Abstract

A synthetic circular-harmonic phase-only filter is described. With this filter and a Fourier-transform correlator it is possible to obtain shift, rotation, and scaling-invariant correlations.

© 1995 Optical Society of America

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References

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  1. A. VanderLugt, “Signal detection by complex spatial filtering,” IRE Trans. Inf. Theory IT-10, 139–145 (1964).
    [CrossRef]
  2. C. F. Hester, D. Casasent, “Multivariant technique for multiclass pattern recognition,” Appl. Opt. 19, 1758–1761 (1980).
    [CrossRef] [PubMed]
  3. D. Casasent, “Unified synthetic discriminant function computation formulation,” Appl. Opt. 23, 1620–1627 (1984).
    [CrossRef] [PubMed]
  4. B. V. K. Vijaya Kumar, “Minimum variance synthetic discrimination correlation,” J. Opt. Soc. Am. A 3, 1579–1584 (1986).
    [CrossRef]
  5. A. Mahalanobis, B. V. K. Vijaya Kumar, D. Casasent, “Minimum average correlation energy filter,” Appl. Opt. 26, 3633–3640 (1987).
    [CrossRef] [PubMed]
  6. Y. N. Hsu, H. H. Arsenault, “Optical pattern reconition using circular harmonic expansion,” Appl. Opt. 21, 4016–4019 (1982).
    [CrossRef] [PubMed]
  7. R. Wu, H. Stark, “Rotation-invariant pattern recognition using a vector reference,” Appl. Opt. 23, 838–840 (1984).
    [CrossRef] [PubMed]
  8. Y. N. Hsu, H. H. Arsenault, “Pattern discrimination by multiple circular harmonic components,” Appl. Opt. 23, 841–844 (1984).
    [CrossRef] [PubMed]
  9. G. F. Schils, D. W. Sweeney, “Iterative technique for the synthesis of optical-correlation filters,” J. Opt. Soc. Am. A 3, 1433–1442 (1986).
    [CrossRef]
  10. L. Hassebrook, B. V. K. Vijaya Kumar, L. Hostler, “Linear phase coefficient composite filter banks for distortion-invariant optical pattern recognition,” Appl. Opt. 29, 1033–1043 (1990).
  11. L. Hassebrook, M. Rahmati, “Training set selection with multiple out-of-plane rotation parameters,” in Optical Pattern Recognition IV, D. P. Casasent, ed., Proc. Soc. Photo-Opt. Instrum. Eng. 1959, 32–43 (1993).
  12. L. Hassebrook, B. V. K. Vijaya Kumar, “Hybrid composite filter banks for distortion-invariant optical pattern recognition,” Opt. Eng. 31, 923–933 (1992).
    [CrossRef]
  13. B. V. K. Vijaya Kumar, “Tutorial survey of composite filter designs for optical correlators,” Appl. Opt. 31, 4773–4801 (1992).
    [CrossRef]
  14. J. L. Horner, P. D. Gianino, “Phase-only matched filtering,” Appl. Opt. 23, 812–816 (1984).
    [CrossRef] [PubMed]
  15. D. Psaltis, E. G. Paek, S. S. Venkatesh, “Optical image correlatin with binary spatial light modulator,” Opt. Eng. 23, 698–704 (1984).
  16. D. Jared, D. Ennis, “Inclusion of filter modulation in synthetic discriminant function construction,” Appl. Opt. 28, 232–239 (1989).
    [CrossRef] [PubMed]
  17. Z. Bahri, B. V. K. Vijaya Kumar, “Algorithm for designing phase-only synthetic discriminant function,” in Optical Information Processing Systems and Architectures, B. Javidi, ed., Proc. Soc. Photo-Opt. Instrum. Eng. 1151, 138–147 (1989).
  18. L. Leclerc, Y. Sheng, H. H. Arsenault, “Rotation invariant phase-only and binary phase-only correlation,” Appl. Opt. 28, 1251–1256 (1989).
    [CrossRef] [PubMed]
  19. L. Leclerc, Y. Sheng, H. H. Arsenault, “Optical binary phase-only filter for circular harmonic correlation,” Appl. Opt. 32, 4643–4649 (1991).
    [CrossRef]
  20. L. Hassebrook, R. W. Cohn, M. Liang, M. E. Lhamon, R. C. Daley, “Using pseudorandom phase-only encoding to approximate fully complex distortion-invariant filters,” in Optical Pattern Recognition V, D. P. Casasent, T.-H. Chao, eds., Proc. Soc. Photo-Opt. Instrum. Eng. 2237, 204–211 (1994).
  21. G. Ravichandran, D. Casasent, “Generalized in-plane rotation-invariant minimum average correlation energy filter,” Opt. Eng. 30, 1601–1606 (1991).
    [CrossRef]

