Abstract

We address the problem of small-target detection with a polarimetric imager that provides orthogonal state contrast images. Such active systems allow one to measure the degree of polarization of the light backscattered by purely depolarizing isotropic materials. To be independent of the spatial nonuniformities of the illumination beam, small-target detection on the orthogonal state contrast image must be performed without using the image of backscattered intensity. We thus propose and develop a simple and efficient target detection algorithm based on a nonlinear pointwise transformation of the orthogonal state contrast image followed by a maximum-likelihood algorithm optimal for additive Gaussian perturbations. We demonstrate the efficiency of this suboptimal technique in comparison with the optimal one, which, however, assumes a priori knowledge about the scene that is not available in practice. We illustrate the performance of this approach on both simulated and real polarimetric images.

© 2001 Optical Society of America

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References

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  1. F. A. Sadjadi, ed., Automatic Target Recognition XI, Proc. SPIE4379, (2001).
  2. G. R. Osche, D. S. Young, “Imaging laser radar in the near and far infrared,” Proc. IEEE 84, 103–125 (1996).
    [CrossRef]
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    [CrossRef]
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    [CrossRef]
  5. R. B. Holmes, “Applications of lasers to imaging of distant objects,” in Intense Laser Beams and Applications, W. E. McDermott, ed., Proc. SPIE1871, 306–315 (1993).
    [CrossRef]
  6. B. Johnson, R. Joseph, M. L. Nischan, A. Newbury, J. P. Kerekes, H. T. Barclay, B. C. Willard, J. J. Zayhowski, “Compact active hyperspectral imaging system for the detection of concealed targets,” in Detection and Remediation Technologies for Mines and Minelike Targets IV, A. C. Dubey, J. F. Harvey, J. T. Broach, R. E. Dugan, eds., Proc. SPIE3710, 144–153 (1999).
    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef]
  9. R. A. Chipman, “Polarization diversity active imaging,” in Image Reconstruction and Restoration II, T. J. Schulz, ed., Proc. SPIE3170, 68–73 (1997).
    [CrossRef]
  10. S. Breugnot, Ph. Clémenceau, “Modeling and performances of a polarization active imager at lambda = 806 nm,” in Laser Radar Technology and Applications IV, G. W. Kamerman, C. Werner, eds., Proc. SPIE3707, 449–460 (1999).
    [CrossRef]
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    [CrossRef]
  12. T. S. Ferguson, “Exponential families of distributions,” in Mathematical Statistics, a Decision Theoretic Approach (Academic, New York, 1967), pp. 125–132.
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    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef]
  23. C. J. Oliver, I. Mc Connell, D. Blacknell, R. G. White, “Optimum edge detection in SAR,” in Synthetic Aperture Radar and Passive Microwave Imaging, G. Franceschetti, C. J. Oliver, J. C. Shiue, S. Tajbakhsh, Proc. SPIE2584, 152–163 (1995).
    [CrossRef]
  24. C. Chesnaud, Ph. Refregier, V. Boulet, “Statistical region snake-based segmentation adapted to different physical noise models,” IEEE Trans. Pattern Anal. Mach. Intell. 21, 1145–1157 (1999).
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2000

1999

C. Chesnaud, Ph. Refregier, V. Boulet, “Statistical region snake-based segmentation adapted to different physical noise models,” IEEE Trans. Pattern Anal. Mach. Intell. 21, 1145–1157 (1999).
[CrossRef]

1996

F. Goudail, Ph. Réfrégier, “Optimal and suboptimal detection of a target with random gray levels imbedded in non-overlapping noise,” Opt. Commun. 125, 211–216 (1996).
[CrossRef]

G. R. Osche, D. S. Young, “Imaging laser radar in the near and far infrared,” Proc. IEEE 84, 103–125 (1996).
[CrossRef]

F. Goudail, Ph. Réfrégier, “Optimal detection of a target with random gray levels on a spatially disjoint noise,” Opt. Lett. 21, 495–497 (1996).
[CrossRef] [PubMed]

J. S. Tyo, M. P. Rowe, E. N. Pugh, N. Engheta, “Target detection in optical scattering media by polarization-difference imaging,” Appl. Opt. 35, 1855–1870 (1996).
[CrossRef] [PubMed]

1995

J. L. Pezzaniti, R. A. Chipman, “Mueller matrix imaging polarimetry,” Opt. Eng. 34, 1558–1568 (1995).
[CrossRef]

1993

1992

1991

1981

Barclay, H. T.

