Abstract

In many imaging applications, the measured optical images are perturbed by strong fluctuations or noise. This can be the case, for example, for coherent-active or low-flux imagery. In such cases, the noise is not Gaussian additive and the definition of a contrast parameter between two regions in the image is not always a straightforward task. We show that for noncorrelated noise, the Bhattacharyya distance can be an efficient candidate for contrast definition when one uses statistical algorithms for detection, location, or segmentation. We demonstrate with numerical simulations that different images with the same Bhattacharyya distance lead to equivalent values of the performance criterion for a large number of probability laws. The Bhattacharyya distance can thus be used to compare different noisy situations and to simplify the analysis and the specification of optical imaging systems.

© 2004 Optical Society of America

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References

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2003

2002

2001

O. Ruch, Ph. Réfrégier, “Minimal-complexity segmentation with a polygonal snake adapted to different optical noise models,” Opt. Lett. 41, 977–979 (2001).
[CrossRef]

2000

1999

1998

1989

M. Basseville, “Distance measures for signal processing and pattern recognition,” Signal Process. 18, 349–369 (1989).
[CrossRef]

1978

J. Rissanen, “Modeling by shortest data description,” Automatica 14, 465–471 (1978).
[CrossRef]

1977

S. A. Kassam, “Optimal quantization for signal detection,” IEEE Trans. Commun. 25, 479–484 (1977).
[CrossRef]

H. V. Poor, J. B. Thomas, “Application of Ali-Silvey distance measures in the design of general quantizers for binary decision systems,” IEEE Trans. Commun. 25, 893–900 (1977).
[CrossRef]

1965

J. W. Goodman, “Some effects of target-induced scintillation on optical radar performance,” Proc. IEEE 53, 1688–1700 (1965).
[CrossRef]

Abbey, C. K.

Barrett, H. H.

Basseville, M.

M. Basseville, “Distance measures for signal processing and pattern recognition,” Signal Process. 18, 349–369 (1989).
[CrossRef]

Blahut, R. E.

J. A. O’Sullivan, R. E. Blahut, D. L. Snyder, “Information-theoretic image formation,” IEEE Trans. Inf. Theory 44, 2094–2123 (1998).
[CrossRef]

Clarkson, E.

Cover, T. M.

T. M. Cover, J. A. Thomas, Elements of Information Theory (Wiley, New York, 1991).

Ferguson, T. S.

T. S. Ferguson, “Exponential families of distributions,” in Mathematical Statistics, a Decision Theoretic Approach (Academic, New York, 1967), pp. 125–132.

Goodman, J. W.

J. W. Goodman, “Some effects of target-induced scintillation on optical radar performance,” Proc. IEEE 53, 1688–1700 (1965).
[CrossRef]

J. W. Goodman, “The speckle effect in coherence imaging,” in Statistical Optics (Wiley, New York, 1985), pp. 347–356.

Goudail, F.

Guillouet, C.

A. O. Hero, C. Guillouet, “Robust detection of SAR/IR targets via invariance,” in Proceedings of the Sixth IEEE International Conference on Image Processing and its Applications (Institute of Electrical and Electronics Engineers, New York, 1997), pp. 472–475.

Hero, A. O.

A. O. Hero, C. Guillouet, “Robust detection of SAR/IR targets via invariance,” in Proceedings of the Sixth IEEE International Conference on Image Processing and its Applications (Institute of Electrical and Electronics Engineers, New York, 1997), pp. 472–475.

Jain, A.

A. Jain, P. Moulin, M. I. Miller, K. Ramchandran, “Information-theoretic bounds on target recognition performance based on degraded image data,” IEEE Trans. Pattern Anal. Mach. Intell. 24, 1153–1166 (2002).
[CrossRef]

Kassam, S. A.

S. A. Kassam, “Optimal quantization for signal detection,” IEEE Trans. Commun. 25, 479–484 (1977).
[CrossRef]

Kay, S. M.

S. M. Kay, “Statistical decision theory II,” in Fundamentals of Statistical Signal Processing, Vol. II: Detection Theory (Prentice Hall, Upper Saddle River, N.J., 1998), pp. 186–247.

Lanterman, A. D.

A. D. Lanterman, A. J. O’Sullivan, M. I. Miller, “Kullback-Leibler distances for quantifying clutter and models,” Opt. Eng. (Bellingham) 38, 2134–2146 (1999).
[CrossRef]

Miller, M. I.

A. Jain, P. Moulin, M. I. Miller, K. Ramchandran, “Information-theoretic bounds on target recognition performance based on degraded image data,” IEEE Trans. Pattern Anal. Mach. Intell. 24, 1153–1166 (2002).
[CrossRef]

A. D. Lanterman, A. J. O’Sullivan, M. I. Miller, “Kullback-Leibler distances for quantifying clutter and models,” Opt. Eng. (Bellingham) 38, 2134–2146 (1999).
[CrossRef]

Moulin, P.

