Abstract

Almost all coherent pattern recognition architectures are based on optical correlation of the input with a designed filter. However, the filter can be implemented via many different media, and each medium will impose different realizability constraints on the filter. That is, different media will have different regions of physical realizability. In the past, there has not been much work addressing the problem of designing an optimal filter given an arbitrary region of realizability. This paper presents the theory for just such an optimal filter design. A fast algorithm is presented to implement the theory. The algorithm is demonstrated with two examples.

© 1989 Optical Society of America

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References

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  1. K. Johnson, A Review of Spatial Light Modulators and Their Applications, short course at O-E/Lase ’88 SPIE conference, Los Angeles, CA (1988).
  2. B. R. Brown, A. W. Lohmann, “Computer-generated Binary Holograms,” IBM J. Res. Dev. 13, 160–168 (1969).
    [CrossRef]
  3. G. Tricoles, “Computer Generated Holography: an Historical Review,” Appl. Opt. 26, 4351–4360 (1987).
    [CrossRef] [PubMed]
  4. H. Farhoosh, M. R. Feldman, S. H. Lee, C. C. Guest, Y. Fainman, R. Eschbach, “Comparison of Binary Encoding Schemes for Electron-Beam Fabrication of Computer Generated Holograms,” Appl. Opt. 26, 4361–4372 (1987).
    [CrossRef] [PubMed]
  5. S. H. Lee, “Computer Generated Holography: An Introduction,” Appl. Opt. 26, 4350–4350 (1987).
    [CrossRef] [PubMed]
  6. S. M. Arnold, “Electron Beam Fabrication of Computer-Generated Holograms,” Opt. Eng. 24, 803–807 (1985).
    [CrossRef]
  7. H. Dammann, “Synthetic Digital-Phase Gratings—Design Features, Applications,” Proc. Soc. Photo-Opt. Instrum. Eng. 437, 312–315 (1983).
  8. R. D. Juday, “Optical Correlation with a Cross-Coupled Spatial Light Modulator,” in Technical Digest, Topical Meeting on Spatial Light Modulators and Applications (Optical Society of America, Washington, DC, 1988), p. 238.
  9. B. V. K. Vijaya Kumar, Z. Bahri, “Phase-Only Filters with Improved Signal to Noise Ratio,” Appl. Opt. 28, 250–257 (1989).
    [CrossRef]
  10. B. V. K. Vijaya Kumar, Z. Bahri, “Efficient Algorithm for Designing Optimal Binary Phase-Only Filters,” submitted to Appl. Opt.
  11. R. R. Kallman, “Optimal Low Noise Phase-only and Binary Phase-only Optical Correlation Filters for Threshold Detectors,” Appl. Opt. 25, 4216–4217 (1986).
    [CrossRef] [PubMed]
  12. B. V. K. Vijaya Kumar, Z. Bahri, “Optimality of Phase-only Filters,” Proc. Soc. Photo-Opt. Instrum. Eng. 884, 146–152 (1988).
  13. M. W. Farn, J. W. Goodman, “Optimal Binary Phase-only Matched Filters,” Appl. Opt. 27, 4431–4437 (1988).
    [CrossRef] [PubMed]
  14. R. N. Bracewell, The Fourier Transform and Its Applications (McGraw-Hill, New York, 1986).

1989 (1)

1988 (2)

B. V. K. Vijaya Kumar, Z. Bahri, “Optimality of Phase-only Filters,” Proc. Soc. Photo-Opt. Instrum. Eng. 884, 146–152 (1988).

M. W. Farn, J. W. Goodman, “Optimal Binary Phase-only Matched Filters,” Appl. Opt. 27, 4431–4437 (1988).
[CrossRef] [PubMed]

1987 (3)

1986 (1)

1985 (1)

S. M. Arnold, “Electron Beam Fabrication of Computer-Generated Holograms,” Opt. Eng. 24, 803–807 (1985).
[CrossRef]

1983 (1)

H. Dammann, “Synthetic Digital-Phase Gratings—Design Features, Applications,” Proc. Soc. Photo-Opt. Instrum. Eng. 437, 312–315 (1983).

1969 (1)

B. R. Brown, A. W. Lohmann, “Computer-generated Binary Holograms,” IBM J. Res. Dev. 13, 160–168 (1969).
[CrossRef]

Arnold, S. M.

S. M. Arnold, “Electron Beam Fabrication of Computer-Generated Holograms,” Opt. Eng. 24, 803–807 (1985).
[CrossRef]

Bahri, Z.