B. Johnson, R. Joseph, M. L. Nischan, A. Newbury, J. P. Kerekes, H. T. Barclay, B. C. Willard, J. J. Zayhowski, “Compact active hyperspectral imaging system for the detection of concealed targets,” in Detection and Remediation Technologies for Mines and Minelike Targets IV, A. C. Dubey, J. F. Harvey, J. T. Broach, R. E. Dugan, eds., Proc. SPIE3710, 144–153 (1999).
[CrossRef]

Beraldin, J. A.

D. G. Laurin, J. A. Beraldin, F. Blais, M. Rioux, L. Cournoyer, “Three-dimensional tracking and imaging laser scanner for space operations,” in Laser Radar Technology and Applications, G. W. Kamerman, Ch. Werner, eds., Proc. SPIE3707, 278–289 (1999).
[CrossRef]

Blacknell, D.

C. J. Oliver, I. Mc Connell, D. Blacknell, R. G. White, “Optimum edge detection in SAR,” in Synthetic Aperture Radar and Passive Microwave Imaging, G. Franceschetti, C. J. Oliver, J. C. Shiue, S. Tajbakhsh, Proc. SPIE2584, 152–163 (1995).
[CrossRef]

Blais, F.

D. G. Laurin, J. A. Beraldin, F. Blais, M. Rioux, L. Cournoyer, “Three-dimensional tracking and imaging laser scanner for space operations,” in Laser Radar Technology and Applications, G. W. Kamerman, Ch. Werner, eds., Proc. SPIE3707, 278–289 (1999).
[CrossRef]

Boulet, V.

C. Chesnaud, Ph. Refregier, V. Boulet, “Statistical region snake-based segmentation adapted to different physical noise models,” IEEE Trans. Pattern Anal. Mach. Intell. 21, 1145–1157 (1999).
[CrossRef]

Breugnot, S.

S. Breugnot, Ph. Clémenceau, “Modeling and performances of a polarization active imager at lambda = 806 nm,” in Laser Radar Technology and Applications IV, G. W. Kamerman, C. Werner, eds., Proc. SPIE3707, 449–460 (1999).
[CrossRef]

P. Clémenceau, S. Breugnot, L. Collot, “Polarization diversity imaging,” in Laser Radar Technology and Applica-tions III, G. W. Kamerman, ed., Proc. SPIE3380, 284–291 (1998).
[CrossRef]

Chesnaud, C.

C. Chesnaud, Ph. Refregier, V. Boulet, “Statistical region snake-based segmentation adapted to different physical noise models,” IEEE Trans. Pattern Anal. Mach. Intell. 21, 1145–1157 (1999).
[CrossRef]

Chipman, R. A.

J. L. Pezzaniti, R. A. Chipman, “Mueller matrix imaging polarimetry,” Opt. Eng. 34, 1558–1568 (1995).
[CrossRef]

R. A. Chipman, “Polarization diversity active imaging,” in Image Reconstruction and Restoration II, T. J. Schulz, ed., Proc. SPIE3170, 68–73 (1997).
[CrossRef]

Clémenceau, P.

P. Clémenceau, S. Breugnot, L. Collot, “Polarization diversity imaging,” in Laser Radar Technology and Applica-tions III, G. W. Kamerman, ed., Proc. SPIE3380, 284–291 (1998).
[CrossRef]

Clémenceau, Ph.

S. Breugnot, Ph. Clémenceau, “Modeling and performances of a polarization active imager at lambda = 806 nm,” in Laser Radar Technology and Applications IV, G. W. Kamerman, C. Werner, eds., Proc. SPIE3707, 449–460 (1999).
[CrossRef]

Collot, L.

P. Clémenceau, S. Breugnot, L. Collot, “Polarization diversity imaging,” in Laser Radar Technology and Applica-tions III, G. W. Kamerman, ed., Proc. SPIE3380, 284–291 (1998).
[CrossRef]

Cournoyer, L.

D. G. Laurin, J. A. Beraldin, F. Blais, M. Rioux, L. Cournoyer, “Three-dimensional tracking and imaging laser scanner for space operations,” in Laser Radar Technology and Applications, G. W. Kamerman, Ch. Werner, eds., Proc. SPIE3707, 278–289 (1999).
[CrossRef]

Egan, W. G.

Engheta, N.

Evans, M.

