Abstract

A nonlinear matched filter based image correlator is investigated. The linear matched filter is expressed as a bandpass function containing the amplitude and phase of the Fourier transform of the reference signal. The bandpass filter function is then applied to a kth law nonlinear device to produce the nonlinear matched filter function. Analytical expressions for the nonlinear matched filter are provided. The effects of the nonlinear transfer characteristics on the correlation signals at the output plane are investigated. The correlation signals are determined in terms of the nonlinear characteristics used to transform the filter. We show that the nonlinear filter results in a sum of infinite harmonic terms. Each harmonic term is envelope modulated due to the nonlinear characteristics of the device, and phase modulated by m times the phase modulation of the linear filter function. The correct phase information of the filter is recovered for the first-order harmonic of the series. The envelope of each harmonic term is proportional to the kth power of the Fourier transform magnitude of the reference signal. We show that various types of filter such as the continuous phase-only filters can be produced simply by varying the severity of the nonlinearity. Nonlinear filters provide higher correlation peak intensity and a better defined correlation spot.

© 1990 Optical Society of America

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References

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  1. B. Javidi, C-J. Kuo, “Joint Transform Image Correlation Using a Binary Spatial Light Modulator at the Fourier Plane,” Appl. Opt. 27, 663–665 (1988); J. Opt. Soc. Am. A 4(13), P86 (1987).
    [CrossRef] [PubMed]
  2. B. Javidi, J. L. Horner, “Multi-Function Nonlinear Signal Processor,” Opt. Eng. 28, 837–843 (1989).
  3. B. Javidi, J. L. Horner, “Single Spatial Light Modulator Joint Transform Correlator,” Appl. Opt. 28, 1027–1032 (1989).
    [CrossRef] [PubMed]
  4. B. Javidi, “Comparison of Bipolar Joint Transform Image Correlators and Phase-Only Matched Filter Correlators,” Opt. Eng. 28, 267–272 (1989).
    [CrossRef]
  5. B. Javidi, “Nonlinear Joint Power Spectrum Based Optical Correlation,” Appl. Opt. 28, 2358–2367 (1989).
    [CrossRef] [PubMed]
  6. C. S. Weaver, J. W. Goodman, “A Technique for Optically Convolving Two Functions,” Appl. Opt. 5, 1248–1249 (1966).
    [CrossRef] [PubMed]
  7. B. Javidi, “Nonlinear Matched Filter Based Optical Correlation,” Appl. Opt. 28, 4518–4520 (1989).
    [CrossRef] [PubMed]
  8. W. B. Davenport, J. W. Root, An Introduction to the Theory of Random Signals and Noise (McGraw-Hill, New York, 1960).
  9. D. Middleton, “Some General Results in the Theory of Noise Through Nonlinear Devices,” Q. Appl. Math. 5, 445–453 (1948).
  10. D. Middleton, An Introduction to Statistical Communication Theory (McGraw-Hill, New York, 1960).
  11. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1967).
  12. A. Kozma, “Photographic Recording of Spatially Modulated Coherent Light,” J. Opt. Soc. Am. 56, 428–432 (1966).
    [CrossRef]
  13. A. Kozma, D. Kelly, “Spatial Filtering for Detection of Signals Submerged in Noise,” Appl. Opt. 4, 387–392 (1965).
    [CrossRef]
  14. J. L. Horner, P. D. Gianino, “Phase-Only Matched Filtering,” Appl. Opt. 23, 812–816 (1984).
    [CrossRef] [PubMed]
  15. J. L. Horner, J. R. Leger, “Pattern Recognition with Binary Phase-Only Filters,” Appl. Opt. 24, 609–611 (1985).
    [CrossRef] [PubMed]
  16. D. Psaltis, E. G. Paek, S. S. Venkatesh, “Optical Image Correlation with a Binary Spatial Light Modulator,” Opt. Eng. 23, 698–704 (1984).
    [CrossRef]
  17. D. L. Flannery, J. S. Loomis, M. E. Milkovich, P. E. Keller, “Application of Binary Phase-Only Correlation to Machine Vision,” Opt. Eng. 27, 309–320 (1988).
    [CrossRef]

1989 (5)

1988 (2)

