Abstract

Three types of nonlinear transformations of the joint spectrum in nonlinear joint transform correlators (NLJTC’s) are investigated with the purpose of achieving the highest discrimination capability in target location in a cluttered background: logarithmic transformation and the (1/k)th law transformation in combination with the limitation of the signal dynamic range and binarization by thresholding. By computer simulation carried out on a set of test images, it is shown that application of these transformations in NLJTC’s may considerably improve the correlator’s capacity to locate and recognize properly small objects on a cluttered background, provided there is proper selection of nonlinearity parameters. It is also shown that a moderate blur of the joint spectrum in such NLJTC’s before nonlinear transformation is permissible, which simplifies the requirements of correlator optical alignment, the resolution power of correlator electronic components, or both.

© 1997 Optical Society of America

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References

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  1. B. Javidi, “Nonlinear joint power spectrum based optical correlation,” Appl. Opt. 28, 2358–2367 (1989).
    [Crossref] [PubMed]
  2. B. Javidi, C.-J. Kuo, “Joint transform image correlation using a binary spatial light modulator at the Fourier plane,” Appl. Opt. 27, 663–665 (1988).
    [Crossref] [PubMed]
  3. F. T. S. Yu, T. Nagata, “Binary phase-only joint transform correlator,” Microwave Opt. Technol. Lett. 2, 15–17 (1989).
    [Crossref]
  4. W. B. Hahn, D. L. Flannery, “Design elements of binary joint transform correlation and selected optimization techniques,” Opt. Eng. 31, 896–905 (1992).
    [Crossref]
  5. P. Refregier, 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]
  6. H. Inbar, N. Konforti, E. Marom, “Modified joint transform correlator binarized by error diffusion. I. Spatially constant noise-dependent range limit,” Appl. Opt. 33, 4434–4441 (1994).
    [Crossref] [PubMed]
  7. H. Inbar, E. Marom, “Modified joint transform correlator bynarized by error diffusion. II. Spatially variant range limit,” Appl. Opt. 33, 4444–4451 (1994).
    [Crossref] [PubMed]
  8. L. P. Yaroslavsky, “The Theory of Optimal Methods for Localization of Objects in Pictures,” Vol. 32 in Progress in Optics Series (Elsevier, Amsterdam, 1993), pp. 147–201.
  9. B. Javidi, J. Wang, “Optimum filter for detection of a target in nonoverlapping scene noise,” Appl. Opt. 33, 4454–4458 (1994).
    [Crossref] [PubMed]
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    [Crossref]
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    [Crossref] [PubMed]
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    [Crossref]
  13. L. P. Yaroslavsky, “Optical correlators with (-k)th law nonlinearity: Optimal and suboptimal solutions,” Appl. Opt. 34, 3924–3932 (1995).
    [Crossref] [PubMed]
  14. The SNRM is sometimes called the peak-to-clutter ratio (see, for instance, Ref. 4).

1996 (1)

1995 (2)

1994 (4)

1992 (1)

W. B. Hahn, D. L. Flannery, “Design elements of binary joint transform correlation and selected optimization techniques,” Opt. Eng. 31, 896–905 (1992).
[Crossref]

1989 (2)

F. T. S. Yu, T. Nagata, “Binary phase-only joint transform correlator,” Microwave Opt. Technol. Lett. 2, 15–17 (1989).
[Crossref]

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

1988 (1)

1944 (1)

Epperson, J. F.

Flannery, D. L.

W. B. Hahn, D. L. Flannery, “Design elements of binary joint transform correlation and selected optimization techniques,” Opt. Eng. 31, 896–905 (1992).
[Crossref]

Hahn, W. B.

W. B. Hahn, D. L. Flannery, “Design elements of binary joint transform correlation and selected optimization techniques,” Opt. Eng. 31, 896–905 (1992).
[Crossref]

Inbar, H.

Javidi, B.

Konforti, N.

Kuo, C.-J.

Laude, V.

Mahalanobis, A.

Marom, E.

Nagata, T.