M. Evans, N. Hastings, B. Peacock, “Beta law,” in Statistical Distributions (Wiley, New York, 1993), p. 37.

Ferguson, T. S.

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

Garthwaite, P. H.

P. H. Garthwaite, I. T. Jolliffe, B. Jones, Statistical Inference (Prentice-Hall Europe, London, 1995).

Goodman, J. W.

J. W. Goodman, “Laser speckle and related phenomena,” in Statistical Properties of Laser Speckle Patterns, Vol. 9 of Topics in Applied Physics (Springer-Verlag, Heidelberg, 1975), pp. 9–75.
[CrossRef]

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

Goudail, F.

F. Goudail, Ph. Réfrégier, “Optimal detection of a target with random gray levels on a spatially disjoint noise,” Opt. Lett. 21, 495–497 (1996).
[CrossRef] [PubMed]

F. Goudail, Ph. Réfrégier, “Optimal and suboptimal detection of a target with random gray levels imbedded in non-overlapping noise,” Opt. Commun. 125, 211–216 (1996).
[CrossRef]

Hastings, N.

M. Evans, N. Hastings, B. Peacock, “Beta law,” in Statistical Distributions (Wiley, New York, 1993), p. 37.

Holmes, R. B.

R. B. Holmes, “Applications of lasers to imaging of distant objects,” in Intense Laser Beams and Applications, W. E. McDermott, ed., Proc. SPIE1871, 306–315 (1993).
[CrossRef]

Javidi, B.

Johnson, B.

B. Johnson, R. Joseph, M. L. Nischan, A. Newbury, J. P. Kerekes, H. T. Barclay, B. C. Willard, J. J. Zayhowski, “Compact active hyperspectral imaging system for the detection of concealed targets,” in Detection and Remediation Technologies for Mines and Minelike Targets IV, A. C. Dubey, J. F. Harvey, J. T. Broach, R. E. Dugan, eds., Proc. SPIE3710, 144–153 (1999).
[CrossRef]

Johnson, W. R.

Jolliffe, I. T.

P. H. Garthwaite, I. T. Jolliffe, B. Jones, Statistical Inference (Prentice-Hall Europe, London, 1995).

Jones, B.

P. H. Garthwaite, I. T. Jolliffe, B. Jones, Statistical Inference (Prentice-Hall Europe, London, 1995).

Joseph, R.

B. Johnson, R. Joseph, M. L. Nischan, A. Newbury, J. P. Kerekes, H. T. Barclay, B. C. Willard, J. J. Zayhowski, “Compact active hyperspectral imaging system for the detection of concealed targets,” in Detection and Remediation Technologies for Mines and Minelike Targets IV, A. C. Dubey, J. F. Harvey, J. T. Broach, R. E. Dugan, eds., Proc. SPIE3710, 144–153 (1999).
[CrossRef]

Kerekes, J. P.

B. Johnson, R. Joseph, M. L. Nischan, A. Newbury, J. P. Kerekes, H. T. Barclay, B. C. Willard, J. J. Zayhowski, “Compact active hyperspectral imaging system for the detection of concealed targets,” in Detection and Remediation Technologies for Mines and Minelike Targets IV, A. C. Dubey, J. F. Harvey, J. T. Broach, R. E. Dugan, eds., Proc. SPIE3710, 144–153 (1999).
[CrossRef]

Laurin, D. G.

D. G. Laurin, J. A. Beraldin, F. Blais, M. Rioux, L. Cournoyer, “Three-dimensional tracking and imaging laser scanner for space operations,” in Laser Radar Technology and Applications, G. W. Kamerman, Ch. Werner, eds., Proc. SPIE3707, 278–289 (1999).
[CrossRef]

Mc Connell, I.

C. J. Oliver, I. Mc Connell, D. Blacknell, R. G. White, “Optimum edge detection in SAR,” in Synthetic Aperture Radar and Passive Microwave Imaging, G. Franceschetti, C. J. Oliver, J. C. Shiue, S. Tajbakhsh, Proc. SPIE2584, 152–163 (1995).
[CrossRef]

Newbury, A.

B. Johnson, R. Joseph, M. L. Nischan, A. Newbury, J. P. Kerekes, H. T. Barclay, B. C. Willard, J. J. Zayhowski, “Compact active hyperspectral imaging system for the detection of concealed targets,” in Detection and Remediation Technologies for Mines and Minelike Targets IV, A. C. Dubey, J. F. Harvey, J. T. Broach, R. E. Dugan, eds., Proc. SPIE3710, 144–153 (1999).
[CrossRef]

Nischan, M. L.