B. Javidi, C-J. Kuo, “Joint Transform Image Correlation Using a Binary Spatial Light Modulator at the Fourier Plane,” Appl. Opt. 27, 663–665 (1988); J. Opt. Soc. Am. A 4(13), P86 (1987).
[CrossRef] [PubMed]

D. L. Flannery, J. S. Loomis, M. E. Milkovich, P. E. Keller, “Application of Binary Phase-Only Correlation to Machine Vision,” Opt. Eng. 27, 309–320 (1988).
[CrossRef]

1985 (1)

1984 (2)

D. Psaltis, E. G. Paek, S. S. Venkatesh, “Optical Image Correlation with a Binary Spatial Light Modulator,” Opt. Eng. 23, 698–704 (1984).
[CrossRef]

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

1966 (2)

1965 (1)

1948 (1)

D. Middleton, “Some General Results in the Theory of Noise Through Nonlinear Devices,” Q. Appl. Math. 5, 445–453 (1948).

Davenport, W. B.

W. B. Davenport, J. W. Root, An Introduction to the Theory of Random Signals and Noise (McGraw-Hill, New York, 1960).

Flannery, D. L.

D. L. Flannery, J. S. Loomis, M. E. Milkovich, P. E. Keller, “Application of Binary Phase-Only Correlation to Machine Vision,” Opt. Eng. 27, 309–320 (1988).
[CrossRef]

Gianino, P. D.

Goodman, J. W.

Horner, J. L.

Javidi, B.

Keller, P. E.

D. L. Flannery, J. S. Loomis, M. E. Milkovich, P. E. Keller, “Application of Binary Phase-Only Correlation to Machine Vision,” Opt. Eng. 27, 309–320 (1988).
[CrossRef]

Kelly, D.

Kozma, A.

Kuo, C-J.

Leger, J. R.

Loomis, J. S.

D. L. Flannery, J. S. Loomis, M. E. Milkovich, P. E. Keller, “Application of Binary Phase-Only Correlation to Machine Vision,” Opt. Eng. 27, 309–320 (1988).
[CrossRef]

Middleton, D.

D. Middleton, “Some General Results in the Theory of Noise Through Nonlinear Devices,” Q. Appl. Math. 5, 445–453 (1948).

D. Middleton, An Introduction to Statistical Communication Theory (McGraw-Hill, New York, 1960).

Milkovich, M. E.

D. L. Flannery, J. S. Loomis, M. E. Milkovich, P. E. Keller, “Application of Binary Phase-Only Correlation to Machine Vision,” Opt. Eng. 27, 309–320 (1988).
[CrossRef]

Paek, E. G.

D. Psaltis, E. G. Paek, S. S. Venkatesh, “Optical Image Correlation with a Binary Spatial Light Modulator,” Opt. Eng. 23, 698–704 (1984).
[CrossRef]

Psaltis, D.

D. Psaltis, E. G. Paek, S. S. Venkatesh, “Optical Image Correlation with a Binary Spatial Light Modulator,” Opt. Eng. 23, 698–704 (1984).
[CrossRef]

Root, J. W.

W. B. Davenport, J. W. Root, An Introduction to the Theory of Random Signals and Noise (McGraw-Hill, New York, 1960).

Venkatesh, S. S.

D. Psaltis, E. G. Paek, S. S. Venkatesh, “Optical Image Correlation with a Binary Spatial Light Modulator,” Opt. Eng. 23, 698–704 (1984).
[CrossRef]

Weaver, C. S.

Appl. Opt. (8)

J. Opt. Soc. Am. (1)

Opt. Eng. (4)

D. Psaltis, E. G. Paek, S. S. Venkatesh, “Optical Image Correlation with a Binary Spatial Light Modulator,” Opt. Eng. 23, 698–704 (1984).
[CrossRef]

D. L. Flannery, J. S. Loomis, M. E. Milkovich, P. E. Keller, “Application of Binary Phase-Only Correlation to Machine Vision,” Opt. Eng. 27, 309–320 (1988).
[CrossRef]

B. Javidi, “Comparison of Bipolar Joint Transform Image Correlators and Phase-Only Matched Filter Correlators,” Opt. Eng. 28, 267–272 (1989).
[CrossRef]

B. Javidi, J. L. Horner, “Multi-Function Nonlinear Signal Processor,” Opt. Eng. 28, 837–843 (1989).

Q. Appl. Math. (1)

D. Middleton, “Some General Results in the Theory of Noise Through Nonlinear Devices,” Q. Appl. Math. 5, 445–453 (1948).

Other (3)

D. Middleton, An Introduction to Statistical Communication Theory (McGraw-Hill, New York, 1960).

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1967).

