Abstract

We show that optimal regions of support for correlation filters in the frequency domain can be approximated by relatively small convolution kernels in the spatial domain. We present an optimal approach for generating regions of support, as well as a fast nonoptimal approach for conventional optical correlators. Because the convolution kernels are similar to low-pass filters, the resulting input image to a correlator is always positive valued. We show that the performance of the convolution-based approach is comparable with the optimal frequency-domain approach. An important advantage of our method is that it can be implemented on low-cost arithmetic frame grabbers that can perform convolution with small kernels in real time. In addition, our method can be used in conjunction with a filter spatial light modulator that cannot produce a zero state.

© 1997 Optical Society of America

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References

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  1. Ph. Refregier, “Optimal trade-off filters for noise robustnes, sharpness of the correlation peak and Horner efficiency,” Opt. Lett. 16, 829–831 (1991).
    [CrossRef] [PubMed]
  2. Ph. Refregier, B. V. K. Vijaya Kumar, C. Hendrix, “Multicriteria optimal binary amplitude phase-only filters,” J. Opt. Soc. Am. A 9, 2118–2125 (1992).
    [CrossRef]
  3. C. D. Hendrix, B. V. K. Vijaya Kumar, “Design and evaluation of three-level composite filters obtained by optimizing a compromise average performance measure,” Opt. Eng. 33, 1767–1773 (1994).
    [CrossRef]
  4. B. V. K. Vijaya Kumar, L. Hasselbrook, “Performance measures for correlation filters,” Appl. Opt. 29, 2997–3006 (1990).
    [CrossRef]
  5. B. A. Kast, M. Giles, S. Lindell, D. Flannery, “Implementation of ternary phase amplitude filters for improved correlation discrimination,” Appl. Opt. 28, 1044–1046 (1989).
    [CrossRef] [PubMed]
  6. N. H. Farhat, Z. Y. Shae, “Scheme for enhancing the frame rate of magnetooptic spatial light modulators,” Appl. Opt. 28, 4792–4800 (1989).
    [CrossRef] [PubMed]
  7. S. P. Kozaitis, S. Halby, W. Foor, “Ground exploitation using a binary phase-only optical correlator,” in Optical Information Processing Systems and Architectures II, B. Javidi, ed., Proc. SPIE1347, 286–296 (1990).
    [CrossRef]
  8. A. Grunnet-Jepsen, S. Tonda, V. Laude, “Convolution-kernel-based optimal trade-off filters for optical pattern recognition,” Appl. Opt. 35, 3874–3879 (1996).
    [CrossRef] [PubMed]
  9. V. Laude, P. Chavel, Ph. Refregier, “Implementation of arbitrary real-valued correlation filters for the shadow-casting incoherent correlator,” Appl. Opt. 35, 5267–5274 (1996).
    [CrossRef] [PubMed]
  10. B. V. K. Vijaya Kumar, Z. Bahri, “Phase-only filters with improved signal to noise ratio,” Appl. Opt. 28, 250–257 (1989).
    [CrossRef]
  11. R. W. Schutten, G. F. Vermeij, “The approximation of image blur restoration filters by finite impulse responses,” IEEE Patt. Anal. Mach. Intel. PAMI-2, 176–180 (1980).
    [CrossRef]
  12. R. W. Hamming, Digital Filters, 3rd ed. (Prentice-Hall, Englewood Cliffs, N.J., 1989), p. 217.

1996 (2)

A. Grunnet-Jepsen, S. Tonda, V. Laude, “Convolution-kernel-based optimal trade-off filters for optical pattern recognition,” Appl. Opt. 35, 3874–3879 (1996).
[CrossRef] [PubMed]

V. Laude, P. Chavel, Ph. Refregier, “Implementation of arbitrary real-valued correlation filters for the shadow-casting incoherent correlator,” Appl. Opt. 35, 5267–5274 (1996).
[CrossRef] [PubMed]

1994 (1)

C. D. Hendrix, B. V. K. Vijaya Kumar, “Design and evaluation of three-level composite filters obtained by optimizing a compromise average performance measure,” Opt. Eng. 33, 1767–1773 (1994).
[CrossRef]

1992 (1)

