Abstract

Active imaging systems that illuminate a scene with polarized light and acquire two images in two orthogonal polarizations yield information about the intensity contrast and the orthogonal state contrast (OSC) in the scene. Both contrasts are relevant for target detection. However, in real systems, the illumination is often spatially or temporally nonuniform. This creates artificial intensity contrasts that can lead to false alarms. We derive generalized likelihood ratio test (GLRT) detectors, for which intensity information is taken into account or not and determine the relevant expressions of the contrast in these two situations. These results are used to determine in which cases considering intensity information in addition to polarimetric information is relevant or not.

© 2009 Optical Society of America

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References

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    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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2008 (5)

M. Anastasiadou, A. De Martino, D. Clement, F. Lige, B. Laude-Boulesteix, N. Quang, J. Dreyfuss, B. Huynh, A. Nazac, L. Schwartz, and H. Cohen, “Polarimetric imaging for the diagnosis of cervical cancer,” Phys. Status Solidi C 5, 991-999 (2008).
[CrossRef]

A. Bénière, F. Goudail, M. Alouini, and D. Dolfi, “Estimation precision of degree of polarization in the presence of signal-dependent and additive Poisson noises,” J. Eur. Opt. Soc. Rapid Publ. 3, 08,002 (2008).
[CrossRef]

M. Alouini, F. Goudail, N. Roux, L. Le Hors, P. Hartemann, S. Breugnot, and D. Dolfi, “Active spectro-polarimetric imaging: signature modeling, imaging demonstrator and target detection,” Eur. Phys. J.: Appl. Phys. 42, 129-139 (2008).
[CrossRef]

P. Terrier, V. Devlaminck, and J. M. Charbois, “Segmentation of rough surfaces using a polarization imaging system,” J. Opt. Soc. Am. A 25, 423-430 (2008).
[CrossRef]

A. Bénière, F. Goudail, M. Alouini, and D. Dolfi, “Degree of polarization estimation in the presence of nonuniform illumination and additive Gaussian noise,” J. Opt. Soc. Am. A 25, 919-929 (2008).
[CrossRef]

2007 (2)

2006 (4)

R. E. Nothdurft and G. Yao, “Effects of turbid media optical properties on object visibility in subsurface polarization imaging,” Appl. Opt. 45, 5532-5541 (2006).
[CrossRef] [PubMed]

P. Réfrégier, M. Roche, and F. Goudail, “Cramer-Rao lower bound for the estimation of the degree of polarization in active coherent imagery at low photon level,” Opt. Lett. 31, 3565-3567 (2006).
[CrossRef] [PubMed]

P. J. Wu, J. Joseph, and T. Walsh, “Stokes polarimetry imaging of rat tail tissue in a turbid medium: degree of linear polarization image maps using incident linearly polarized light,” J. Biomed. Opt. 11, 014031 (2006).
[CrossRef] [PubMed]

M. Alouini, F. Goudail, A. Grisard, J. Bourderionnet, D. Dolfi, I. Baarstad, T. Løke, P. Kaspersen, and X. Normandin, “Active polarimetric and multispectral laboratory demonstrator: contrast enhancement for target detection,” Proc. SPIE 6396, 63960B (2006).
[CrossRef]

2003 (1)

J. C. Ramella-Roman, K. Lee, S. A. Prahl, and S. L. Jacques, “Polarized light imaging with a handheld camera,” Proc. SPIE 5068, 284-293 (2003).
[CrossRef]

2002 (2)

V. L. Gamiz and J. F. Belsher, “Performance limitations of a four-channel polarimeter in the presence of detection noise,” Opt. Eng. 41, 973-980 (2002).
[CrossRef]

F. Goudail and P. Réfrégier, “Target segmentation in active polarimetric images by use of statistical active contours,” Appl. Opt. 41, 874-883 (2002).
[CrossRef] [PubMed]

2001 (2)

2000 (1)

S. Breugnot and P. Clémenceau, “Modeling and performances of a polarization active imager at λ=806 nm,” Opt. Eng. 39, 2681-2688 (2000).
[CrossRef]

1999 (1)

1997 (1)

L. B. Wolff, “Polarization vision: a new sensory approach to image understanding,” Image Vis. Comput. 15, 81-93 (1997).
[CrossRef]

Alouini, M.