W. B. Davenport, J. W. Root, An Introduction to the Theory of Random Signals and Noise (McGraw-Hill, New York, 1960).

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Figures (7)

Fig. 1
Fig. 1

Nonlinear matched filter based optical correlator.

Fig. 2
Fig. 2

Nonlinear characteristics used to generate the nonlinear filter.

Fig. 3
Fig. 3

Values of the harmonic terms {1/[Γ(1 − v/2)Γ(1 + v/2)]} present in the hard clipped baseband nonlinear filter [see Eq. (36)].

Fig. 4
Fig. 4

Reference function used in the correlation tests.

Fig. 5
Fig. 5

(a) Bandpass linear filter function [see Eq. (3)] generated for the image shown in Fig. 4. (b) Impulse response of the bandpass linear filter shown in (a).

Fig. 6
Fig. 6

Nonlinear filter using a kth law device [see Eq. (23)]. The severity of the nonlinearity increases as k decreases. k = 0 corresponds to a hard clipping nonlinearity: (a) k = 0.8; (b) k = 0.5; (c) k = 0.

Fig. 7
Fig. 7

Correlation signals obtained by the nonlinear bandpass filter using a kth law device. The convolution function is also indicated. k = 1 corresponds to a linear filter and the severity of the nonlinearity increases as k decreases: (a) k = 1; (b) k = 0.8; (c) k = 0.5; (d) k = 0.

Tables (1)

Tables Icon

Table I Correlation Resultsa

Equations (59)

