Abstract

An analytic solution for real optimal filters is known, and the special case of optimal binary phase-only filters can be solved by a fast binning algorithm but no analytic solution is known. We establish a geometric solution for the design of optimal binary amplitude filters (OBAF’s) and optimal binary phase-only filters (OBPOF’s) for any object. The optimal filter is found in terms of maximizing the field strength at the origin in the correlation plane. We found that it is possible to construct a unique convex polygon by using an ordered set of phasors from the filter object’s Fourier transform. This process leads eventually to an exact solution for the filter-design problem. We show that the maximum distance across the polygon divides the phasors into two groups: For the OBAF, it determines the group that is passed or blocked; for the OBPOF, it determines which group is passed with a zero or a π phase shift. The shape of the convex polygon gives qualitative information on the criticalness and the tightness needed in the design process. It provides good insight into the binning-process algorithm and permits us to bound the error in the binning process. Design examples through computer simulation and applications in fingerprint identification are presented.

© 1998 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).
  2. D. O. North, “An analysis of the factors which determine signal/noise discriminations in pulsed carrier systems,” Proc. IEEE 51, 1016–1027 (1963).
    [CrossRef]
  3. J. L. Horner, “Light utilization in optical correlators,” Appl. Opt. 21, 4511–4514 (1982).
    [CrossRef] [PubMed]
  4. J. L. Horner, J. R. Leger, “Pattern recognition with binary phase-only filters,” Appl. Opt. 24, 609–611 (1985).
    [CrossRef] [PubMed]
  5. J. Horner, H. Bartelt, “Two-bit correlation,” Appl. Opt. 24, 2889–2893 (1985).
    [CrossRef] [PubMed]
  6. R. D. Juday, “Optical correlation with a cross-coupled spatial light modulator,” in Spatial Light Modulators and Applications, Vol. 8 of 1988 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1988), pp. 238–241.
  7. J. L. Horner, P. D. Gianino, “Phase-only matched filtering,” Appl. Opt. 23, 812–816 (1984).
    [CrossRef] [PubMed]
  8. B. V. K. Vijaya Kumar, Z. Bahri, “Phase-only filters with improved signal to noise ratio,” Appl. Opt. 28, 250–257 (1989).
  9. B. V. K. Vijaya Kumar, Z. Bahri, “Efficient algorithm for designing optimal binary phase-only filters,” Appl. Opt. 28, 1919–1925 (1989).
  10. B. V. K. Vijaya Kumar, R. D. Juday, “Design of phase-only, binary phase-only, and complex ternary matched filters with increased signal-to-noise ratio for colored noise,” Opt. Lett. 16, 1025–1027 (1991).
  11. D. Psaltis, E. Paek, S. Venkatesh, “Optical image correlation with a binary spatial light modulator,” Opt. Eng. 23, 698–704 (1984).
    [CrossRef]
  12. M. A. Flavin, J. L. Horner, “Amplitude encoded phase-only filters,” Appl. Opt. 28, 1692–1696 (1989).
    [CrossRef] [PubMed]
  13. S. M. Arnold, “Electron beam fabrication of computer generated holograms,” Opt. Eng. 24, 803–807 (1985).
    [CrossRef]
  14. R. B. Brown, A. W. Lohmann, “Computer-generated binary holograms,” IBM J. Res. Dev. 13, 160–168 (1969).
    [CrossRef]
  15. H. Dammann, “Synthetic digital-phase gratings—design features: applications,” in International Conference on Computer-Generated Holography, S. H. Lee, ed., Proc. SPIE437, 312–315 (1983).
  16. 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]
  17. S. H. Lee, “Computer generated holography: an introduction,” Appl. Opt. 26, 4350–4350 (1987).
    [CrossRef] [PubMed]
  18. G. Tricoles, “Computer generated holography: a historical review,” Appl. Opt. 26, 4351–4360 (1987).
    [CrossRef] [PubMed]
  19. J. Knopp, “Optical calculation of correlation filters,” in Real-Time Image Processing II, R. Juday, ed., Proc. SPIE1295, 68–75 (1990).
    [CrossRef]
  20. R. D. Juday, B. V. K. Vijaya Kumar, P. K. Rajan, “Optimal real correlation filters,” Appl. Opt. 24, 520–522 (1991).
    [CrossRef]
  21. M. W. Farn, J. W. Goodman, “Optimal binary phase-only matched filters,” Appl. Opt. 27, 4431–4437 (1988).
    [CrossRef] [PubMed]
  22. M. W. Farn, J. W. Goodman, “Optimal maximum correlation filter for arbitrary constrained devices,” Appl. Opt. 28, 3362–3366 (1989).
    [CrossRef]
  23. M. M. Matalgah, “Geometric theory for designing optical binary amplitude and binary phase-only filters,” Ph.D. dissertation (Department of Electrical and Computer Engineering, The University of Missouri-Columbia, Columbia, Mo., 1996).
  24. E. Hille, Analytic Function Theory (Ginn, Boston, Mass., 1959), Vol. 1.
  25. J. O’Rourke, Computational Geometry in C (Cambridge U. Press, Cambridge, 1994).