B. Johnson, R. Joseph, M. L. Nischan, A. Newbury, J. P. Kerekes, H. T. Barclay, B. C. Willard, J. J. Zayhowski, “Compact active hyperspectral imaging system for the detection of concealed targets,” in Detection and Remediation Technologies for Mines and Minelike Targets IV, A. C. Dubey, J. F. Harvey, J. T. Broach, R. E. Dugan, eds., Proc. SPIE3710, 144–153 (1999).
[CrossRef]

Oliver, C. J.

C. J. Oliver, I. Mc Connell, D. Blacknell, R. G. White, “Optimum edge detection in SAR,” in Synthetic Aperture Radar and Passive Microwave Imaging, G. Franceschetti, C. J. Oliver, J. C. Shiue, S. Tajbakhsh, Proc. SPIE2584, 152–163 (1995).
[CrossRef]

Osche, G. R.

G. R. Osche, D. S. Young, “Imaging laser radar in the near and far infrared,” Proc. IEEE 84, 103–125 (1996).
[CrossRef]

Peacock, B.

M. Evans, N. Hastings, B. Peacock, “Beta law,” in Statistical Distributions (Wiley, New York, 1993), p. 37.

Pezzaniti, J. L.

J. L. Pezzaniti, R. A. Chipman, “Mueller matrix imaging polarimetry,” Opt. Eng. 34, 1558–1568 (1995).
[CrossRef]

Poor, H. V.

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

Pugh, E. N.

Refregier, Ph.

C. Chesnaud, Ph. Refregier, V. Boulet, “Statistical region snake-based segmentation adapted to different physical noise models,” IEEE Trans. Pattern Anal. Mach. Intell. 21, 1145–1157 (1999).
[CrossRef]

Réfrégier, Ph.

Rioux, M.

D. G. Laurin, J. A. Beraldin, F. Blais, M. Rioux, L. Cournoyer, “Three-dimensional tracking and imaging laser scanner for space operations,” in Laser Radar Technology and Applications, G. W. Kamerman, Ch. Werner, eds., Proc. SPIE3707, 278–289 (1999).
[CrossRef]

Rowe, M. P.

Solomon, J. E.

Tajahuerce, E.

Tyo, J. S.

Wang, J.

White, R. G.

C. J. Oliver, I. Mc Connell, D. Blacknell, R. G. White, “Optimum edge detection in SAR,” in Synthetic Aperture Radar and Passive Microwave Imaging, G. Franceschetti, C. J. Oliver, J. C. Shiue, S. Tajbakhsh, Proc. SPIE2584, 152–163 (1995).
[CrossRef]

Whitehead, V. S.

Willard, B. C.

B. Johnson, R. Joseph, M. L. Nischan, A. Newbury, J. P. Kerekes, H. T. Barclay, B. C. Willard, J. J. Zayhowski, “Compact active hyperspectral imaging system for the detection of concealed targets,” in Detection and Remediation Technologies for Mines and Minelike Targets IV, A. C. Dubey, J. F. Harvey, J. T. Broach, R. E. Dugan, eds., Proc. SPIE3710, 144–153 (1999).
[CrossRef]

Willet, P.

Young, D. S.

G. R. Osche, D. S. Young, “Imaging laser radar in the near and far infrared,” Proc. IEEE 84, 103–125 (1996).
[CrossRef]

Zayhowski, J. J.

B. Johnson, R. Joseph, M. L. Nischan, A. Newbury, J. P. Kerekes, H. T. Barclay, B. C. Willard, J. J. Zayhowski, “Compact active hyperspectral imaging system for the detection of concealed targets,” in Detection and Remediation Technologies for Mines and Minelike Targets IV, A. C. Dubey, J. F. Harvey, J. T. Broach, R. E. Dugan, eds., Proc. SPIE3710, 144–153 (1999).
[CrossRef]

Appl. Opt.

IEEE Trans. Pattern Anal. Mach. Intell.

C. Chesnaud, Ph. Refregier, V. Boulet, “Statistical region snake-based segmentation adapted to different physical noise models,” IEEE Trans. Pattern Anal. Mach. Intell. 21, 1145–1157 (1999).
[CrossRef]

Opt. Commun.