Ph. Refregier, B. V. K. Vijaya Kumar, C. Hendrix, “Multicriteria optimal binary amplitude phase-only filters,” J. Opt. Soc. Am. A 9, 2118–2125 (1992).
[CrossRef]

1991 (1)

Ph. Refregier, “Optimal trade-off filters for noise robustnes, sharpness of the correlation peak and Horner efficiency,” Opt. Lett. 16, 829–831 (1991).
[CrossRef] [PubMed]

1990 (1)

B. V. K. Vijaya Kumar, L. Hasselbrook, “Performance measures for correlation filters,” Appl. Opt. 29, 2997–3006 (1990).
[CrossRef]

1989 (3)

B. A. Kast, M. Giles, S. Lindell, D. Flannery, “Implementation of ternary phase amplitude filters for improved correlation discrimination,” Appl. Opt. 28, 1044–1046 (1989).
[CrossRef] [PubMed]

N. H. Farhat, Z. Y. Shae, “Scheme for enhancing the frame rate of magnetooptic spatial light modulators,” Appl. Opt. 28, 4792–4800 (1989).
[CrossRef] [PubMed]

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

1980 (1)

R. W. Schutten, G. F. Vermeij, “The approximation of image blur restoration filters by finite impulse responses,” IEEE Patt. Anal. Mach. Intel. PAMI-2, 176–180 (1980).
[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]

Chavel, P.

V. Laude, P. Chavel, Ph. Refregier, “Implementation of arbitrary real-valued correlation filters for the shadow-casting incoherent correlator,” Appl. Opt. 35, 5267–5274 (1996).
[CrossRef] [PubMed]

Farhat, N. H.

N. H. Farhat, Z. Y. Shae, “Scheme for enhancing the frame rate of magnetooptic spatial light modulators,” Appl. Opt. 28, 4792–4800 (1989).
[CrossRef] [PubMed]

Flannery, D.

B. A. Kast, M. Giles, S. Lindell, D. Flannery, “Implementation of ternary phase amplitude filters for improved correlation discrimination,” Appl. Opt. 28, 1044–1046 (1989).
[CrossRef] [PubMed]

Foor, W.

S. P. Kozaitis, S. Halby, W. Foor, “Ground exploitation using a binary phase-only optical correlator,” in Optical Information Processing Systems and Architectures II, B. Javidi, ed., Proc. SPIE1347, 286–296 (1990).
[CrossRef]

Giles, M.

B. A. Kast, M. Giles, S. Lindell, D. Flannery, “Implementation of ternary phase amplitude filters for improved correlation discrimination,” Appl. Opt. 28, 1044–1046 (1989).
[CrossRef] [PubMed]

Grunnet-Jepsen, A.

A. Grunnet-Jepsen, S. Tonda, V. Laude, “Convolution-kernel-based optimal trade-off filters for optical pattern recognition,” Appl. Opt. 35, 3874–3879 (1996).
[CrossRef] [PubMed]

Halby, S.

S. P. Kozaitis, S. Halby, W. Foor, “Ground exploitation using a binary phase-only optical correlator,” in Optical Information Processing Systems and Architectures II, B. Javidi, ed., Proc. SPIE1347, 286–296 (1990).
[CrossRef]

Hamming, R. W.

R. W. Hamming, Digital Filters, 3rd ed. (Prentice-Hall, Englewood Cliffs, N.J., 1989), p. 217.

Hasselbrook, L.

B. V. K. Vijaya Kumar, L. Hasselbrook, “Performance measures for correlation filters,” Appl. Opt. 29, 2997–3006 (1990).
[CrossRef]

Hendrix, C.

Ph. Refregier, B. V. K. Vijaya Kumar, C. Hendrix, “Multicriteria optimal binary amplitude phase-only filters,” J. Opt. Soc. Am. A 9, 2118–2125 (1992).
[CrossRef]

Hendrix, C. D.

C. D. Hendrix, B. V. K. Vijaya Kumar, “Design and evaluation of three-level composite filters obtained by optimizing a compromise average performance measure,” Opt. Eng. 33, 1767–1773 (1994).
[CrossRef]

Kast, B. A.

B. A. Kast, M. Giles, S. Lindell, D. Flannery, “Implementation of ternary phase amplitude filters for improved correlation discrimination,” Appl. Opt. 28, 1044–1046 (1989).
[CrossRef] [PubMed]

Kozaitis, S. P.

