Abstract

We develop a processor for pattern recognition that is optimum in terms of discrimination and is tolerant to variations of the object to be recognized. This optimum processor is found to be adaptive nonlinear joint-transform correlator.

© 1994 Optical Society of America

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References

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  1. A. Vander Lugt, IEEE Trans. Inform. Theory IT-10, 139 (1964).
    [CrossRef]
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    [CrossRef] [PubMed]
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  12. To find the optimal trade-off between the two functions σm2[h] [Eq. (2)] and Es[h] [Eq. (3)] under the constraint of Eq. (1), one needs to minimize8: Ψ[h]=σm2[h]+μEs[h]-λ ∑t=1Nh(t)*r(t). Setting [∂/∂ĥ(k)]Ψ[h] = 0 leads to ĥ(k) = λr̂(k)/[σr2 + μ|ŝ(k)|2], which can be written as (λ/μ)r̂(k)/[σ2 + |ŝ(k)|2] with σ2 = σr2/μ. Since C0 is an arbitrary constraint, it can be chosen such that μ = λ. One then obtains the filter of Eq. (4). It is necessary to emphasize that σ2 is a parameter chosen for optimal balance between noise robustness and discrimination.
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    [CrossRef] [PubMed]

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1966

1964

A. Vander Lugt, IEEE Trans. Inform. Theory IT-10, 139 (1964).
[CrossRef]

Arsenault, H. H.

Casasent, D.

Caulfield, H. J.

Goodman, J. W.

Hassebrook, L.

Horner, J. L.

Hsu, Y. N.

Javidi, B.

Maloney, W. T.

Pasaltis, D.

Réfrégier, Ph.

Vander Lugt, A.

A. Vander Lugt, IEEE Trans. Inform. Theory IT-10, 139 (1964).
[CrossRef]

Vijaya Kumar, B. V. K.

Wang, J.

Weaver, C. S.

Yaroslavsky, L. P.

Appl. Opt.

IEEE Trans. Inform. Theory

A. Vander Lugt, IEEE Trans. Inform. Theory IT-10, 139 (1964).
[CrossRef]

Opt. Lett.

Other

To find the optimal trade-off between the two functions σm2[h] [Eq. (2)] and Es[h] [Eq. (3)] under the constraint of Eq. (1), one needs to minimize8: Ψ[h]=σm2[h]+μEs[h]-λ ∑t=1Nh(t)*r(t). Setting [∂/∂ĥ(k)]Ψ[h] = 0 leads to ĥ(k) = λr̂(k)/[σr2 + μ|ŝ(k)|2], which can be written as (λ/μ)r̂(k)/[σ2 + |ŝ(k)|2] with σ2 = σr2/μ. Since C0 is an arbitrary constraint, it can be chosen such that μ = λ. One then obtains the filter of Eq. (4). It is necessary to emphasize that σ2 is a parameter chosen for optimal balance between noise robustness and discrimination.

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Figures (4)

Fig. 1
Fig. 1

Image of a car used as the reference object for numerical simulations.

Fig. 2
Fig. 2

Input image used for correlation tests.

Fig. 3
Fig. 3

Correlation function obtained with the proposed nonlinear filtering (nonlinear JTC) technique.

Fig. 4
Fig. 4

Correlation function obtained with an optimal trade-off linear filter.

Equations (6)

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t = 1 N h ( t ) * r ( t ) = C 0 .
σ m 2 [ h ] = k h ^ ( k ) 2 σ r 2 ,
E s [ h ] = k h ^ ( k ) 2 s ^ ( k ) 2 ,
h ^ ( k ) = r ^ ( k ) σ 2 + s ^ ( k ) 2 ,
C ^ ( k ) = r ^ ( k ) * s ^ ( k ) σ 2 + s ^ ( k ) 2 .
C ^ ( k ) = r ^ ( k ) * s ^ ( k ) 2 σ 2 + r ^ ( k ) 2 + s ^ ( k ) 2 .

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