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FT { s ( x , y ) } = S ( α , β ) exp [ i ϕ S ( α , β ) ] ,
FT { r ( x , y ) } = R ( α , β ) exp [ i ϕ R ( α , β ) ] ,
H ( α , β ) = R ( α , β ) exp [ - i ϕ R ( α , β ) ] .
E ( α , β ) = R ( α , β ) cos [ x 0 α - ϕ R ( α , β ) ] ,
2 h ( x , y ) = R 12 ( x + x 0 , y ) + M 12 ( x - x 0 , y ) ,
R 12 ( x , y ) = s ( ξ , ζ ) r ( ξ - x , ζ - y ) d ξ d ζ ,
M 12 ( x , y ) = s ( ξ , ζ ) r ( x - ξ , y - ζ ) d ξ d ζ .
G ( ω ) = - g ( E ) exp ( - i ω E ) d E .
g ( E ) = 1 2 π - G ( ω ) exp ( i ω E ) d ω .
g ( E ) = 1 2 π - G ( ω ) exp { i ω R ( α , β ) cos [ x 0 α - ϕ R ( α , β ) ] } d ω .
exp [ i ω R ( α , β ) cos [ x 0 α - ϕ R ( α , β ) ] ] = v = 0 v ( i ) v J v [ ω R ( α , β ) ] cos [ v x 0 α - v ϕ R ( α , β ) ] ,
v = { 1 , v = 0 , 2 , v > 0 ,
g ( E ) = v = 0 v 2 π ( i ) v G ( ω ) J v [ ω R ( α , β ) ] cos [ v x 0 α - v ϕ R ( α , β ) ] d ω .
g ( E ) = v = 0 H v [ R ( α , β ) ] cos [ v x 0 α - v ϕ R ( α , β ) ] ,
H v [ R ( α , β ) ] = v 2 π ( i ) v G ( ω ) J v [ ω R ( α , β ) ] d ω .
h v ( x , y ) = FT - 1 { g ( E ) } = v = 0 v 2 π ( i ) v { G ( ω ) J v [ ω R ( α , β ) ] × cos [ v x 0 α - v ϕ R ( α , β ) ] d ω } × exp { i ( x α + y β ) ] d α d β .
T ( α , β ) = S ( α , β ) exp [ i ϕ S ( α , β ) ] g ( E ) = v = 0 v 2 π ( i ) v S ( α , β ) exp [ i ϕ S ( α , β ) ] × G ( ω ) J v [ ω R ( α , β ) ] cos [ v x 0 α - v ϕ R ( α , β ) ] d ω } .
y ( x , y ) = s ( x , y ) * h v ( x , y ) .
E ( α , β ) = S 2 ( α , β ) + R 2 ( α , β ) + 2 R ( α , β ) S ( α , β ) cos [ 2 x 0 α + ϕ S ( α , β ) - ϕ R ( α , β ) ] .
g ( E ) = v = 0 H v [ R ( α , β ) , S ( α , β ) ] × cos [ 2 v x 0 α + v ϕ S ( α , β ) - v ϕ R ( α , β ) ] ,
H v [ R ( α , β ) , S ( α , β ) ] = v 2 π ( i ) v G ( ω ) exp { i ω [ R 2 ( α , β ) + S 2 ( α , β ) ] } J v [ 2 ω R ( α , β ) S ( α , β ) ] d ω .
g ( E ) = { a E k , E 0 , - a E k , E < 0 ,
G ( ω ) = 2 a ( i ω ) k + 1 Γ ( k + 1 ) , k 1 ,
H v [ R ( α , β ) ] = Γ ( k + 1 ) v π ( i ) v - k - 1 1 ω k + 1 J v [ ω R ( α , β ) ] d ω .
H v [ R ( α , β ) ] = 2 Γ ( k + 1 ) v [ R ( α , β ) ] k 2 k + 1 Γ ( 1 - v - k 2 ) Γ ( 1 + v + k 2 ) ,
g v ( E ) = v = 1 ( v odd ) v Γ ( k + 1 ) [ R ( α , β ) ] k 2 K Γ ( 1 - v - k 2 ) Γ ( 1 + v + k 2 ) × cos [ v x 0 α - v ϕ R ( α , β ) ] ,
g k ( E ) = k = 1 v Γ ( k + 1 ) [ R ( α , β ) ] k 2 k + 1 Γ ( 1 - v - k 2 ) Γ ( 1 + v + k 2 ) × ( exp { i [ v x 0 α - v ϕ R ( α , β ) ] } + exp { - i [ v x 0 α - v ϕ R ( α , β ) ] } ) .
g k c ( E ) = v = 1 ( v odd ) v Γ ( k + 1 ) 2 k + 1 Γ ( 1 - v - k 2 ) Γ ( 1 + v + k 2 ) × [ R ( α , β ) ] k exp { i [ v x 0 α - v ϕ R ( α , β ) ] } ,
h 1 c ( x , y ) = r ( - x 0 - x , - y ) .
g 1 k c ( E ) = Γ ( k + 1 ) 2 k + 1 Γ ( 1 - l - k 2 ) Γ ( 1 + l + k 2 ) × [ R ( α , β ) ] k exp { i [ x 0 α - ϕ R ( α , β ) ] } ,
g 0 c ( E ) = v = 1 ( v odd ) v 2 Γ ( 1 - v 2 ) Γ ( 1 + v 2 ) × exp { i [ v x 0 α - v ϕ R ( α , β ) ] } .