F. T. S. Yu, T. Nagata, “Binary phase-only joint transform correlator,” Microwave Opt. Technol. Lett. 2, 15–17 (1989).
[Crossref]

Refregier, P.

Sims, S. R. F.

Song, S.

Vijaya Kumar, B. W. K.

Wang, J.

Yaroslavsky, L. P.

L. P. Yaroslavsky, “Optical correlators with (-k)th law nonlinearity: Optimal and suboptimal solutions,” Appl. Opt. 34, 3924–3932 (1995).
[Crossref] [PubMed]

L. P. Yaroslavsky, “The Theory of Optimal Methods for Localization of Objects in Pictures,” Vol. 32 in Progress in Optics Series (Elsevier, Amsterdam, 1993), pp. 147–201.

Yu, F. T. S.

F. T. S. Yu, T. Nagata, “Binary phase-only joint transform correlator,” Microwave Opt. Technol. Lett. 2, 15–17 (1989).
[Crossref]

Appl. Opt. (7)

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

Microwave Opt. Technol. Lett. (1)

F. T. S. Yu, T. Nagata, “Binary phase-only joint transform correlator,” Microwave Opt. Technol. Lett. 2, 15–17 (1989).
[Crossref]

Opt. Eng. (1)

W. B. Hahn, D. L. Flannery, “Design elements of binary joint transform correlation and selected optimization techniques,” Opt. Eng. 31, 896–905 (1992).
[Crossref]

Opt. Lett. (2)

Other (2)

L. P. Yaroslavsky, “The Theory of Optimal Methods for Localization of Objects in Pictures,” Vol. 32 in Progress in Optics Series (Elsevier, Amsterdam, 1993), pp. 147–201.

The SNRM is sometimes called the peak-to-clutter ratio (see, for instance, Ref. 4).

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Figures (8)

Fig. 1
Fig. 1

Schematic diagram of the JTC with a nonlinear transformation of the joint spectrum.

Fig. 2
Fig. 2

(a) Example of the input image of the JTC. (b) Stereoscopic images used in the experiments.

Fig. 3
Fig. 3

Frequency response of the blur operator (blr) used in the experiments.

Fig. 4
Fig. 4

Plots of the average SNRV and SNRM at the output of the logarithmic JTC versus the limitation threshold.

Fig. 5
Fig. 5

Plots of the SNRV and SNRM at the output of the logarithmic JTC versus the blur parameter (blr) in the Fourier plane.

Fig. 6
Fig. 6

Illustration of the similarity between the logarithmic and the (1/k)th law nonlinearities.

Fig. 7
Fig. 7

Average SNRV (SNRVav) at the output of the JTC with the (1/k)th law nonlinearity versus the nonlinearity index k for (a) the set of test images and (b) the set of stereoscopic images (SNRVstereo).

Fig. 8
Fig. 8

Plot of the SNRV at the output of the BJTC (SNRVbin) versus the fraction of the joint-spectrum energy under the binarization threshold for the set of test images.

Equations (19)

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Hoptf=RO*fAVimsysAVxoBf2,
AVimsysAVx0Bf2IMf2.
bx=wx-x0imx,
bx=imx-rox-x0,
wx-x0=0within the target object1elsewhere,
AVimsysAVx0Bf2AVimsysIMf2Wf2/S,
AVimsysAVx0Bf2AVimsysIMf2+ROf2.
HoptfRO*fAVimsysIMf2Wf2
HoptRO*fAVimsysIMf2+ROf2.
OUTf=IMf·RO*fAVimsysIMf2+ROf2.
OUTfNLJTC=logIMf+ROf2=logIMf2+ROf2+IMf·RO*f+IM*f·ROf.
IMf2+ROf2IMf·ROf.
OUTNLJTCflogIMf2+ROf2+IMf*·ROfIMf2+ROf2+IMf·RO*fIMf2+ROf2.
LDRx=xxlimlimotherwise.
coroutput=IFTCONVLDRlogFTcorinput2, hblr2.
Φ=1/k,
coroutput=IFTLDRFTcorinput2-k2.
LDRhx=0x<lim1otherwise.
coroutput=IFTLDRhFTcorinput22.

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