1992

L. Hassebrook, B. V. K. Vijaya Kumar, “Hybrid composite filter banks for distortion-invariant optical pattern recognition,” Opt. Eng. 31, 923–933 (1992).
[CrossRef]

B. V. K. Vijaya Kumar, “Tutorial survey of composite filter designs for optical correlators,” Appl. Opt. 31, 4773–4801 (1992).
[CrossRef]

1991

L. Leclerc, Y. Sheng, H. H. Arsenault, “Optical binary phase-only filter for circular harmonic correlation,” Appl. Opt. 32, 4643–4649 (1991).
[CrossRef]

G. Ravichandran, D. Casasent, “Generalized in-plane rotation-invariant minimum average correlation energy filter,” Opt. Eng. 30, 1601–1606 (1991).
[CrossRef]

1990

L. Hassebrook, B. V. K. Vijaya Kumar, L. Hostler, “Linear phase coefficient composite filter banks for distortion-invariant optical pattern recognition,” Appl. Opt. 29, 1033–1043 (1990).

1989

1987

1986

1984

1982

1980

1964

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

Arsenault, H. H.

Bahri, Z.

Z. Bahri, B. V. K. Vijaya Kumar, “Algorithm for designing phase-only synthetic discriminant function,” in Optical Information Processing Systems and Architectures, B. Javidi, ed., Proc. Soc. Photo-Opt. Instrum. Eng. 1151, 138–147 (1989).

Casasent, D.

Cohn, R. W.

L. Hassebrook, R. W. Cohn, M. Liang, M. E. Lhamon, R. C. Daley, “Using pseudorandom phase-only encoding to approximate fully complex distortion-invariant filters,” in Optical Pattern Recognition V, D. P. Casasent, T.-H. Chao, eds., Proc. Soc. Photo-Opt. Instrum. Eng. 2237, 204–211 (1994).

Daley, R. C.

L. Hassebrook, R. W. Cohn, M. Liang, M. E. Lhamon, R. C. Daley, “Using pseudorandom phase-only encoding to approximate fully complex distortion-invariant filters,” in Optical Pattern Recognition V, D. P. Casasent, T.-H. Chao, eds., Proc. Soc. Photo-Opt. Instrum. Eng. 2237, 204–211 (1994).

Ennis, D.

Gianino, P. D.

Hassebrook, L.

L. Hassebrook, B. V. K. Vijaya Kumar, “Hybrid composite filter banks for distortion-invariant optical pattern recognition,” Opt. Eng. 31, 923–933 (1992).
[CrossRef]

L. Hassebrook, B. V. K. Vijaya Kumar, L. Hostler, “Linear phase coefficient composite filter banks for distortion-invariant optical pattern recognition,” Appl. Opt. 29, 1033–1043 (1990).

L. Hassebrook, M. Rahmati, “Training set selection with multiple out-of-plane rotation parameters,” in Optical Pattern Recognition IV, D. P. Casasent, ed., Proc. Soc. Photo-Opt. Instrum. Eng. 1959, 32–43 (1993).

L. Hassebrook, R. W. Cohn, M. Liang, M. E. Lhamon, R. C. Daley, “Using pseudorandom phase-only encoding to approximate fully complex distortion-invariant filters,” in Optical Pattern Recognition V, D. P. Casasent, T.-H. Chao, eds., Proc. Soc. Photo-Opt. Instrum. Eng. 2237, 204–211 (1994).

Hester, C. F.

Horner, J. L.

Hostler, L.