1991 (2)

1989 (4)

1988 (1)

1987 (3)

1985 (3)

1984 (2)

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

D. Psaltis, E. Paek, S. Venkatesh, “Optical image correlation with a binary spatial light modulator,” Opt. Eng. 23, 698–704 (1984).
[CrossRef]

1982 (1)

1969 (1)

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

1964 (1)

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

1963 (1)

D. O. North, “An analysis of the factors which determine signal/noise discriminations in pulsed carrier systems,” Proc. IEEE 51, 1016–1027 (1963).
[CrossRef]

Arnold, S. M.

S. M. Arnold, “Electron beam fabrication of computer generated holograms,” Opt. Eng. 24, 803–807 (1985).
[CrossRef]

Bahri, Z.

Bartelt, H.

Brown, R. B.

R. B. 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,” in International Conference on Computer-Generated Holography, S. H. Lee, ed., Proc. SPIE437, 312–315 (1983).

Eschbach, R.

Fainman, Y.

Farhoosh, H.

Farn, M. W.

Feldman, M. R.

Flavin, M. A.

Gianino, P. D.

Goodman, J. W.

Guest, C. C.

Hille, E.

E. Hille, Analytic Function Theory (Ginn, Boston, Mass., 1959), Vol. 1.

Horner, J.

Horner, J. L.

Juday, R. D.

B. V. K. Vijaya Kumar, R. D. Juday, “Design of phase-only, binary phase-only, and complex ternary matched filters with increased signal-to-noise ratio for colored noise,” Opt. Lett. 16, 1025–1027 (1991).

R. D. Juday, B. V. K. Vijaya Kumar, P. K. Rajan, “Optimal real correlation filters,” Appl. Opt. 24, 520–522 (1991).
[CrossRef]

R. D. Juday, “Optical correlation with a cross-coupled spatial light modulator,” in Spatial Light Modulators and Applications, Vol. 8 of 1988 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1988), pp. 238–241.

Knopp, J.

J. Knopp, “Optical calculation of correlation filters,” in Real-Time Image Processing II, R. Juday, ed., Proc. SPIE1295, 68–75 (1990).
[CrossRef]

Lee, S. H.

Leger, J. R.

Lohmann, A. W.

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

Matalgah, M. M.

M. M. Matalgah, “Geometric theory for designing optical binary amplitude and binary phase-only filters,” Ph.D. dissertation (Department of Electrical and Computer Engineering, The University of Missouri-Columbia, Columbia, Mo., 1996).

North, D. O.

D. O. North, “An analysis of the factors which determine signal/noise discriminations in pulsed carrier systems,” Proc. IEEE 51, 1016–1027 (1963).
[CrossRef]

O’Rourke, J.

J. O’Rourke, Computational Geometry in C (Cambridge U. Press, Cambridge, 1994).

Paek, E.

D. Psaltis, E. Paek, S. Venkatesh, “Optical image correlation with a binary spatial light modulator,” Opt. Eng. 23, 698–704 (1984).
[CrossRef]

Psaltis, D.

D. Psaltis, E. Paek, S. Venkatesh, “Optical image correlation with a binary spatial light modulator,” Opt. Eng. 23, 698–704 (1984).
[CrossRef]

Rajan, P. K.

