Abstract

Ordinarily, filters are derived from the optimization of certain expressions with respect to the mean squared metric. We construct a family of linear and nonlinear processors (filters) for image recognition that is lp-norm optimum in terms of tolerance to input noise and discrimination capabilities. The lp norm is the generalization of the usual mean squared (l2) norm, which we obtain by replacing the exponent 2 with any positive constant p (usually p1). These processors are developed by minimizing the lp norm of the filter output that is due to the input scene and the output that is due to input noise. We use the lp norm to measure the size of the filter output that is due to noise so that we can obtain greater freedom in adjusting the noise robustness and discrimination capabilities. We give a unified theoretical basis for developing these filters. This family of filters includes some of the existing linear and nonlinear filters, giving us a subfamilies of processors, which we denote by Hqσ and Hq. The values of q control the discrimination capabilities and the robustness of the processors. The parameter σ is the standard deviation of the noise process.

© 1999 Optical Society of America

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References

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  1. J. L. Turin, “An introduction to matched filters,” IRE Trans. Inf. Theory IT-6, 311–329 (1960).
    [CrossRef]
  2. A. Vanderlugt, “Signal detection by complex filters,” IEEE Trans. Inf. Theory IT-10, 139–145 (1964).
  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]
  17. P. Maragos, “Optimal morphological approaches to image matching and object detection,” in Proceedings of the Second International Conference on Computer Vision and Pattern Recognition (IEEE Computer Society Press, Los Alamitos, Calif., 1988), pp. 695–699.
  18. Y. Katznelson, An Introduction to Harmonic Analysis (Dover, New York, 1968).
  19. L. C. Young, “An inequality of the Hölder type, connected with Stieltjes integration,” Acta Math. 67, 251–282 (1936).
    [CrossRef]

1994 (1)

1992 (1)

1991 (3)

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

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 (2)

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 (2)

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

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

1987 (1)

1984 (2)

1976 (1)

1969 (1)

1964 (1)

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

1960 (1)

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

1936 (1)

L. C. Young, “An inequality of the Hölder type, connected with Stieltjes integration,” Acta Math. 67, 251–282 (1936).
[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.

Katznelson, Y.

Y. Katznelson, An Introduction to Harmonic Analysis (Dover, New York, 1968).

Laude, V.

Mahalanobis, A.

Maloney, W. T.

Maragos, P.

P. Maragos, “Optimal morphological approaches to image matching and object detection,” in Proceedings of the Second International Conference on Computer Vision and Pattern Recognition (IEEE Computer Society Press, Los Alamitos, Calif., 1988), pp. 695–699.

Psaltis, D.

Réfrégier, Ph.

Turin, J. L.

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

Vanderlugt, A.

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

Vijaya Kumar, B. V. K.

Young, L. C.

L. C. Young, “An inequality of the Hölder type, connected with Stieltjes integration,” Acta Math. 67, 251–282 (1936).
[CrossRef]

Acta Math. (1)

L. C. Young, “An inequality of the Hölder type, connected with Stieltjes integration,” Acta Math. 67, 251–282 (1936).
[CrossRef]

Appl. Opt. (7)

IEEE Trans. Inf. Theory (1)

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

IRE Trans. Inf. Theory (1)

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

Opt. Commun. (1)

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

Opt. Comput. Process. (1)

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. (1)

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

Opt. Lett. (3)

Proc. IEEE (1)

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

Other (2)

P. Maragos, “Optimal morphological approaches to image matching and object detection,” in Proceedings of the Second International Conference on Computer Vision and Pattern Recognition (IEEE Computer Society Press, Los Alamitos, Calif., 1988), pp. 695–699.

Y. Katznelson, An Introduction to Harmonic Analysis (Dover, New York, 1968).

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Equations (89)