F. Goudail, Ph. Réfrégier, “Optimal and suboptimal detection of a target with random gray levels imbedded in non-overlapping noise,” Opt. Commun. 125, 211–216 (1996).
[CrossRef]

Opt. Eng.

J. L. Pezzaniti, R. A. Chipman, “Mueller matrix imaging polarimetry,” Opt. Eng. 34, 1558–1568 (1995).
[CrossRef]

Opt. Lett.

Proc. IEEE

G. R. Osche, D. S. Young, “Imaging laser radar in the near and far infrared,” Proc. IEEE 84, 103–125 (1996).
[CrossRef]

Other

D. G. Laurin, J. A. Beraldin, F. Blais, M. Rioux, L. Cournoyer, “Three-dimensional tracking and imaging laser scanner for space operations,” in Laser Radar Technology and Applications, G. W. Kamerman, Ch. Werner, eds., Proc. SPIE3707, 278–289 (1999).
[CrossRef]

R. B. Holmes, “Applications of lasers to imaging of distant objects,” in Intense Laser Beams and Applications, W. E. McDermott, ed., Proc. SPIE1871, 306–315 (1993).
[CrossRef]

B. Johnson, R. Joseph, M. L. Nischan, A. Newbury, J. P. Kerekes, H. T. Barclay, B. C. Willard, J. J. Zayhowski, “Compact active hyperspectral imaging system for the detection of concealed targets,” in Detection and Remediation Technologies for Mines and Minelike Targets IV, A. C. Dubey, J. F. Harvey, J. T. Broach, R. E. Dugan, eds., Proc. SPIE3710, 144–153 (1999).
[CrossRef]

M. Evans, N. Hastings, B. Peacock, “Beta law,” in Statistical Distributions (Wiley, New York, 1993), p. 37.

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

P. H. Garthwaite, I. T. Jolliffe, B. Jones, Statistical Inference (Prentice-Hall Europe, London, 1995).

R. A. Chipman, “Polarization diversity active imaging,” in Image Reconstruction and Restoration II, T. J. Schulz, ed., Proc. SPIE3170, 68–73 (1997).
[CrossRef]

S. Breugnot, Ph. Clémenceau, “Modeling and performances of a polarization active imager at lambda = 806 nm,” in Laser Radar Technology and Applications IV, G. W. Kamerman, C. Werner, eds., Proc. SPIE3707, 449–460 (1999).
[CrossRef]

J. W. Goodman, “Laser speckle and related phenomena,” in Statistical Properties of Laser Speckle Patterns, Vol. 9 of Topics in Applied Physics (Springer-Verlag, Heidelberg, 1975), pp. 9–75.
[CrossRef]

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

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

P. Clémenceau, S. Breugnot, L. Collot, “Polarization diversity imaging,” in Laser Radar Technology and Applica-tions III, G. W. Kamerman, ed., Proc. SPIE3380, 284–291 (1998).
[CrossRef]

F. A. Sadjadi, ed., Automatic Target Recognition XI, Proc. SPIE4379, (2001).

C. J. Oliver, I. Mc Connell, D. Blacknell, R. G. White, “Optimum edge detection in SAR,” in Synthetic Aperture Radar and Passive Microwave Imaging, G. Franceschetti, C. J. Oliver, J. C. Shiue, S. Tajbakhsh, Proc. SPIE2584, 152–163 (1995).
[CrossRef]

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Figures (8)

Fig. 1
Fig. 1

Principle of the active polarimetric imager.

Fig. 2
Fig. 2

Example of a real polarimetric image, with parallel channel s1 and orthogonal channel s2, and the OSCI in standard representation ρ and in natural representation β (see Subsection 4.B). The white square in the ρ image indicates the position of the target.

Fig. 3
Fig. 3

pdf of the OSCI when the channels are distributed with a Gamma pdf [see Eq. (2)] for different values of the order L and of the parameter u: u=0 (solid line and curves), u=0.5 (dotted curves), u=0.8 (dotted curves with diamonds).

Fig. 4
Fig. 4

ROCs of different types of algorithms on the OSCI ρ and β representations: MLRTOSCI on image ρ (⋄), GLRTρ on image ρ (+), GLRTβ on image β (□) MLRTOSCI (see Fig. 6 below). The pdfs of the channels s1 and s2 are (a) Gamma of order L=1 with SNRdB=6 dB [see Eq. (16)] and (b) Gamma of order L=50 with SNRdB=1.2 dB. Nw=9 pixels, and NF=25 pixels. The curves have been estimated with Monte Carlo simulations with 106 realizations.

