRef] [PubMed]
  10. D. Mendlovic, E. Marom, N. Konforti, “Shift and Scale Invariant Pattern Recognition Using Mellin Radial Harmonics,” Opt. Commun. 67, 172–176 (1988).
    [CrossRef]
  11. J. Rosen, J. Shamir, “Scale Invariant Pattern Recognition with Logarithmic Radial Harmonic Filters,” Appl. Opt. 28, 240–244 (1989).
    [CrossRef] [PubMed]
  12. D. Mendlovic, N. Konforti, E. Marom, “Scale and Projection Invariant Pattern Recognition,” Appl. Opt. 28, 4982–4986 (1989).
    [CrossRef] [PubMed]
  13. F. T. S. Yu, Optical Information Processing (Wiley–Interscience, New York, 1983), pp. 198–202.
  14. S. M. Arnold, “Electron Beam Fabrication of Computer Generated Hologram,” Opt. Eng. 24, 803–807 (1985).

1989 (2)

1988 (4)

1985 (1)

S. M. Arnold, “Electron Beam Fabrication of Computer Generated Hologram,” Opt. Eng. 24, 803–807 (1985).

1984 (1)

1982 (2)

1981 (1)

A. V. Oppenheim, J. S. Lim, “The Importance of Phase in Signals,” Proc. IEEE 69, 529–541 (1981).
[CrossRef]

1976 (1)

D. Casasent, D. Psaltis, “Scale Invariant Optical Correlation Using Mellin Transforms,” Opt. Commun. 17, 59–63 (1976).
[CrossRef]

1964 (1)

A. B. VanderLugt, “Signal Detection By Complex Spatial Filtering,” IEEE Trans. Inf. Theory IT-10, 139–145 (1964).
[CrossRef]

April, G.

Arnold, S. M.

S. M. Arnold, “Electron Beam Fabrication of Computer Generated Hologram,” Opt. Eng. 24, 803–807 (1985).

Arsenault, H. H.

Casasent, D.

D. Casasent, D. Psaltis, “Scale Invariant Optical Correlation Using Mellin Transforms,” Opt. Commun. 17, 59–63 (1976).
[CrossRef]

Flannery, D. L.

Gianino, P. D.

Horner, J. L.

Hsu, Y-N.

Konforti, N.

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 Mellin Radial Harmonics,” Opt. Commun. 67, 172–176 (1988).
[CrossRef]

Lim, J. S.

A. V. Oppenheim, J. S. Lim, “The Importance of Phase in Signals,” Proc. IEEE 69, 529–541 (1981).
[CrossRef]

Loomis, J. S.

Marom, E.

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 Mellin Radial Harmonics,” Opt. Commun. 67, 172–176 (1988).
[CrossRef]

Mendlovic, D.

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 Mellin Radial Harmonics,” Opt. Commun. 67, 172–176 (1988).
[CrossRef]

Milkovich, M. E.

Mu, G. G.

Oppenheim, A. V.

A. V. Oppenheim, J. S. Lim, “The Importance of Phase in Signals,” Proc. IEEE 69, 529–541 (1981).
[CrossRef]

Psaltis, D.

D. Casasent, D. Psaltis, “Scale Invariant Optical Correlation Using Mellin Transforms,” Opt. Commun. 17, 59–63 (1976).
[CrossRef]

Rosen, J.

Shamir, J.

VanderLugt, A. B.

A. B. VanderLugt, “Signal Detection By Complex Spatial Filtering,” IEEE Trans. Inf. Theory IT-10, 139–145 (1964).
[CrossRef]

Wang, X.-M.

Wang, Z.-Q.

Yu, F. T. S.

F. T. S. Yu, Optical Information Processing (Wiley–Interscience, New York, 1983), pp. 198–202.

Appl. Opt. (8)

IEEE Trans. Inf. Theory (1)

A. B. VanderLugt, “Signal Detection By Complex Spatial Filtering,” IEEE Trans. Inf. Theory IT-10, 139–145 (1964).
[CrossRef]

Opt. Commun. (2)

D. Mendlovic, E. Marom, N. Konforti, “Shift and Scale Invariant Pattern Recognition Using Mellin Radial Harmonics,” Opt. Commun. 67, 172–176 (1988).
[CrossRef]

D. Casasent, D. Psaltis, “Scale Invariant Optical Correlation Using Mellin Transforms,” Opt. Commun. 17, 59–63 (1976).
[CrossRef]

Opt. Eng. (1)

S. M. Arnold, “Electron Beam Fabrication of Computer Generated Hologram,” Opt. Eng. 24, 803–807 (1985).

Proc. IEEE (1)

A. V. Oppenheim, J. S. Lim, “The Importance of Phase in Signals,” Proc. IEEE 69, 529–541 (1981).
[CrossRef]

Other (1)

F. T. S. Yu, Optical Information Processing (Wiley–Interscience, New York, 1983), pp. 198–202.

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Figures (8)

Fig. 1
Fig. 1

Autocorrelation with amplitude compensated circular harmonic expansion matched filter.