R. D. Juday, B. V. K. Vijaya Kumar, P. K. Rajan, “Optimal real correlation filters,” Appl. Opt. 24, 520–522 (1991).
[CrossRef]

Tricoles, G.

VanderLugt, A.

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

Venkatesh, S.

D. Psaltis, E. Paek, S. Venkatesh, “Optical image correlation with a binary spatial light modulator,” Opt. Eng. 23, 698–704 (1984).
[CrossRef]

Vijaya Kumar, B. V. K.

Appl. Opt. (13)

J. L. Horner, “Light utilization in optical correlators,” Appl. Opt. 21, 4511–4514 (1982).
[CrossRef] [PubMed]

J. L. Horner, J. R. Leger, “Pattern recognition with binary phase-only filters,” Appl. Opt. 24, 609–611 (1985).
[CrossRef] [PubMed]

J. Horner, H. Bartelt, “Two-bit correlation,” Appl. Opt. 24, 2889–2893 (1985).
[CrossRef] [PubMed]

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

B. V. K. Vijaya Kumar, Z. Bahri, “Phase-only filters with improved signal to noise ratio,” Appl. Opt. 28, 250–257 (1989).

B. V. K. Vijaya Kumar, Z. Bahri, “Efficient algorithm for designing optimal binary phase-only filters,” Appl. Opt. 28, 1919–1925 (1989).

M. A. Flavin, J. L. Horner, “Amplitude encoded phase-only filters,” Appl. Opt. 28, 1692–1696 (1989).
[CrossRef] [PubMed]

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]

S. H. Lee, “Computer generated holography: an introduction,” Appl. Opt. 26, 4350–4350 (1987).
[CrossRef] [PubMed]

G. Tricoles, “Computer generated holography: a historical review,” Appl. Opt. 26, 4351–4360 (1987).
[CrossRef] [PubMed]

R. D. Juday, B. V. K. Vijaya Kumar, P. K. Rajan, “Optimal real correlation filters,” Appl. Opt. 24, 520–522 (1991).
[CrossRef]

M. W. Farn, J. W. Goodman, “Optimal binary phase-only matched filters,” Appl. Opt. 27, 4431–4437 (1988).
[CrossRef] [PubMed]

M. W. Farn, J. W. Goodman, “Optimal maximum correlation filter for arbitrary constrained devices,” Appl. Opt. 28, 3362–3366 (1989).
[CrossRef]

IBM J. Res. Dev. (1)

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

IEEE Trans. Inf. Theory (1)

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

Opt. Eng. (2)

S. M. Arnold, “Electron beam fabrication of computer generated holograms,” Opt. Eng. 24, 803–807 (1985).
[CrossRef]

D. Psaltis, E. Paek, S. Venkatesh, “Optical image correlation with a binary spatial light modulator,” Opt. Eng. 23, 698–704 (1984).
[CrossRef]

Opt. Lett. (1)

Proc. IEEE (1)

D. O. North, “An analysis of the factors which determine signal/noise discriminations in pulsed carrier systems,” Proc. IEEE 51, 1016–1027 (1963).
[CrossRef]

Other (6)

J. Knopp, “Optical calculation of correlation filters,” in Real-Time Image Processing II, R. Juday, ed., Proc. SPIE1295, 68–75 (1990).
[CrossRef]

R. D. Juday, “Optical correlation with a cross-coupled spatial light modulator,” in Spatial Light Modulators and Applications, Vol. 8 of 1988 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1988), pp. 238–241.

H. Dammann, “Synthetic digital-phase gratings—design features: applications,” in International Conference on Computer-Generated Holography, S. H. Lee, ed., Proc. SPIE437, 312–315 (1983).

M. M. Matalgah, “Geometric theory for designing optical binary amplitude and binary phase-only filters,” Ph.D. dissertation (Department of Electrical and Computer Engineering, The University of Missouri-Columbia, Columbia, Mo., 1996).

E. Hille, Analytic Function Theory (Ginn, Boston, Mass., 1959), Vol. 1.

J. O’Rourke, Computational Geometry in C (Cambridge U. Press, Cambridge, 1994).

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Figures (9)

Fig. 1
Fig. 1

Geometry explaining the half-plane sum ζ = z k + ⋯ + z m . In this example, z 1 = 3 + j1, z 2 = 1 + j3, z 3 = -4 + j2, z 4 = -1 - j5, and ζ = z 1 + z 2 + z 3 = j6. Here k = 1 and m = 3.

