Abstract

Through the Rayleigh quotient (the ratio of intensity responses of a filter to different objects) we may generalize a great number of metrics used in optical pattern recognition. The Rayleigh quotient has been optimized in linear digital systems under the constraint of unit-energy filters. In optical pattern recognition at least two considerations violate the conditions under which the quotient has been digitally optimized: the noise background of the measurement invokes nonlinearity, and filters are constrained other than to unit energy. I show a solution that optimizes the ratio of biased measurements, subject to constraining filter values to arbitrary subsets of the complex plane. Previous solutions are discussed as special cases. A metric’s numerator and denominator may now both include the objects’ phase.

© 1998 Optical Society of America

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References

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  1. O. Duda, P. E. Hart, Pattern Classification and Scene Analysis (Wiley, New York, 1973).
  2. For example, C. W. Therrien, Discrete Random Signals and Statistical Signal Processing (Prentice-Hall, Englewood Cliffs, N.J., 1992), Sec. 5.4.
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    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
  8. L. P. Yaroslavsky, “Is [sic] the phase-only filter and its modifications optimal in terms of the discrimination capability in pattern recognition?” Appl. Opt. 31, 1677–1679 (1992).
    [CrossRef] [PubMed]
  9. L. P. Yaroslavsky, “The theory of optimal methods for localization of objects in pictures,” in Progress in Optics, E. Wolf, ed. (Elsevier, Amsterdam, 1993), Vol. XXXII, pp. 145–201.
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]

1996

1994

1993

1992

1989

1984

Bahri, Z.

Carlson, D. W.

Duda, O.

O. Duda, P. E. Hart, Pattern Classification and Scene Analysis (Wiley, New York, 1973).

Farn, M. W.

Gianino, P. D.

Goodman, J. W.

Hart, P. E.

O. Duda, P. E. Hart, Pattern Classification and Scene Analysis (Wiley, New York, 1973).

Horner, J. L.

Javidi, B.

Juday, R. D.

Karivaratha Rajan, P.

Laude, V.

Mahalanobis, A.

Parchekani, F.

Réfrégier, Ph.

Therrien, C. W.

For example, C. W. Therrien, Discrete Random Signals and Statistical Signal Processing (Prentice-Hall, Englewood Cliffs, N.J., 1992), Sec. 5.4.

Vijaya Kumar, B. V. K.

Willett, P.

Yaroslavsky, L. P.

L. P. Yaroslavsky, “Is [sic] the phase-only filter and its modifications optimal in terms of the discrimination capability in pattern recognition?” Appl. Opt. 31, 1677–1679 (1992).
[CrossRef] [PubMed]

L. P. Yaroslavsky, “The theory of optimal methods for localization of objects in pictures,” in Progress in Optics, E. Wolf, ed. (Elsevier, Amsterdam, 1993), Vol. XXXII, pp. 145–201.

Zhang, G.

Appl. Opt.

J. Opt. Soc. Am. A

Opt. Lett.

Other

O. Duda, P. E. Hart, Pattern Classification and Scene Analysis (Wiley, New York, 1973).

For example, C. W. Therrien, Discrete Random Signals and Statistical Signal Processing (Prentice-Hall, Englewood Cliffs, N.J., 1992), Sec. 5.4.

L. P. Yaroslavsky, “The theory of optimal methods for localization of objects in pictures,” in Progress in Optics, E. Wolf, ed. (Elsevier, Amsterdam, 1993), Vol. XXXII, pp. 145–201.

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).

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Figures (6)

Fig. 1
Fig. 1

Graphical description of optical measurements that lead to the value of σ2 to use in digital computations of the optimal filter. In this section of the optical correlation plane we see the optical correlation intensity Io and two different philosophies for specifying σo2, the optical variance. The first of these is chosen from off-center secondary peaks, and the second is the average level of off-center energy. By building Io from a maximum-intensity filter, we reduce the light energy for the rest of the plane as much as possible and thus approach the concept for σ2: that it is the background variation not further reducible by choice of filter.

Fig. 2
Fig. 2

In the complex plane of values for the mth filter element Hm, we would select the realizable value that lies highest on the curves that produce equal values of the metric. Tangents of the isometric contours and of the realizable region are parallel there. The situation in which there is a local maximum of the metric relates to the metric with a non-phase-sensitive denominator. (In fact, the isometric contours are circular.)

Fig. 3
Fig. 3

For the metrics with a phase-sensitive denominator, there is no local maximum of the metric anywhere in the complex plane of values for Hm. The gradient is uniform, and we choose the value of Hm with the largest projection in the direction of the gradient.

Fig. 4
Fig. 4

The gradient of the phase-sensitive part of the metric, as drawn in Fig. 2, is the difference between two vectors as indicated here. (See text.)

