Abstract

We examined the performance of linear and nonlinear processors (filters) for image recognition that are lp-norm optimum in terms of tolerance to input noise and discrimination capabilities. These processors were developed by minimizing the lp norm of the filter output due to the input scene and the output due to the noise. We tested the performance of the lp-norm optimum filters by measuring the average peak-to-sidelobe ratio of the output of the filters for different values of p. We also tested the performance of these filters by placing a target in a scene containing additive noise and a realistic background. For the images presented here, the filters detected the target in the presence of additive noise and a realistic background. The tests conducted show that the discrimination capabilities of the lp-norm filters improve as p decreases (p>1). This is shown by sharper peaks at the target location and higher average peak-to-sidelobe ratios for smaller values of p.

© 1999 Optical Society of America

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References

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  1. N. Towghi, B. Javidi, “lp-norm optimum filters for image recognition. Part I: algorithms” J. Opt. Soc. Am. A 16, 1928–1935 (1999).
    [CrossRef]
  2. J. L. Turin, “An introduction to matched filters,” IRE Trans. Inf. Theory IT-6, 311–329 (1960).
    [CrossRef]
  3. D. Casasent, D. Psaltis, “Position, rotation, and scale invariant optical correlation,” Appl. Opt. 15, 1795–1799 (1976).
    [CrossRef] [PubMed]
  4. J. L. Horner, P. D. Gianino, “Phase-only matched filtering,” Appl. Opt. 23, 812–816 (1984).
    [CrossRef] [PubMed]
  5. H. J. Caufield, W. T. Maloney, “Improved discrimination in optical character recognition,” Appl. Opt. 8, 2354 (1969).
    [CrossRef]
  6. D. L. Flannery, J. L. Horner, “Fourier optical signal processor,” Proc. IEEE 77, 1511 (1989).
    [CrossRef]
  7. D. Casasent, “Unified synthetic discrimination function computational formulation,” Appl. Opt. 23, 1620–1627 (1984).
    [CrossRef]
  8. Ph. Réfrégier, J. Figue, “Optimal trade-off filters for pattern recognition and their comparison with Wiener approach,” Opt. Comput. Process. 1, 245–265 (1991).
  9. K. H. Fielding, J. L. Horner, “1-f binary joint transform correlator,” Opt. Eng. 29, 1081–1087 (1990).
    [CrossRef]
  10. B. Javidi, “Nonlinear joint power spectrum based optical correlation,” Appl. Opt. 28, 2358–2367 (1989).
    [CrossRef] [PubMed]
  11. Ph. Réfrégier, V. Laude, B. Javidi, “Nonlinear joint-transform correlation: an optimal solution for adaptive image discrimination and input noise robustness,” Opt. Lett. 19, 405–407 (1994).
    [PubMed]
  12. B. V. K. Vijaya Kumar, “Tutorial survey of composite filter designs for optical correlators,” Appl. Opt. 31, 4773–4800 (1992).
    [CrossRef]
  13. A. Mahalanobis, B. V. K. Vijaya Kumar, D. Casasent, “Minimum average correlation energy filters,” Appl. Opt. 26, 3633–3640 (1987).
    [CrossRef] [PubMed]
  14. Ph. Réfrégier, “Filter design for optical pattern recognition: multicriteria approach,” Opt. Lett. 15, 854–856 (1990).
    [CrossRef]
  15. Ph. Réfrégier, “Optimal trade-off filters for noise robustness, sharpness of the correlation peak, and Horner efficiency,” Opt. Lett. 16, 829–831 (1991).
    [CrossRef] [PubMed]
  16. Ph. Réfrégier, “Optical pattern recognition: optimal trade-off circular harmonic filters,” Opt. Commun. 86, 113–118 (1991).
    [CrossRef]

1999

1994

1992

1991

Ph. Réfrégier, “Optical pattern recognition: optimal trade-off circular harmonic filters,” Opt. Commun. 86, 113–118 (1991).
[CrossRef]

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

Ph. Réfrégier, J. Figue, “Optimal trade-off filters for pattern recognition and their comparison with Wiener approach,” Opt. Comput. Process. 1, 245–265 (1991).

1990

K. H. Fielding, J. L. Horner, “1-f binary joint transform correlator,” Opt. Eng. 29, 1081–1087 (1990).
[CrossRef]

Ph. Réfrégier, “Filter design for optical pattern recognition: multicriteria approach,” Opt. Lett. 15, 854–856 (1990).
[CrossRef]

1989

B. Javidi, “Nonlinear joint power spectrum based optical correlation,” Appl. Opt. 28, 2358–2367 (1989).
[CrossRef] [PubMed]

D. L. Flannery, J. L. Horner, “Fourier optical signal processor,” Proc. IEEE 77, 1511 (1989).
[CrossRef]

1987

1984

1976

1969

1960

J. L. Turin, “An introduction to matched filters,” IRE Trans. Inf. Theory IT-6, 311–329 (1960).
[CrossRef]

Casasent, D.