B. V. K. Vijaya Kumar, Z. Bahri, “Phase-Only Filters with Improved Signal to Noise Ratio,” Appl. Opt. 28, 250–257 (1989).
[CrossRef]

B. V. K. Vijaya Kumar, Z. Bahri, “Optimality of Phase-only Filters,” Proc. Soc. Photo-Opt. Instrum. Eng. 884, 146–152 (1988).

B. V. K. Vijaya Kumar, Z. Bahri, “Efficient Algorithm for Designing Optimal Binary Phase-Only Filters,” submitted to Appl. Opt.

Bracewell, R. N.

R. N. Bracewell, The Fourier Transform and Its Applications (McGraw-Hill, New York, 1986).

Brown, B. R.

B. R. Brown, A. W. Lohmann, “Computer-generated Binary Holograms,” IBM J. Res. Dev. 13, 160–168 (1969).
[CrossRef]

Dammann, H.

H. Dammann, “Synthetic Digital-Phase Gratings—Design Features, Applications,” Proc. Soc. Photo-Opt. Instrum. Eng. 437, 312–315 (1983).

Eschbach, R.

Fainman, Y.

Farhoosh, H.

Farn, M. W.

Feldman, M. R.

Goodman, J. W.

Guest, C. C.

Johnson, K.

K. Johnson, A Review of Spatial Light Modulators and Their Applications, short course at O-E/Lase ’88 SPIE conference, Los Angeles, CA (1988).

Juday, R. D.

R. D. Juday, “Optical Correlation with a Cross-Coupled Spatial Light Modulator,” in Technical Digest, Topical Meeting on Spatial Light Modulators and Applications (Optical Society of America, Washington, DC, 1988), p. 238.

Kallman, R. R.

Lee, S. H.

Lohmann, A. W.

B. R. Brown, A. W. Lohmann, “Computer-generated Binary Holograms,” IBM J. Res. Dev. 13, 160–168 (1969).
[CrossRef]

Tricoles, G.

Vijaya Kumar, B. V. K.

B. V. K. Vijaya Kumar, Z. Bahri, “Phase-Only Filters with Improved Signal to Noise Ratio,” Appl. Opt. 28, 250–257 (1989).
[CrossRef]

B. V. K. Vijaya Kumar, Z. Bahri, “Optimality of Phase-only Filters,” Proc. Soc. Photo-Opt. Instrum. Eng. 884, 146–152 (1988).

B. V. K. Vijaya Kumar, Z. Bahri, “Efficient Algorithm for Designing Optimal Binary Phase-Only Filters,” submitted to Appl. Opt.

Appl. Opt. (6)

IBM J. Res. Dev. (1)

B. R. Brown, A. W. Lohmann, “Computer-generated Binary Holograms,” IBM J. Res. Dev. 13, 160–168 (1969).
[CrossRef]

Opt. Eng. (1)

S. M. Arnold, “Electron Beam Fabrication of Computer-Generated Holograms,” Opt. Eng. 24, 803–807 (1985).
[CrossRef]

Proc. Soc. Photo-Opt. Instrum. Eng. (2)

H. Dammann, “Synthetic Digital-Phase Gratings—Design Features, Applications,” Proc. Soc. Photo-Opt. Instrum. Eng. 437, 312–315 (1983).

B. V. K. Vijaya Kumar, Z. Bahri, “Optimality of Phase-only Filters,” Proc. Soc. Photo-Opt. Instrum. Eng. 884, 146–152 (1988).

Other (4)

R. N. Bracewell, The Fourier Transform and Its Applications (McGraw-Hill, New York, 1986).

R. D. Juday, “Optical Correlation with a Cross-Coupled Spatial Light Modulator,” in Technical Digest, Topical Meeting on Spatial Light Modulators and Applications (Optical Society of America, Washington, DC, 1988), p. 238.

K. Johnson, A Review of Spatial Light Modulators and Their Applications, short course at O-E/Lase ’88 SPIE conference, Los Angeles, CA (1988).

B. V. K. Vijaya Kumar, Z. Bahri, “Efficient Algorithm for Designing Optimal Binary Phase-Only Filters,” submitted to Appl. Opt.

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Figures (3)

Fig. 1
Fig. 1

Examples of regions of realizability: (a) ideal, unconstrained filter; (b) phase-only filter; (c) binary phase-only filter; (d) cross-coupled phase and amplitude filter.

Fig. 2
Fig. 2

Examples of function G: (a) operating region; (b) operating curve.

Fig. 3
Fig. 3

Division of complex Ŝ plane.