S. P. Kozaitis, S. Halby, W. Foor, “Ground exploitation using a binary phase-only optical correlator,” in Optical Information Processing Systems and Architectures II, B. Javidi, ed., Proc. SPIE1347, 286–296 (1990).
[CrossRef]

Laude, V.

A. Grunnet-Jepsen, S. Tonda, V. Laude, “Convolution-kernel-based optimal trade-off filters for optical pattern recognition,” Appl. Opt. 35, 3874–3879 (1996).
[CrossRef] [PubMed]

V. Laude, P. Chavel, Ph. Refregier, “Implementation of arbitrary real-valued correlation filters for the shadow-casting incoherent correlator,” Appl. Opt. 35, 5267–5274 (1996).
[CrossRef] [PubMed]

Lindell, S.

B. A. Kast, M. Giles, S. Lindell, D. Flannery, “Implementation of ternary phase amplitude filters for improved correlation discrimination,” Appl. Opt. 28, 1044–1046 (1989).
[CrossRef] [PubMed]

Refregier, Ph.

V. Laude, P. Chavel, Ph. Refregier, “Implementation of arbitrary real-valued correlation filters for the shadow-casting incoherent correlator,” Appl. Opt. 35, 5267–5274 (1996).
[CrossRef] [PubMed]

Ph. Refregier, B. V. K. Vijaya Kumar, C. Hendrix, “Multicriteria optimal binary amplitude phase-only filters,” J. Opt. Soc. Am. A 9, 2118–2125 (1992).
[CrossRef]

Ph. Refregier, “Optimal trade-off filters for noise robustnes, sharpness of the correlation peak and Horner efficiency,” Opt. Lett. 16, 829–831 (1991).
[CrossRef] [PubMed]

Schutten, R. W.

R. W. Schutten, G. F. Vermeij, “The approximation of image blur restoration filters by finite impulse responses,” IEEE Patt. Anal. Mach. Intel. PAMI-2, 176–180 (1980).
[CrossRef]

Shae, Z. Y.

N. H. Farhat, Z. Y. Shae, “Scheme for enhancing the frame rate of magnetooptic spatial light modulators,” Appl. Opt. 28, 4792–4800 (1989).
[CrossRef] [PubMed]

Tonda, S.

A. Grunnet-Jepsen, S. Tonda, V. Laude, “Convolution-kernel-based optimal trade-off filters for optical pattern recognition,” Appl. Opt. 35, 3874–3879 (1996).
[CrossRef] [PubMed]

Vermeij, G. F.

R. W. Schutten, G. F. Vermeij, “The approximation of image blur restoration filters by finite impulse responses,” IEEE Patt. Anal. Mach. Intel. PAMI-2, 176–180 (1980).
[CrossRef]

Vijaya Kumar, B. V. K.

C. D. Hendrix, B. V. K. Vijaya Kumar, “Design and evaluation of three-level composite filters obtained by optimizing a compromise average performance measure,” Opt. Eng. 33, 1767–1773 (1994).
[CrossRef]

Ph. Refregier, B. V. K. Vijaya Kumar, C. Hendrix, “Multicriteria optimal binary amplitude phase-only filters,” J. Opt. Soc. Am. A 9, 2118–2125 (1992).
[CrossRef]

B. V. K. Vijaya Kumar, L. Hasselbrook, “Performance measures for correlation filters,” Appl. Opt. 29, 2997–3006 (1990).
[CrossRef]

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

Appl. Opt. (6)

B. V. K. Vijaya Kumar, L. Hasselbrook, “Performance measures for correlation filters,” Appl. Opt. 29, 2997–3006 (1990).
[CrossRef]

B. A. Kast, M. Giles, S. Lindell, D. Flannery, “Implementation of ternary phase amplitude filters for improved correlation discrimination,” Appl. Opt. 28, 1044–1046 (1989).
[CrossRef] [PubMed]

N. H. Farhat, Z. Y. Shae, “Scheme for enhancing the frame rate of magnetooptic spatial light modulators,” Appl. Opt. 28, 4792–4800 (1989).
[CrossRef] [PubMed]