A. Jain, P. Moulin, M. I. Miller, K. Ramchandran, “Information-theoretic bounds on target recognition performance based on degraded image data,” IEEE Trans. Pattern Anal. Mach. Intell. 24, 1153–1166 (2002).
[CrossRef]

O’Sullivan, A. J.

A. D. Lanterman, A. J. O’Sullivan, M. I. Miller, “Kullback-Leibler distances for quantifying clutter and models,” Opt. Eng. (Bellingham) 38, 2134–2146 (1999).
[CrossRef]

O’Sullivan, J. A.

J. A. O’Sullivan, R. E. Blahut, D. L. Snyder, “Information-theoretic image formation,” IEEE Trans. Inf. Theory 44, 2094–2123 (1998).
[CrossRef]

Pagé, V.

Poor, H. V.

H. V. Poor, J. B. Thomas, “Application of Ali-Silvey distance measures in the design of general quantizers for binary decision systems,” IEEE Trans. Commun. 25, 893–900 (1977).
[CrossRef]

H. V. Poor, “Elements of hypothesis testing,” in An Introduction to Signal Detection and Estimation (Springer-Verlag, New York, 1994), pp. 5–39.

Ramchandran, K.

A. Jain, P. Moulin, M. I. Miller, K. Ramchandran, “Information-theoretic bounds on target recognition performance based on degraded image data,” IEEE Trans. Pattern Anal. Mach. Intell. 24, 1153–1166 (2002).
[CrossRef]

Réfrégier, Ph.

Rissanen, J.

J. Rissanen, “Modeling by shortest data description,” Automatica 14, 465–471 (1978).
[CrossRef]

J. Rissanen, Stochastic Complexity in Statistical Inquiry (World Scientific, Singapore, 1989).

Roux, N.

Ruch, O.

O. Ruch, Ph. Réfrégier, “Minimal-complexity segmentation with a polygonal snake adapted to different optical noise models,” Opt. Lett. 41, 977–979 (2001).
[CrossRef]

Shapiro, J.

Snyder, D. L.

J. A. O’Sullivan, R. E. Blahut, D. L. Snyder, “Information-theoretic image formation,” IEEE Trans. Inf. Theory 44, 2094–2123 (1998).
[CrossRef]

Thomas, J. A.

T. M. Cover, J. A. Thomas, Elements of Information Theory (Wiley, New York, 1991).

Thomas, J. B.

H. V. Poor, J. B. Thomas, “Application of Ali-Silvey distance measures in the design of general quantizers for binary decision systems,” IEEE Trans. Commun. 25, 893–900 (1977).
[CrossRef]

Appl. Opt.

Automatica

J. Rissanen, “Modeling by shortest data description,” Automatica 14, 465–471 (1978).
[CrossRef]

IEEE Trans. Commun.

S. A. Kassam, “Optimal quantization for signal detection,” IEEE Trans. Commun. 25, 479–484 (1977).
[CrossRef]

H. V. Poor, J. B. Thomas, “Application of Ali-Silvey distance measures in the design of general quantizers for binary decision systems,” IEEE Trans. Commun. 25, 893–900 (1977).
[CrossRef]

IEEE Trans. Inf. Theory

J. A. O’Sullivan, R. E. Blahut, D. L. Snyder, “Information-theoretic image formation,” IEEE Trans. Inf. Theory 44, 2094–2123 (1998).
[CrossRef]

IEEE Trans. Pattern Anal. Mach. Intell.

A. Jain, P. Moulin, M. I. Miller, K. Ramchandran, “Information-theoretic bounds on target recognition performance based on degraded image data,” IEEE Trans. Pattern Anal. Mach. Intell. 24, 1153–1166 (2002).
[CrossRef]

J. Opt. Soc. Am. A

Opt. Eng. (Bellingham)

A. D. Lanterman, A. J. O’Sullivan, M. I. Miller, “Kullback-Leibler distances for quantifying clutter and models,” Opt. Eng. (Bellingham) 38, 2134–2146 (1999).
[CrossRef]

Opt. Lett.

Proc. IEEE

J. W. Goodman, “Some effects of target-induced scintillation on optical radar performance,” Proc. IEEE 53, 1688–1700 (1965).
[CrossRef]

Signal Process.

M. Basseville, “Distance measures for signal processing and pattern recognition,” Signal Process. 18, 349–369 (1989).
[CrossRef]

Other

S. M. Kay, “Statistical decision theory II,” in Fundamentals of Statistical Signal Processing, Vol. II: Detection Theory (Prentice Hall, Upper Saddle River, N.J., 1998), pp. 186–247.