Equations (26)

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I = | s ( x ) * h ( x ) at x = 0 | 2 = | S ( p ) H ( p ) d p | 2
I = | B S ( p ) H ( p ) d p | 2
H ( p ) Ω for all p .
C = B S ( p ) H ( p ) d p = | C | exp ( j α ) . S ( p ) = | S ( p ) | exp ( j θ S ( p ) ) H ( p ) = | H ( p ) | exp ( j θ H ( p ) ) .
| C ( α ) | = Re [ ( B S ( p ) H ( p ) d p ) exp ( j α ) ] = Re [ B S ( p ) H ( p ) exp ( j α ) d p ] = B Re [ S ( p ) H ( p ) exp ( j α ) ] d p ,
max H ( p ) | C ( α ) | = max H ( p ) [ B Re [ H ( p ) S ( p ) exp ( j α ) ] d p ] = B max H ( p ) { Re [ H ( p ) S ( p ) exp ( j α ) ] } d p = B | S ( p ) | max H ( p ) { | H ( p ) | cos [ θ H ( p ) + θ S ( p ) α ] } d p ,
G ( ϕ ) = max H ( p ) { Re [ H ( p ) exp ( j ϕ ) ] } = max H ( p ) { | H ( p ) | cos [ θ H ( p ) ϕ ] } ,
max H ( p ) | C ( α ) | = B | S ( p ) | G [ α θ S ( p ) ] d p .
N = 2 π / δ = number of phase bins , δ = angular resolution of each phase bin , | S ˆ ( k ) | = sampled | S ( p ) | , n S ( k ) δ = sampled and quantized arg [ S ( p ) ] , S ˆ ( k ) = | S ˆ ( k ) | exp [ j n S ( k ) δ ] , | H ˆ ( k ) | = sampled | H ( p ) | , n H ( k ) δ = sampled and quantized arg [ H ( p ) ] , H ˆ ( k ) = | H ˆ ( k ) | exp [ j n H ( k ) δ ] , G ˆ ( n ϕ ) = sampled G ( ϕ ) = G ( n ϕ δ ) , C ˆ ( n α ) = sampled max H ( p ) | C ( α ) | = max H ˆ ( k ) | C ( n α δ ) | .
S ˆ bin ( n ) = n S ( k ) = n S ˆ ( k ) for 0 n ( N 1 ) .
C ˆ ( n α ) = k | S ˆ ( k ) | G ˆ [ n α n S ( k ) ] = n | S ˆ bin ( n ) | G ˆ ( n α n ) .
G ˆ ( n ) = maximum projection of region Ω in direction exp ( j n δ ) = max H ˆ ( k ) { | H ˆ ( k ) | cos [ ( n H ( k ) n ) δ ] } ,
S ˆ bin ( n ) = n S ( k ) = n S ˆ ( k ) .
C ˆ ( n α ) = n = 0 N 1 | S ˆ bin ( n ) | G ˆ ( n α n ) = FFT 1 { FFT { | S ˆ bin ( n ) | } · FFT { G ˆ ( n ) } } .
G ˆ bin ( n ) = G ˆ ( n o n ) .
H ˆ ( k ) = exp [ j n S ( k ) δ ] .
M 1 : exp ( j n δ ) G ˆ ( n ) .
M 2 : S ˆ ( k ) S ˆ bin [ n S ( k ) ] .
C ˆ ( n α ) = n | S ˆ bin ( n ) | = constant .
G ˆ bin ( n ) = G ˆ ( n ) .
H ˆ bin ( n ) = M 1 1 { G ˆ bin ( n ) } = M 1 1 { G ˆ ( n ) } = exp ( j n δ ) .
H ˆ ( k ) = M 2 1 { H ˆ bin ( n ) } = M 2 1 { exp ( j n δ ) } = exp ( j n S ( k ) δ ) .
C ˆ ( n α ) = k max H ˆ ( k ) { Re [ H ˆ ( k ) S ˆ ( k ) exp ( j n α δ ) ] } .
C ˆ ( n α ) = S ˆ ( k ) region 1 | S ˆ ( k ) | cos { [ n s ( k ) n α ] δ } S ˆ ( k ) region 2 | S ˆ ( k ) | cos { [ n s ( k ) n α ] δ } = k | S ˆ ( k ) | | cos { [ n s ( k ) n α ] δ } | = n | S ˆ bin ( n ) | | cos [ ( n α n ) δ ] | .
M 1 : 1 G ˆ ( n ) for π / 2 n   < 3 π / 2 + 1 G ˆ ( n ) for all other n .
C ˆ ( n α ) = n | S ˆ bin ( n ) | | cos [ ( n α n ) δ ] | .

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