Fig. 2
Fig. 2

Geometry explaining the half-plane sum ζ = ω n - (z k + ⋯ + z m ). In this example, z 1 = 3 + j1, z 2 = 1 + j3, z 3 = -4 + j2, z 4 = -1 - j5, z 5 = -j2, and ζ = z 4 + z 5 = -1 - j7 = ω n - (z 1 + z 2 + z 3). Here k = 1 and m = 3.

Fig. 3
Fig. 3

Geometry for finding the maximum distance across the UPS by use of the algorithm presented in Section 5. In this figure, the relation max1 < max2 < optimal max holds.

Fig. 4
Fig. 4

(a) Gray-level plot for the test object of the letter E. (b) UPS of the test object E. (c) Gray-level plot obtained with the BAF for the test object E. Gray levels: white passes light; black blocks light. (d) Filter correlation with a strength at the origin of 3.8775 × 104.

Fig. 5
Fig. 5

(a) Gray-level plot for the test object E. (b) UPS for the test object E obtained by use of a BPOF design algorithm. (c) Gray-level plot obtained with the BPOF for E. Gray levels: white passes light as it is; black passes light with a 180° phase shift. (d) Filter correlation with a strength at the origin of 6.1166 × 104.

Fig. 6
Fig. 6

(a) Gray-level plot for a fingerprint as the test object. (b) UPS for the fingerprint. (c) Gray-level plot obtained with the BAF for the fingerprint. Gray levels: white passes light; black blocks light. (d) Filter correlation with a strength at the origin of 4.9844 × 107.

Fig. 7
Fig. 7

(a) Gray-level plot for the fingerprint test object. (b) UPS for the fingerprint obtained by use of the BPOF design algorithm. (c) Gray-level plot of the fingerprint obtained with the BPOF. Gray levels: white passes light as it is; black passes light with a 180° phase shift. (d) Filter correlation with a strength at the origin of 9.9688 × 107.

Fig. 8
Fig. 8

(a) Gray-level plot for the letter E shifted to the right by 1 pixel. (b) UPS for the shifted letter E. (c) Gray-level plot obtained with the BAF for the shifted letter E. Gray levels: white passes light; black blocks light. (d) Filter correlation with a strength at the origin of 3.9505 × 104.

Fig. 9
Fig. 9

Approximate UPS diagrams for the test object of the letter E by use of different numbers of bins: (a) 10 bins, (b) 50 bins, (c) 100 bins, (d) 200 bins.

Equations (19)

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H POF f = exp - j ϕ s f ,
H BPOF f = exp - j ϕ B f ,
H BAF f = 1 2 1 + H BPOF f = 1 2 1 + exp - j ϕ B f ,
h BPOF x ,   y = - 1 exp - j ϕ B u ,   ν ,
h BAF x ,   y = - 1 1 2 + 1 2 exp - j ϕ B u ,   ν = 1 2 δ 0,0 + 1 2 h BPOF x ,   y .
C BAF τ x ,   τ y = 1 2 s x ,   y + 1 2 C BPOF τ x ,   τ y .
S Δ f = DFT s Δ x , S k = S k Δ f = A k   exp j ϕ k ,
C 0 = B   exp j β = k = 1 N   H k A k   exp j ϕ k ,
H m = u cos ϕ m - β ,
β = arg k = 1 N   u cos ϕ k - β A k   exp j ϕ k .
β = arg k A k P nk   cos ϕ k - β A k   exp j ϕ k .
u cos ϕ k - α = 1     k = 1 M   u cos ϕ k - α A k   exp j ϕ k = k = 1 M   A k   exp j ϕ k = D   exp j α .
α = arg k = 1 N   u cos ϕ k - α A k   exp j ϕ k .
OBAF S = max k = 1 n   b k z k   :   b 1 , ,   b n 0 ,   1 .
α = r   exp j θ   :   r > 0 ,   | θ - α | < π / 2 .
σ α = k = 1 n   b k α z k .
ζ = z k + + z m ,
ζ = ω n - z k + + z m .
OBAF S = | ω m - ω k - 1 | .

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