g POF ( α , β ) = g 10 c ( α , β ) = 2 π exp { - i ϕ R ( α , β ) ] exp ( i x 0 α ) ,
H POF ( α , β ) = exp [ - i ϕ R ( α , β ) ] .
T k ( α , β ) = v = 1 v odd ) v Γ ( k + 1 ) 2 k Γ ( 1 - v - k 2 ) Γ ( 1 + v + k 2 ) S ( α , β ) × exp [ i ϕ S ( α , β ) ] [ R ( α , β ) ] k cos [ v x 0 α - v ϕ R ( α , β ) ] .
T 1 k c ( α , β ) = Γ ( k + 1 ) 2 k Γ ( 1 - 1 - k 2 ) Γ ( 1 + 1 + k 2 ) [ R ( α , β ) ] k S ( α , β ) × exp { i [ x 0 α + ϕ S ( α , β ) - ϕ R ( α , β ) ] } ,
T 0 ( α , β ) = 2 π S ( α , β ) exp { i [ ϕ S ( α , β ) - ϕ R ( α , β ) ] } exp ( i x 0 α ) + 2 π S ( α , β ) exp [ i ϕ S ( α , β ) ] exp [ i ϕ R ( α , β ) ] exp ( - i x 0 α ) .
E ¯ ( α , β ) = R ( α , β ) cos [ ϕ R ( α , β ) ] ,
g ¯ k ( E ) = v = 1 ( v odd ) v Γ ( k + 1 ) 2 k Γ ( 1 - v - k 2 ) Γ ( 1 + v + k 2 ) × [ R ( α , β ) ] k cos [ v ϕ R ( α , β ) ] ,
g BPOF ( α , β ) = { + 1 , E ¯ ( α , β ) 0 , - 1 , E ¯ ( α , β ) < 0.
g BPOF ( α , β ) = g ¯ 0 ( E ¯ ) = v = 1 ( v odd ) 1 Γ ( 1 - v 2 ) Γ ( 1 + v 2 ) × { exp [ - i v ϕ R ( α , β ) ] + exp [ i v ϕ R ( α , β ) ] } ,
g ¯ 0 ( E ¯ ) = 2 π exp [ - i ϕ R ( α , β ) ] + 2 π exp [ i ϕ R ( α , β ) ] + v = 3 ( v odd ) 1 Γ ( 1 - v 2 ) Γ ( 1 + v 2 ) × { exp [ - i v ϕ R ( α , β ) ] + exp [ i v ϕ R ( α , β ) ] .
T ¯ k ( α , β ) = v = 1 ( v odd ) v Γ ( k + 1 ) 2 k Γ ( 1 - v - k 2 ) Γ ( 1 + v + k 2 ) × S ( α , β ) [ R ( α , β ) ] k cos [ v ϕ R ( α , β ) ] exp [ i ϕ S ( α , β ) ] ,
T ¯ 0 ( α , β ) = 2 π S ( α , β ) exp { i [ ϕ S ( α , β ) - ϕ R ( α , β ) ] } + 2 π S ( α , β ) exp [ i ϕ S ( α , β ) ] exp [ i ϕ R ( α , β ) ] + v = 3 ( v odd ) 1 Γ ( 1 - v 2 ) Γ ( 1 + v 2 ) S ( α , β ) × exp [ i ϕ S ( α , β ) ] exp [ - i v ϕ R ) ( α , β ) ] .
g ( α , β ) = { 1 , R ( α , β ) cos [ x 0 α - ϕ R ( α , β ) ] V T , 0 , R ( α , β ) cos [ x 0 α - ϕ R ( α , β ) ] < V T ,
R ( α , β ) cos [ x 0 d 2 - ϕ R ( α , β ) ] = V T .
d = 2 x 0 cos - 1 [ V T R ( α , β ) ] for | V T R ( α , β ) | 1.
R ( α , β ) V T .
g ( α , β ) = g ( α ) = { v = - K v exp ( i v α x 0 ) | V T R ( α , β ) | 1 , 0 , else ,
k v = x 0 2 π - 1 2 x 0 1 2 x 0 g ( α ) exp ( - i x 0 v α ) d α .
k v = { x 0 d , v = 0 1 π v exp { i v [ - ϕ R ( α , β ) ] } sin ( v x 0 d 2 ) , v 0 ,
g ( α , β ) = x 0 d + { v = - H v [ R ( α , β ) ] cos [ v α x 0 - v ϕ R ( α , β ) ] , | V T R ( α , β ) | 1 , 0 , else ,
H v [ R ( α , β ) ] = 2 π v sin [ v cos - 1 [ - v R ( α , β ) ] ] .
g 1 ( α , β ) = 2 π 1 - [ V T R ( α , β ) ] 2 cos [ x 0 α - ϕ R ( α , β ) ] ,
sin ( cos - 1 x ) = 1 - x 2 .
g 2 ( α , β ) = 1 π sin { 2 cos - 1 [ V T R ( α , β ) ] } cos [ 2 x 0 α - 2 ϕ R ( α , β ) ] ,
g 2 ( α , β ) = 2 π [ V T R ( α , β ) ] 1 - [ V T R ( α , β ) ] 2 cos [ 2 x 0 α - 2 ϕ R ( α , β ) ] .
g 1 c ( α , β ) 2 π 1 - [ V T R ( α , β ) ] 2 exp [ - i ϕ R ( α , β ) ] exp ( i x 0 α ) .
g 1 ( α , β ) = 2 π exp [ x 0 α - ϕ R ( α , β ) ] .
A ( α , β ) = 2 π 1 - [ V T R ( α , β ) ] 2 .

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