Fig. 2
Fig. 2

Input pattern (tank).

Fig. 3
Fig. 3

Autocorrelation with classical circular harmonic expansion matched filter.

Fig. 4
Fig. 4

Reference pattern O.

Fig. 5
Fig. 5

Autocorrelation with classical Mellin transform matched filter.

Fig. 6
Fig. 6

Autocorrelation with amplitude compensated Mellin transform matched filter.

Fig. 7
Fig. 7

Crosscorrelation with classical Mellin transform matched filter.

Fig. 8
Fig. 8

Crosscorrelation with amplitude compensated Mellin transform matched filter.

Equations (25)

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f ( r , θ ) = N = - f N ( r ) exp ( i N θ ) ,
f N ( r ) = 1 2 π 0 2 π f ( r , θ ) exp ( - i N θ ) d θ ,
F [ f ( r , θ ) ] = N = - F N ( p , q ) exp [ i φ N ( p , q ) ] .
{ N = - F N ( p , q ) exp [ i φ N ( p , q ) ] } · F M ( p , q ) exp [ - i φ M ( p , q ) ] = { F M ( p , q ) + N M F N ( p , q ) exp [ i φ N - φ M ] } · F M ( p , q ) .
g M ( α , β ) = F { F M ( p , q ) + N M F N ( p , q ) exp [ i ( φ N - φ N ) ] } * F [ F M ( p , q ) ] ,
F a ( p , q ) = 1 / F ( p , q ) = exp [ - i φ ( p , q ) ] / F ( p , q ) .
N = - F N ( p , q ) exp [ i φ N ( p , q ) ] · exp [ - i φ M ( p , q ) ] / F M ( p , q ) = 1 + N M F N ( p , q ) · exp [ i ( φ N - φ M ) ] / F M ( p , q ) .
g a ( α , β ) = δ ( α , β ) + F { N M F N ( p , q ) exp [ i ( φ N - φ M ) ] / F M ( p , q ) }
F a ( p , q ) = { F 0 exp [ - i φ M ( p , q ) ] / F M ( p , q ) [ F M ( p , q ) > F 0 ] exp [ - i φ M ( p , q ) ] [ F M ( p , q ) < F 0 ] ,
f ( r , θ + φ ) = N = - f N ( r ) exp ( i N θ ) exp ( i N φ ) .
F [ f ( r , θ + φ ) ] = N = - F N ( p , q ) exp [ i φ N ( p , q ) ] exp ( i N φ ) .
N = - F N ( p , q ) exp [ i φ N ( p , q ) ] exp ( i N φ ) · exp [ - i φ M ( p , q ) ] / F M ( p , q ) ,
g a ( φ ) ( α , β ) = F { N = - F N ( p , q ) exp [ i φ N ( p , q ) ] exp ( i N φ ) } F { exp [ i φ M ( p , q ) ] / F M ( p , q ) } = f ( r , θ + φ ) [ g M ( r ) exp ( i M θ ) ] ,
R M ( φ ) = 0 r d r 0 2 π f ( r , θ + φ ) g M * ( r ) exp ( - i M θ ) d θ .
R M ( φ ) = 2 π [ 0 r f M ( r ) g M * ( r ) d r ] exp ( i M φ ) .
R M ( ρ ) = 2 π [ 0 r f M ( r ) g M * ( r ) d r ] .
M ( p , q ) = M [ f ( ζ , η ) ] = M ( p , q ) exp [ i φ ( p , q ) ] ,
M [ f ( a ζ , a η ) ] = a i ( p + q ) M ( p , q ) .
K 1 M ( p , q ) M * ( p , q ) = K 1 M ( p , q ) 2 .
M ( p , q ) · K 2 exp [ - i φ ( p , q ) ] / M ( p , q ) = K 2 ,
g M ( α , β ) = K 1 F [ M ( p , q ) 2 ] ,
g a ( α , β ) = K 2 δ ( α , β ) .
F a ( p , q ) = { F 0 exp [ - i φ ( p , q ) ] / M ( p , q ) [ M ( p , q ) > F 0 ] exp [ - i φ ( p , q ) ] [ M ( p , q ) < F 0 ] ,
K 1 a i ( p + q ) M ( p , q ) 2 ,
K 2 a i ( p + q ) .

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