L. Hassebrook, B. V. K. Vijaya Kumar, L. Hostler, “Linear phase coefficient composite filter banks for distortion-invariant optical pattern recognition,” Appl. Opt. 29, 1033–1043 (1990).

Hsu, Y. N.

Jared, D.

Leclerc, L.

L. Leclerc, Y. Sheng, H. H. Arsenault, “Optical binary phase-only filter for circular harmonic correlation,” Appl. Opt. 32, 4643–4649 (1991).
[CrossRef]

L. Leclerc, Y. Sheng, H. H. Arsenault, “Rotation invariant phase-only and binary phase-only correlation,” Appl. Opt. 28, 1251–1256 (1989).
[CrossRef] [PubMed]

Lhamon, M. E.

L. Hassebrook, R. W. Cohn, M. Liang, M. E. Lhamon, R. C. Daley, “Using pseudorandom phase-only encoding to approximate fully complex distortion-invariant filters,” in Optical Pattern Recognition V, D. P. Casasent, T.-H. Chao, eds., Proc. Soc. Photo-Opt. Instrum. Eng. 2237, 204–211 (1994).

Liang, M.

L. Hassebrook, R. W. Cohn, M. Liang, M. E. Lhamon, R. C. Daley, “Using pseudorandom phase-only encoding to approximate fully complex distortion-invariant filters,” in Optical Pattern Recognition V, D. P. Casasent, T.-H. Chao, eds., Proc. Soc. Photo-Opt. Instrum. Eng. 2237, 204–211 (1994).

Mahalanobis, A.

Paek, E. G.

D. Psaltis, E. G. Paek, S. S. Venkatesh, “Optical image correlatin with binary spatial light modulator,” Opt. Eng. 23, 698–704 (1984).

Psaltis, D.

D. Psaltis, E. G. Paek, S. S. Venkatesh, “Optical image correlatin with binary spatial light modulator,” Opt. Eng. 23, 698–704 (1984).

Rahmati, M.

L. Hassebrook, M. Rahmati, “Training set selection with multiple out-of-plane rotation parameters,” in Optical Pattern Recognition IV, D. P. Casasent, ed., Proc. Soc. Photo-Opt. Instrum. Eng. 1959, 32–43 (1993).

Ravichandran, G.

G. Ravichandran, D. Casasent, “Generalized in-plane rotation-invariant minimum average correlation energy filter,” Opt. Eng. 30, 1601–1606 (1991).
[CrossRef]

Schils, G. F.

Sheng, Y.

L. Leclerc, Y. Sheng, H. H. Arsenault, “Optical binary phase-only filter for circular harmonic correlation,” Appl. Opt. 32, 4643–4649 (1991).
[CrossRef]

L. Leclerc, Y. Sheng, H. H. Arsenault, “Rotation invariant phase-only and binary phase-only correlation,” Appl. Opt. 28, 1251–1256 (1989).
[CrossRef] [PubMed]

Stark, H.

Sweeney, D. W.

VanderLugt, A.

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

Venkatesh, S. S.

D. Psaltis, E. G. Paek, S. S. Venkatesh, “Optical image correlatin with binary spatial light modulator,” Opt. Eng. 23, 698–704 (1984).

Vijaya Kumar, B. V. K.

L. Hassebrook, B. V. K. Vijaya Kumar, “Hybrid composite filter banks for distortion-invariant optical pattern recognition,” Opt. Eng. 31, 923–933 (1992).
[CrossRef]

B. V. K. Vijaya Kumar, “Tutorial survey of composite filter designs for optical correlators,” Appl. Opt. 31, 4773–4801 (1992).
[CrossRef]

L. Hassebrook, B. V. K. Vijaya Kumar, L. Hostler, “Linear phase coefficient composite filter banks for distortion-invariant optical pattern recognition,” Appl. Opt. 29, 1033–1043 (1990).

A. Mahalanobis, B. V. K. Vijaya Kumar, D. Casasent, “Minimum average correlation energy filter,” Appl. Opt. 26, 3633–3640 (1987).
[CrossRef] [PubMed]

B. V. K. Vijaya Kumar, “Minimum variance synthetic discrimination correlation,” J. Opt. Soc. Am. A 3, 1579–1584 (1986).
[CrossRef]

Z. Bahri, B. V. K. Vijaya Kumar, “Algorithm for designing phase-only synthetic discriminant function,” in Optical Information Processing Systems and Architectures, B. Javidi, ed., Proc. Soc. Photo-Opt. Instrum. Eng. 1151, 138–147 (1989).