Caufield, H. J.

Fielding, K. H.

K. H. Fielding, J. L. Horner, “1-f binary joint transform correlator,” Opt. Eng. 29, 1081–1087 (1990).
[CrossRef]

Figue, J.

Ph. Réfrégier, J. Figue, “Optimal trade-off filters for pattern recognition and their comparison with Wiener approach,” Opt. Comput. Process. 1, 245–265 (1991).

Flannery, D. L.

D. L. Flannery, J. L. Horner, “Fourier optical signal processor,” Proc. IEEE 77, 1511 (1989).
[CrossRef]

Gianino, P. D.

Horner, J. L.

K. H. Fielding, J. L. Horner, “1-f binary joint transform correlator,” Opt. Eng. 29, 1081–1087 (1990).
[CrossRef]

D. L. Flannery, J. L. Horner, “Fourier optical signal processor,” Proc. IEEE 77, 1511 (1989).
[CrossRef]

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

Javidi, B.

Laude, V.

Mahalanobis, A.

Maloney, W. T.

Psaltis, D.

Réfrégier, Ph.

Towghi, N.

Turin, J. L.

J. L. Turin, “An introduction to matched filters,” IRE Trans. Inf. Theory IT-6, 311–329 (1960).
[CrossRef]

Vijaya Kumar, B. V. K.

Appl. Opt.

IRE Trans. Inf. Theory

J. L. Turin, “An introduction to matched filters,” IRE Trans. Inf. Theory IT-6, 311–329 (1960).
[CrossRef]

J. Opt. Soc. Am. A

Opt. Commun.

Ph. Réfrégier, “Optical pattern recognition: optimal trade-off circular harmonic filters,” Opt. Commun. 86, 113–118 (1991).
[CrossRef]

Opt. Comput. Process.

Ph. Réfrégier, J. Figue, “Optimal trade-off filters for pattern recognition and their comparison with Wiener approach,” Opt. Comput. Process. 1, 245–265 (1991).

Opt. Eng.

K. H. Fielding, J. L. Horner, “1-f binary joint transform correlator,” Opt. Eng. 29, 1081–1087 (1990).
[CrossRef]

Opt. Lett.

Proc. IEEE

D. L. Flannery, J. L. Horner, “Fourier optical signal processor,” Proc. IEEE 77, 1511 (1989).
[CrossRef]

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Figures (4)

Fig. 1
Fig. 1

Input scene: (a) two targets (jet planes) with background scene; (b) scene of (a) with additive zero-mean white Gaussian noise with standard deviation of 0.7.

Fig. 2
Fig. 2

Output of the Hqσ nonlinear filters of Eqs. (5) and (6). These filters are obtained by minimizing the lp norm of the output due to input scene and the output due to noise. (a) Output of the filter when q=2 [Eq. (5)], (b) output of the filter when q=10 [Eq. (5)], (c) output of the filter when q= [Eq. (6)].

Fig. 3
Fig. 3

Output of the Hq0 nonlinear filters of Eqs. (7) and (8). These filters are obtained by minimizing the lp norm of the output due to input scene. (a) Output of the filter when q=2 [Eq. (7)], (b) output of the filter when q=10 [Eq. (7)], (c) output of the filter when q= [Eq. (8)].

Fig. 4
Fig. 4

Output of the Hq linear filters of Eqs. (9) and (10). These filters are obtained by minimizing the lp norm of the output due to noise. (a) Output of the filter when q=2 [Eq. (9)], (b) output of the filter when q=10 [Eq. (9)], (c) output of the filter when q= [Eq. (10)].  

Equations (11)

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

s(j)=r(j)+n(j).
j=0J-1h(j)*r(j)=C=C(0),
aj=0J-1El=0J-1h(j-l)n(l)p+bj=0J-1lh(j-l)s(l)p,
j=0J-1|H(j)|q[σ^q+|S(j)|q],
j=0J-1H(j)*R(j)=JC(0),
Hqσ(j)=|R(j)|σ^q+|S(j)|q1/(q-1) exp[iΦR(j)],
Hσ(j)=1max{Jσ,|S(j)|}exp[iΦR(j)].
Hq0(j)=|R(j)|1/(q-1)|S(j)|1/(1-q)-1 exp[iΦR(j)],
H0(j)=|S(j)|-1 exp[iΦR(j)].
Hq(j)=|R(j)|1/(q-1) exp[iΦR(j)],
H(j)=exp[iΦR(j)].

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