Fig. 5
Fig. 5

In the section of the Hm plane through the origin and in the direction of the gradient of Λ, plotted here are the metric potentials for the phase-sensitive portion of the denominator, the phase-insensitive portion, and their sum. The balance between the phase-sensitive and the phase-insensitive portions of the metric denominator sets the distance to the ideal value. The phase-insensitive portion moves the ideal location to a finite distance instead of infinity, and the phase sensitive portion moves it to a nonzero location.

Fig. 6
Fig. 6

In the plane of Hm, plots of metric potentials as in Fig. 5. Adding a uniform gradient to the parabolic potential surface merely offsets the location of the apex of the parabola. Isopotential loci remain circular, so the optimum realizable value of Hm is the one closest to the apex of the parabola.

Equations (40)

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J(w)=wtSBwwtSWw,
SW-1SBw=λw.
A exp(jα)=kHkSk,
J(H)=(ETH)*(ETH)+σe2(FTH)*(FTH)+σf2
ETH=A exp(jα)=kEkMk exp[j(k+θk)],
FTH=B exp(jβ)=kFkMk exp[j(ηk+θk)].
J(H)=A2+σe2B2+σf2.
JMm=(B2+σf2) A2Mm-(A2+σe2) B2Mm(B2+σf2)2.
A2=kEk exp(jk)Mk exp(jθk)×kEk exp(-jk)Mk exp(-jθk).
A2AMm=Em exp(jm)exp(jθm)A exp(-jα)+A exp(jα)Em exp(-jm)exp(-jθm)=AEm{exp[j(m+θm-α)]+exp[-j(m+θm-α)]}=2AEm cos(α-m-θm)=2AEm[cos(α-m)cos θm+sin(α-m)sin θm].
B2Mm=2BFm cos(β-ηm-θm)=2BFm[cos(β-ηm)cos θm+sin(β-ηm)sin θm].
(B2+σf2)22JMm=cos θm[A(B2+σf2)Em cos(α-m)-B(A2+σe2)Fm cos(β-ηm)]+sin θm[A(B2+σf2)Em sin(α-m)-B(A2+σe2)Fm sin(β-ηm)]=Rm cos θm+Qm sin θm,
ϕm=arctan 2(-Qm, Rm),
ϕm=arctan 2[Fm sin(β-ηm)-ρEm×sin(α-m), ρEm cos(α-m)-Fm cos(β-ηm)]
ϕm,po=arctan 2[Fm sin(δ-ηm)+ρEm×sin(m),ρEm cos(m)-Fm cos(δ-ηm)]
JPC(H)=A2+B2+σe2+σf2|A2-B2|+σe2+σf2=NumDen,
ϕPC=arctan 2(-QPC, RPC),
RPC=[Den-Num]AEm cos(α-m)+[Den+Num]BFm cos(β-ηm),
QPC=[Den-Num]AEm sin(α-m)+[Den+Num]BFm sin(β-ηm).
ρPC=(Den-Num)A(Den+Num)B=-min{A2, B2}max{A2, B2}+σe2+σf2,
RPC=ρPCEm cos(α-m)+Fm cos(β-ηm),
QPC=ρPCEm sin(α-m)+Fm sin(β-ηm),
J2=A2/σe2B2+σf2,
Jpair=A2+σe2B2+σf2JSDF=i=1n(Ai2+σei2)j=1m(Bj2+σfj2)=:NumSDFDenSDF.
DenSDF2JSDFMm=i Ai2Mm-(NumSDF)(DenSDF)j Bj2Mm=cos θmiAiEmi cos(αi-mi)-(JSDF)jBjFmj cos(βj-ηmj)+sin θmiAiEmi sin(αi-mi)-(JSDF)jBjFmj sin(βj-ηmj)=Rm,SDF cos θm+Qm,SDF sin θm,
ϕm,SDF=arctan 2(-Qm,SDF, Rm,SDF),
D2+σf2=kPkMk2+σf2,
(D2+σf2) A2θm=(A2+σe2) D2θm=0.
A2θm=2AMmEm sin(α-m-θm),
θm=α-m
(D2+σf2) A2Mm=(A2+σe2) D2Mm.
D2Mm=2PmMm,
(D2+σf2)AEm=(A2+σe2)PmMm,
Hm,ideal=D2+σf2A2+σe2A exp(jα) Em exp(-jm)Pm.
Hm=closest_to{Hm,ideal}.
Jhyb(H)=A2+σe2B2+σf2+kPnkMk2=:NumhybDenhyb,
DenMm=B2Mm+2PnmMnm.
Den22JMm=cos θm[Den AEm cos(α-m)-Num BFm cos(β-ηm)]+sin θm[Den AEm sin(α-m)-Num BFm sin(β-ηm)]-Num PnmMm.
JMm=μ cos θm+ζ sin θm-2NumDen2PnmMm.
Mm=Den22Pnm Num(μ cos θm+ζ sin θm).

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