Fig. 5
Fig. 5

Centered pdf Φ(x) [see Eq. (18)] of the natural representation of the OSCI (image β) for Gamma-distributed channels of different orders L (solid curves) and pdf of the Gaussian with the same variance (dotted curves with diamonds). Note that for any value of the OSCI parameter ν, Pν(β)(x)=Φ(x-ν). Left column: pdf in natural value, right column: logarithm of the pdf.

Fig. 6
Fig. 6

Principle of the proposed detection algorithm for OSCI images.

Fig. 7
Fig. 7

ROCs of different algorithms applied to Gaussian and OSCI data with the same means and variances as those in Fig. 4: Gaussian MLRT on OSCI data in the natural representation β (□), MLRTOSCI on the OSCI image (+), Gaussian MLRT on Gaussian data (⋄), GLRTβ on image β (△) (see Fig. 6), GLRTβ on Gaussian data (×). (a) Image β corresponding to Gamma laws of order L=1, and (b) image β corresponding to Gamma laws of order L=50. Nw=9 pixels, and NF=25 pixels. The curves have been estimated with Monte Carlo simulations with 106 realizations.

Fig. 8
Fig. 8

Results of different detection algorithms applied to images in Fig. 2. The images in the left column represent the likelihood ratio R(i, j) at each point (i, j) of the scene, and those in the right column represent the maximum value of each column of these planes.

Equations (37)

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

ρ(i, j)=s1(i, j)-s2(i, j)s1(i, j)+s2(s, j).
Pu(ρ)(ρ)=122L-1(2L-1)![(L-1)!]2(1-u2)L(1-ρ2)L-1(1-uρ)2L,
u=(μ1-μ2)/(μ1+μ2).
R(θa, θb)=L1(θa, θb)-L0(θb),
R=maxθa,θbL1(θa, θb)-maxθbL0(θb).
if Rλ,H1 ischosen(targetpresent)if R<λ,H0 ischosen(targetabsent).
MLRTOSCI=LNw log1-ua21-ub2-2L×(m, n)Wlog1-uaρ(m, n)1-ubρ(m, n).
GLRTρ=-Nw log(σˆw2)-Nw¯ log(σˆw¯2)+NF log(σˆF2),
σˆα2=1Nα(i, j)α[s(i, j)-μˆα]2,
μˆα=1Nα(i, j)αs(i, j),
s˜1=s1/μ1,s˜2=s2/μ2,
η=(s˜1-s˜2)/(s˜1+s˜2).
ρ=(u+η)/(1+uη).
α=2 tanh-1(η),β=2 tanh-1(ρ),
ν=2 tanh-1(u),
Pν(β)(β)=Φ(β-ν),
χ=lnμ1aμ2bμ2aμ1b.
χdB=10 log10μ1aμ2bμ2aμ1b.
Φ(x)=122L(2L-1)![(L-1)!]21cosh(x/2)2L,
GLRTβ=A(μˆw-μˆw¯)2,
μˆα=1Nα(i, j)αs(i, j),
Ps(x)=(L/μ)LΓ(L)xL-1 exp(-Lx/μ),
η=s˜1-s˜2s˜1+s˜2,
s˜1=s1/μ1,s˜2=s2/μ2.
ρ=(u+η)/(1+uη),
Pu(ρ)(ρ)=1-u2(1-uρ)Ψρ-u1-uρ.
η=x-1x+1 withx=s˜1/s˜2.
Px(x)=1B(L, L)xL-1(1+x)2L,
B(M, N)=Γ(M+N)Γ(M)Γ(N).
Ψ(η)=dxdηPx(x).
x=1+η1-η,dxdη=21-ρ2.
Ψ(η)=122L-1(2L-1)![(L-1)!]2(1-η2)L-1.
Pu(ρ)(ρ)=(2L-1)!22L-1[(L-1)!]2(1-u2)L(1-ρ2)L-1(1-uρ)2L,
Φ(β)=1|dβ/dρ|P0(ρ)(ρ),
P0(ρ)(ρ)=122L-1(2L-1)![(L-1)!]2(1-ρ2)L-1.
ρ=tanh(β/2),β=ln1+ρ1-ρ,dβdρ=21-ρ2.
Φ(β)=122L(2L-1)![(L-1)!]2(1-ρ2)L.

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