Wu, R.

Appl. Opt.

L. Hassebrook, B. V. K. Vijaya Kumar, L. Hostler, “Linear phase coefficient composite filter banks for distortion-invariant optical pattern recognition,” Appl. Opt. 29, 1033–1043 (1990).

L. Leclerc, Y. Sheng, H. H. Arsenault, “Optical binary phase-only filter for circular harmonic correlation,” Appl. Opt. 32, 4643–4649 (1991).
[CrossRef]

C. F. Hester, D. Casasent, “Multivariant technique for multiclass pattern recognition,” Appl. Opt. 19, 1758–1761 (1980).
[CrossRef] [PubMed]

Y. N. Hsu, H. H. Arsenault, “Optical pattern reconition using circular harmonic expansion,” Appl. Opt. 21, 4016–4019 (1982).
[CrossRef] [PubMed]

J. L. Horner, P. D. Gianino, “Phase-only matched filtering,” Appl. Opt. 23, 812–816 (1984).
[CrossRef] [PubMed]

R. Wu, H. Stark, “Rotation-invariant pattern recognition using a vector reference,” Appl. Opt. 23, 838–840 (1984).
[CrossRef] [PubMed]

Y. N. Hsu, H. H. Arsenault, “Pattern discrimination by multiple circular harmonic components,” Appl. Opt. 23, 841–844 (1984).
[CrossRef] [PubMed]

D. Casasent, “Unified synthetic discriminant function computation formulation,” Appl. Opt. 23, 1620–1627 (1984).
[CrossRef] [PubMed]

A. Mahalanobis, B. V. K. Vijaya Kumar, D. Casasent, “Minimum average correlation energy filter,” Appl. Opt. 26, 3633–3640 (1987).
[CrossRef] [PubMed]

D. Jared, D. Ennis, “Inclusion of filter modulation in synthetic discriminant function construction,” Appl. Opt. 28, 232–239 (1989).
[CrossRef] [PubMed]

L. Leclerc, Y. Sheng, H. H. Arsenault, “Rotation invariant phase-only and binary phase-only correlation,” Appl. Opt. 28, 1251–1256 (1989).
[CrossRef] [PubMed]

B. V. K. Vijaya Kumar, “Tutorial survey of composite filter designs for optical correlators,” Appl. Opt. 31, 4773–4801 (1992).
[CrossRef]

IRE Trans. Inf. Theory

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

J. Opt. Soc. Am. A

Opt. Eng.

G. Ravichandran, D. Casasent, “Generalized in-plane rotation-invariant minimum average correlation energy filter,” Opt. Eng. 30, 1601–1606 (1991).
[CrossRef]

L. Hassebrook, B. V. K. Vijaya Kumar, “Hybrid composite filter banks for distortion-invariant optical pattern recognition,” Opt. Eng. 31, 923–933 (1992).
[CrossRef]

D. Psaltis, E. G. Paek, S. S. Venkatesh, “Optical image correlatin with binary spatial light modulator,” Opt. Eng. 23, 698–704 (1984).

Other

Z. Bahri, B. V. K. Vijaya Kumar, “Algorithm for designing phase-only synthetic discriminant function,” in Optical Information Processing Systems and Architectures, B. Javidi, ed., Proc. Soc. Photo-Opt. Instrum. Eng. 1151, 138–147 (1989).

L. Hassebrook, M. Rahmati, “Training set selection with multiple out-of-plane rotation parameters,” in Optical Pattern Recognition IV, D. P. Casasent, ed., Proc. Soc. Photo-Opt. Instrum. Eng. 1959, 32–43 (1993).

L. Hassebrook, R. W. Cohn, M. Liang, M. E. Lhamon, R. C. Daley, “Using pseudorandom phase-only encoding to approximate fully complex distortion-invariant filters,” in Optical Pattern Recognition V, D. P. Casasent, T.-H. Chao, eds., Proc. Soc. Photo-Opt. Instrum. Eng. 2237, 204–211 (1994).