A. Grunnet-Jepsen, S. Tonda, V. Laude, “Convolution-kernel-based optimal trade-off filters for optical pattern recognition,” Appl. Opt. 35, 3874–3879 (1996).
[CrossRef] [PubMed]

V. Laude, P. Chavel, Ph. Refregier, “Implementation of arbitrary real-valued correlation filters for the shadow-casting incoherent correlator,” Appl. Opt. 35, 5267–5274 (1996).
[CrossRef] [PubMed]

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

IEEE Patt. Anal. Mach. Intel. (1)

R. W. Schutten, G. F. Vermeij, “The approximation of image blur restoration filters by finite impulse responses,” IEEE Patt. Anal. Mach. Intel. PAMI-2, 176–180 (1980).
[CrossRef]

J. Opt. Soc. Am. A (1)

Ph. Refregier, B. V. K. Vijaya Kumar, C. Hendrix, “Multicriteria optimal binary amplitude phase-only filters,” J. Opt. Soc. Am. A 9, 2118–2125 (1992).
[CrossRef]

Opt. Eng. (1)

C. D. Hendrix, B. V. K. Vijaya Kumar, “Design and evaluation of three-level composite filters obtained by optimizing a compromise average performance measure,” Opt. Eng. 33, 1767–1773 (1994).
[CrossRef]

Opt. Lett. (1)

Ph. Refregier, “Optimal trade-off filters for noise robustnes, sharpness of the correlation peak and Horner efficiency,” Opt. Lett. 16, 829–831 (1991).
[CrossRef] [PubMed]

Other (2)

R. W. Hamming, Digital Filters, 3rd ed. (Prentice-Hall, Englewood Cliffs, N.J., 1989), p. 217.

S. P. Kozaitis, S. Halby, W. Foor, “Ground exploitation using a binary phase-only optical correlator,” in Optical Information Processing Systems and Architectures II, B. Javidi, ed., Proc. SPIE1347, 286–296 (1990).
[CrossRef]

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Figures (8)

Fig. 1
Fig. 1

Image used in the experiments.

Fig. 2
Fig. 2

Optimal trade-off between the SNR and the PCE by use of a BPOF for the image shown in Fig. 1.

Fig. 3
Fig. 3

Support regions corresponding to three points on the graph of the optimal trade-off between the SNR and the PCE for the image shown in Fig. 1. The white areas indicate transparent regions: (a) mask 1, (b) mask 2, and (c) mask 3.

Fig. 4
Fig. 4

Graph of the SNR and PCE as functions of the transmittance of the area surrounding the support region for masks 1–3 for the autocorrelation of the image shown in Fig. 1: (a) SNR versus the transmittance and (b) PCE versus the transmittance.

Fig. 5
Fig. 5

Graph of the MSE between the FT of a convolution kernel and masks 1–3 as a function of filter number. The convolution kernels corresponding to the filter numbers are listed in Table 2.

Fig. 6
Fig. 6

Graph of the SNR as a function of the filter number for the autocorrelation of Fig. 1. The convolution kernels corresponding to the filter numbers are listed in Table 2.

Fig. 7
Fig. 7

Graph of the PCE as a function of the filter number for the autocorrelation of Fig. 1. The convolution kernels corresponding to the filter numbers are listed in Table 2.

Fig. 8
Fig. 8

Noise performance of a 9 × 9 convolution kernel compared with regions of support generated by the frequency-domain method (crosses): (a) mask 1, (b) mask 2, and (c) mask 3.

Tables (2)

Tables Icon

Table 1 SNR and PCE Values of Selected Support Regions

Tables Icon

Table 2 Kernel Sizes for the First 20 Filter Numbers from Figs. 57

Equations (9)

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

Fμ=C021-μMSE+μCPE,
Gu, ν, μ=Su, ν1-μPu, ν+μSu, ν2.
Ri, μ=u, νSu, ν1-μPu, ν+μSu, ν2>Giu, ν, μ,
Rˆu, ν=xn-1yn-1rˆx, yexp-j2πux+νy/N.
e2=uN-1νN-1Rˆu, ν-Ru, ν2.
e2=Rˆ-R*Rˆ-R=Rˆ-R2=Cˆr-R2,
Ci, k=1Nexp-j2πux+νy/N,
rˆ=C*C-1C*R.
SNR=Su, νHu, ν2 Pu, νHu, ν2,

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