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s( j)=r( j)+n( j).
cp=j=0J-1|cj|p1/p
j=0J-1h(j)*r(j)=C=C(0),
E(h*n)pp=j=0J-1El=0J-1h( j-l)n(l)p,
h*spp=j=0J-1lh( j-l)s(l)p.
aE(h*n)pp+bh*spp
j=0J-1|H( j)|q[σ^q+|S( j)|q],
j=0J-1H( j)*R( j)=JC(0),
σ^q=E|N( j)|q.
Hqσ( j)=|R( j)|σ^q+|S( j)|q1/(q-1) exp[iΦR( j)],
c( j)=1J j=0J-1Hqσ(k)*S(k)expi2πkjJ.
Hqσ( j)=|R( j)|1 /(q-1)max{σ^q1/(q-1), |S( j)|q/(q-1)}×exp[iΦR( j)],q1.
limq|S( j)|q/(q-1)=|S( j)|,
limq σ^q1/(q-1)=N( j),
HM( j)=1max{M, S( j)|}exp[iΦR( j)].
Hσ( j)=1max{Jσ, |S( j)|}exp(iΦR( j)).
E|N( j)|q=(Jσ)q2q/2Γ(q/2+0.5)π.
N( j)=l=0J-1a( j, l)n(l),
E|N( j)|q=12πJσ -|x|q exp-x22Jσ2dx.
E|N( j)|q(σJ)q,q2.
E|N( j)|2=Jσ2.
E|N( j)|q(JM)q.
E|N( j)|qσ2Mq-2qJ2q.
E|N( j)|q[σJ]q.
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)],
cp1cp2,
j=0J-1l=0J-1h( j-l)s(l)p+El=0j-1h( j-l)n(l)p,
j=0J-1h( j)*r( j)=C.
j=1Mwja( j)b( j)j=1Mwj|a( j)|p1/pj=1Mwj|b( j)|q1/q.
j=1Mwj|bj|q=1,j=1Mwj|a( j)|p1/p=j=1Mwjajbj,
b(l)=|a(l)|p-2a(l)/cifa(l)00otherwise,
J(h)=j=0J-1l=0J-1h( j-l)s(l)p+El=0J-1h( j-l)n(l)p1/p.
j=0J-1hk( j)*r( j)=0.
H1(t)( j)=[(t1h1++tJ-1hJ-1+h0)*s]( j),
H2(t)( j)=[(t1h1++tJ-1hJ-1+h0)*n]( j).
G(t)(l)=H1(t)(l)if0lJ-1H2(t)(l- J)ifJl2J-1.
Vp(t)=l=02J-1E|G(t)(l)|p1/p
G(t)(l)=c1H1(t)(l)if0lJ-1c2H2(t)(l- J)ifJl2J-1,
Q(t)=G(t0), G(t)=l=02J-1G(t0)(l)G(t)(l).
E[Q(t)]Vp(t),
E[Q(t0)]=Vp(t0).
Vq(t)= 1J j=0J-1|H^1(t)( j)|q+j=0J-1E[|H^2(t)( j)|q]1/q.
R(t)(l)=H^1(t)(l)if0lJ-1H^2(t)(l- J)ifJl2J-1.
Vq(t)=1J l=02J-1E|R(t)(l)|q1/q,
1J R(t0), R(t)=1J l=02J-1R(t)(l)R(t)(l)=1J l=02J-1|R(t)(l)|q1/q,
1J l=02J-1|R(t)(l)|p1/p=1.
Q(t)=1J R(t0), R(t).
E[Q(t)]Vq(t),
E[Q(t0)]=Vq(t0).
Q(t)=1J [c1H^1(t0), H^1(t)+c2H^2(t0), H^2(t)].
Q(t)=1J [d1H1(t0), H^1(t)+d2H2(t0), H^2(t)],
Qtk (t0)=j=0J-1W( j)h^k( j),
W( j)=1J [c1H^1(t0)( j)S( j)+c2H^2(t0)( j)N( j)],
Qtk (t0)=j=0J-1U( j)h^k( j),
U( j)=1J [d1H1(t0)( j)S( j)+d2H2(t0)( j)N( j)].
E(Q)tk (t0)=j=0J-1E[W( j)]h^k( j)=0,
E(Q)tk (t0)=j=0J-1E[U( j)]h^k( j)=0.
E(W)={E[W(0)],, E[W(J-1)]},
E(U)={E[U(0)],, E[U(J-1)]}
E(U)=λE(W),
Vp(t0)=Vq(t0).
Vp(t)Vq(t).
j=0J-1|H( j)|q[σ^q+|S( j)|q],
j=0J-1H( j)*R( j)=JC(0).
j=0J-1|H( j)||R( j)|cos(ϕj-θj)=JC(0),
j=0J-1|H( j)||R( j)|sin(ϕj-θj)=0.
q|H( j)|(q-1)[σ^q+|S( j)|q]-|R( j)|[λ1 cos(ϕj-θj)
+λ2 sin(ϕj-θj)]=0,
[λ2 cos(ϕj-θj)-λ1 sin(ϕj-θj)]=0.
|H( j)|=λ1|R( j)|q[σ^q+|S( j)|q]1/(q-1).
λ1=q[JC(0)]q-1j=0J-1|R( j)|×|R( j)|σ^q+|S( j)|q21/(q-1)(1-q),
|H( j)|=|R( j)|σ^q+|S( j)|q1/(q-1).
H( j)=|H( j)|exp(iθj)=c|R( j)|σ^q+|S( j)|q1/(q-1) exp(iϕj),
E|N( j)|qσ2Mq-2qJ22.
El=1Kaln(l)qσ2Mq-2[q/2]q/2l=1K|al|2q/2.
E[n(l)]t=0.
 (2m)!v1!v2!vK! a˜1a˜2a˜KE[nv1(1)nvK(K)],
El=1Kaln(l)2m
=k1+k2++kK=m (2m)!(2k1)!(2kK)! |a1|2k1|aK|2kK
×E[n2k1(1)n2kK(K)].
E[n2k1(1)nkK(K)]σ2M2m-2.
k1+k2++kK=m (2m)!(2k1)!(2kK)! a12k1aK2kK
mmk1+k2++kK=m m!(k1)!(kK!) |a1|2k1|aK|2kK.
k1+k2++kK=m (2m!)(2k1)!(2kK)! a12k1aK2kK
mml=1K|al|2)m.
N( j)=l=0J-1a(l, j)n(l),

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