T. S. Ferguson, “Exponential families of distributions,” in Mathematical Statistics, a Decision Theoretic Approach (Academic, New York, 1967), pp. 125–132.

J. Rissanen, Stochastic Complexity in Statistical Inquiry (World Scientific, Singapore, 1989).

J. W. Goodman, “The speckle effect in coherence imaging,” in Statistical Optics (Wiley, New York, 1985), pp. 347–356.

A. O. Hero, C. Guillouet, “Robust detection of SAR/IR targets via invariance,” in Proceedings of the Sixth IEEE International Conference on Image Processing and its Applications (Institute of Electrical and Electronics Engineers, New York, 1997), pp. 472–475.

T. M. Cover, J. A. Thomas, Elements of Information Theory (Wiley, New York, 1991).

H. V. Poor, “Elements of hypothesis testing,” in An Introduction to Signal Detection and Estimation (Springer-Verlag, New York, 1994), pp. 5–39.

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Figures (6)

Fig. 1
Fig. 1

AUC obtained with the ideal observer as a function of different statistical distances: Fisher ratio, Kullback divergence, and Bhattacharyya distance. Different types of noise statistics and parameter combinations have been considered. The considered task was detection of a target with Na=4 pixels. Each ROC has been estimated from 104 random experiments.

Fig. 2
Fig. 2

AUC obtained with the generalized-likelihood-ratio test as a function of different statistical distances: Fisher ratio, Kullback divergence, and Bhattacharyya distance. The considered noise configurations are the same as in Fig. 1. The size of the background region is Nb=20.

Fig. 3
Fig. 3

Standard deviation of the location estimate obtained with L˜(τ) as a function of different statistical distances: Fisher ratio, Kullback divergence, and Bhattacharyya distance. Different types of noise statistics and parameter combinations have been considered. The considered task is location of an edge in a one-dimensional signal with Na=50 and Nb=50 pixels. Each ROC has been estimated from 104 random experiments.

Fig. 4
Fig. 4

Examples of segmentation results of the MDL snake for different types of noise PDF. (a) Normal with identical variances, (b) normal with identical means, (c) Gamma with order 5, (d) Poisson. All configurations correspond to B0.9.

Fig. 5
Fig. 5

ANMP obtained with the snake MDL for the segmentation of the airplane in Fig. 4 (size, 575 pixels) as a function of different statistical distances: Fisher ratio, Kullback divergence, and Bhattacharyya distance. Different types of noise statistics and parameter combinations have been considered. Each ANMP has been estimated from 500 random experiments.

Fig. 6
Fig. 6

Left, SAR image of an agricultural region provided by the French Space Agency and delivered by the European Space Agency. Right, three extracts of the image and images with equivalent additive Gaussian noise. B stands for the Bhattacharyya distance.

Tables (3)

Tables Icon

Table 1 Pdf’s Used in This Paper and Their Corresponding Parameters a

Tables Icon

Table 2 Expression of the Fisher Ratio ℱ , the Kullback Divergence K , and the Bhattacharyya Distance ℬ for Different Gray-Level Pdf’s a

Tables Icon

Table 3 Expression of the Criterion J(x, w) Involved in the MDL Snake for Some of the Pdf’s Listed in Table 1 a

Equations (18)

Equations on this page are rendered with MathJax. Learn more.

F=(ma-mb)2σa2+σb2
D[PpaPpb]=Ppa(x)lnPpa(x)Ppb(x)dx,
K=D[PpaPpb]+D[PpbPpa].
C(s)=-lnPpas(x)Ppb1-s(x)dx,
B=-ln[Ppa(x)Ppb(x)]1/2dx.
HPpa(x)Ppb(x)Ppa(x)dx.
L=iWlog[Pp(a)(xi)]-iWlog[Ppb(xi)].
AUC12+12erf[(2NaB)1/2],
L˜=iWlog[Pp˜a(xi)]+iW¯log[Pp˜b(xi)]-iFlog[Pp˜(c)(xi)].
L˜(τ)=iW(τ)log[Pp˜(a)(xi)]+iW¯(τ)log[Pp˜b(xi)].
Δ=J(x, w)+k ln(N).
(λa-λb)28σP2=12 (λa-λb)2,
Pλu(x)=exp(-λu)δ(x)+[1-exp(-λu)]δ(1-x).
BB=-ln([exp(-λa)exp(-λb)]1/2+{[1-exp(-λa)][1-exp(-λb)]}1/2),
(ma-mb)28σS2=L ln12r+1r,
σS2=(ma-mb)28L ln[12(ma/mb+mb/ma)].
SNR=10 log10(ma-mb)2σ2,
SNR=10 log10(8 B).

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