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Figures (5)

Fig. 1
Fig. 1

Standard object. In the center of the pattern there are 4 E’s of different scale ratios, 11:16:21:26.

Fig. 2
Fig. 2

Calculating block for the computer. FFT, fast Fourier transform; CGH, computer-generated hologram; IFFT, inverse fast Fourier transform.

Fig. 3
Fig. 3

Recognized objects.

Fig. 4
Fig. 4

Result of correlation of the fifth-order synthetic circular-harmonic POF.

Fig. 5
Fig. 5

Result of correlation of the thirtieth-order synthetic circular-harmonic POF.

Equations (17)

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

f ( k ) ( r , θ ) = m = - f m ( k ) ( r ) exp ( j m θ ) ,
f m ( k ) ( r ) = 1 2 π 0 2 π f ( k ) ( r , θ ) exp ( - j m θ ) d θ .
h ( r , θ ) = k = 1 N a k f l ( k ) ( r , θ ) = k = 1 N a k f l ( k ) ( r ) exp ( j l θ ) ,
H ( ρ , ϕ ) = FF [ h ( r , θ ) ] = k = 1 N a k FF [ f l ( k ) ( r , θ ) ] .
FF [ f l ( k ) ( r , θ ) ] = F l ( k ) ( ρ ) exp [ j l ϕ ( k ) ] ,
F l ( k ) ( ρ ) = 2 π ( - j ) l 0 f l ( k ) ( r ) J l ( 2 π r ρ ) r d r
H ( ρ , ϕ ) = k = 1 N a k F l ( k ) ( ρ ) exp [ j α l ( k ) ( ρ ) ] exp [ j l ϕ ( k ) ] = k = 1 N a k exp [ j α l ( k ) ( ρ ) ] exp [ j l ϕ ( k ) ] .
f ( I ) ( r , θ + θ 0 ) = m = - f m ( I ) ( r ) exp [ j m ( θ + θ 0 ) ] , F ( I ) ( ρ , ϕ + θ 0 ) = m = 0 F m ( I ) ( ρ ) × exp [ j α m ( I ) ] exp { j m [ ϕ ( I ) + θ 0 ] } ,
C I = f ( I ) ( r , θ + θ 0 ) h * ( r , θ ) = 0 0 2 π m = 0 F m ( I ) ( ρ ) exp [ j α m ( I ) ( ρ ) ] exp { j m [ ϕ ( I ) + θ 0 ] } × k = 1 N a k * exp [ - j α l ( k ) ( ρ ) ] exp [ - j l ϕ ( k ) ] ρ d ρ d ϕ .
C I = exp ( j l θ 0 ) 0 0 2 π F l ( I ) ( ρ ) exp [ j α l ( I ) ( ρ ) ] exp [ j l ϕ ( I ) ] × k = 1 N a k * exp [ - j α l ( k ) ( ρ ) ] exp [ - j l ϕ ( k ) ] ρ d ρ d ϕ = exp ( j l θ 0 ) k = 1 N a k * 0 0 2 π F l ( I ) ( ρ , ϕ ) × exp [ - j ϕ l ( k ) ( ρ , ϕ ) ] ρ d ρ d ϕ = exp ( j l θ 0 ) k = 1 N a k * R KI .
F l ( I ) ( ρ , ϕ ) = F l ( I ) ( ρ ) exp [ j α l ( I ) ] exp [ j l ϕ ( I ) ]
exp [ - j ϕ l ( k ) ( ρ , ϕ ) ] = exp [ - j α l ( k ) ] exp [ - j l ϕ ( k ) ]
R KI = 0 0 2 π F l ( I ) ( ρ , ϕ ) = exp [ - j ϕ l ( k ) ( ρ , ϕ ) ] ρ d ρ d ϕ
C = a * R .
C = ( c 1 , c 2 , , c 1 , , c N ) T , a = ( a 1 , a 2 , , a 1 , , a N ) T ,
a * R = U = ( 1 , 1 , , 1 ) T .
a * = R - 1 U ,

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