Abstract

We address the problem of localizing small targets with random gray levels that appear in random background clutter. We consider the recently proposed maximum-likelihood ratio test (MLRT) algorithm, which scans the observed scene with an estimation window in which the local statistics are estimated. In the presence of a spatially homogeneous background, we show that if the estimation window is a few times larger than the target itself, the MLRT is quasi-equivalent to the optimal maximum-likelihood (ML) algorithm, which uses the whole scene for estimating the background statistics. The MLRT thus constitutes an efficient alternative to the ML algorithm and is more robust in dealing with spatially nonhomogeneous clutter since it utilizes a small estimation window.

© 1999 Optical Society of America

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References

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  1. D. Casasent, D. Weber, and M. Sipe, IEEE Trans. Image Process. 6, 114 (1997).
    [CrossRef]
  2. A. Mahalanobis and V. Kumar, Opt. Eng. 36, 2642 (1997).
    [CrossRef]
  3. J. W. Goodman, in Laser Speckle and Related Phenomena, J. C. Dainty, ed., Vol. 9 of Topics in Applied Physics (Springer-Verlag, Berlin, 1975), pp. 9–75.
  4. B. Javidi, Ph. Réfrégier, and P. Willet, Opt. Lett. 18, 1660 (1993).
    [CrossRef] [PubMed]
  5. A. H. Fazlollahi and B. Javidi, J. Opt. Soc. Am. A 14, 1024 (1997).
    [CrossRef]
  6. H. Sjöberg, F. Goudail, and Ph. Réfrégier, J. Opt. Soc. Am. A 15, 2976 (1998).
    [CrossRef]
  7. M. Evans, N. Hastings, and B. Peacock, Statistical Distributions (Wiley, New York, 1993).
  8. F. Guérault and Ph. Réfrégier, Opt. Lett. 22, 630 (1997).
    [CrossRef]

1998 (1)

1997 (4)

D. Casasent, D. Weber, and M. Sipe, IEEE Trans. Image Process. 6, 114 (1997).
[CrossRef]

A. Mahalanobis and V. Kumar, Opt. Eng. 36, 2642 (1997).
[CrossRef]

A. H. Fazlollahi and B. Javidi, J. Opt. Soc. Am. A 14, 1024 (1997).
[CrossRef]

F. Guérault and Ph. Réfrégier, Opt. Lett. 22, 630 (1997).
[CrossRef]

1993 (1)

Casasent, D.

D. Casasent, D. Weber, and M. Sipe, IEEE Trans. Image Process. 6, 114 (1997).
[CrossRef]

Evans, M.

M. Evans, N. Hastings, and B. Peacock, Statistical Distributions (Wiley, New York, 1993).

Fazlollahi, A. H.

Goodman, J. W.

J. W. Goodman, in Laser Speckle and Related Phenomena, J. C. Dainty, ed., Vol. 9 of Topics in Applied Physics (Springer-Verlag, Berlin, 1975), pp. 9–75.

Goudail, F.

Guérault, F.

Hastings, N.

M. Evans, N. Hastings, and B. Peacock, Statistical Distributions (Wiley, New York, 1993).

Javidi, B.

Kumar, V.

A. Mahalanobis and V. Kumar, Opt. Eng. 36, 2642 (1997).
[CrossRef]

Mahalanobis, A.

A. Mahalanobis and V. Kumar, Opt. Eng. 36, 2642 (1997).
[CrossRef]

Peacock, B.

M. Evans, N. Hastings, and B. Peacock, Statistical Distributions (Wiley, New York, 1993).

Réfrégier, Ph.

Sipe, M.

D. Casasent, D. Weber, and M. Sipe, IEEE Trans. Image Process. 6, 114 (1997).
[CrossRef]

Sjöberg, H.

Weber, D.

D. Casasent, D. Weber, and M. Sipe, IEEE Trans. Image Process. 6, 114 (1997).
[CrossRef]

Willet, P.

IEEE Trans. Image Process. (1)

D. Casasent, D. Weber, and M. Sipe, IEEE Trans. Image Process. 6, 114 (1997).
[CrossRef]

J. Opt. Soc. Am. A (2)

Opt. Eng. (1)

A. Mahalanobis and V. Kumar, Opt. Eng. 36, 2642 (1997).
[CrossRef]

Opt. Lett. (2)

Other (2)

J. W. Goodman, in Laser Speckle and Related Phenomena, J. C. Dainty, ed., Vol. 9 of Topics in Applied Physics (Springer-Verlag, Berlin, 1975), pp. 9–75.

M. Evans, N. Hastings, and B. Peacock, Statistical Distributions (Wiley, New York, 1993).

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Figures (2)

Fig. 1
Fig. 1

(a) Tank-shaped target of 78-pixel size (enlarged). (b) 128×128 pixel scene containing (a) in the center. The target statistics are exponential, and the background is exponential but nonhomogeneous. (c) Result of processing (b) with the ML algorithm for gamma statistics. (d) Result of processing (b) with the MLRT algorithm for gamma statistics. (c), (d) The maximum of each column of the output plane of the corresponding algorithm.

Fig. 2
Fig. 2

Estimation of the probability of correct location with the MLRT algorithm as a function of the width of the square-shaped estimation window. Each graph corresponds to a different statistics: (a) exponential, (b) Poisson. Inset, Gaussian. In each graph the three curves correspond to different target sizes (in pixels): , 3×3; +, 5×5; , 11×11.

Tables (1)

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Table 1 Probability of Obtaining Correct Location with Four Different Versions of the MLRT Algorithm for Two Different Values of the Contrast Ratio C

Equations (4)

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lj=iwjlog Pθt,si-log Pθb,si,
rj=iwjlog Pθt,si+iw¯jlog Pθb,si-iFjlog Pθb,si,
lkj=rkj=1/θb-1/θtiwjsi,
ruj=-Nw logiwjsi-Nw¯ logiw¯jsi+NF logiFjsi,

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