A. Bénière, F. Goudail, M. Alouini, and D. Dolfi, “Degree of polarization estimation in the presence of nonuniform illumination and additive Gaussian noise,” J. Opt. Soc. Am. A 25, 919-929 (2008).
[CrossRef]

M. Alouini, F. Goudail, N. Roux, L. Le Hors, P. Hartemann, S. Breugnot, and D. Dolfi, “Active spectro-polarimetric imaging: signature modeling, imaging demonstrator and target detection,” Eur. Phys. J.: Appl. Phys. 42, 129-139 (2008).
[CrossRef]

A. Bénière, F. Goudail, M. Alouini, and D. Dolfi, “Estimation precision of degree of polarization in the presence of signal-dependent and additive Poisson noises,” J. Eur. Opt. Soc. Rapid Publ. 3, 08,002 (2008).
[CrossRef]

M. Alouini, F. Goudail, A. Grisard, J. Bourderionnet, D. Dolfi, I. Baarstad, T. Løke, P. Kaspersen, and X. Normandin, “Active polarimetric and multispectral laboratory demonstrator: contrast enhancement for target detection,” Proc. SPIE 6396, 63960B (2006).
[CrossRef]

Anastasiadou, M.

M. Anastasiadou, A. De Martino, D. Clement, F. Lige, B. Laude-Boulesteix, N. Quang, J. Dreyfuss, B. Huynh, A. Nazac, L. Schwartz, and H. Cohen, “Polarimetric imaging for the diagnosis of cervical cancer,” Phys. Status Solidi C 5, 991-999 (2008).
[CrossRef]

Baarstad, I.

M. Alouini, F. Goudail, A. Grisard, J. Bourderionnet, D. Dolfi, I. Baarstad, T. Løke, P. Kaspersen, and X. Normandin, “Active polarimetric and multispectral laboratory demonstrator: contrast enhancement for target detection,” Proc. SPIE 6396, 63960B (2006).
[CrossRef]

Belsher, J. F.

V. L. Gamiz and J. F. Belsher, “Performance limitations of a four-channel polarimeter in the presence of detection noise,” Opt. Eng. 41, 973-980 (2002).
[CrossRef]

Bénière, A.

A. Bénière, F. Goudail, M. Alouini, and D. Dolfi, “Degree of polarization estimation in the presence of nonuniform illumination and additive Gaussian noise,” J. Opt. Soc. Am. A 25, 919-929 (2008).
[CrossRef]

A. Bénière, F. Goudail, M. Alouini, and D. Dolfi, “Estimation precision of degree of polarization in the presence of signal-dependent and additive Poisson noises,” J. Eur. Opt. Soc. Rapid Publ. 3, 08,002 (2008).
[CrossRef]

Bourderionnet, J.

M. Alouini, F. Goudail, A. Grisard, J. Bourderionnet, D. Dolfi, I. Baarstad, T. Løke, P. Kaspersen, and X. Normandin, “Active polarimetric and multispectral laboratory demonstrator: contrast enhancement for target detection,” Proc. SPIE 6396, 63960B (2006).
[CrossRef]

Breugnot, S.

M. Alouini, F. Goudail, N. Roux, L. Le Hors, P. Hartemann, S. Breugnot, and D. Dolfi, “Active spectro-polarimetric imaging: signature modeling, imaging demonstrator and target detection,” Eur. Phys. J.: Appl. Phys. 42, 129-139 (2008).
[CrossRef]

S. Breugnot and P. Clémenceau, “Modeling and performances of a polarization active imager at λ=806 nm,” Opt. Eng. 39, 2681-2688 (2000).
[CrossRef]

Bueno, J. M.

Campbell, M.

Charbois, J. M.

Clémenceau, P.

S. Breugnot and P. Clémenceau, “Modeling and performances of a polarization active imager at λ=806 nm,” Opt. Eng. 39, 2681-2688 (2000).
[CrossRef]

Clement, D.

M. Anastasiadou, A. De Martino, D. Clement, F. Lige, B. Laude-Boulesteix, N. Quang, J. Dreyfuss, B. Huynh, A. Nazac, L. Schwartz, and H. Cohen, “Polarimetric imaging for the diagnosis of cervical cancer,” Phys. Status Solidi C 5, 991-999 (2008).
[CrossRef]

Cohen, H.

M. Anastasiadou, A. De Martino, D. Clement, F. Lige, B. Laude-Boulesteix, N. Quang, J. Dreyfuss, B. Huynh, A. Nazac, L. Schwartz, and H. Cohen, “Polarimetric imaging for the diagnosis of cervical cancer,” Phys. Status Solidi C 5, 991-999 (2008).
[CrossRef]

Cookson, C.

De Martino, A.

M. Anastasiadou, A. De Martino, D. Clement, F. Lige, B. Laude-Boulesteix, N. Quang, J. Dreyfuss, B. Huynh, A. Nazac, L. Schwartz, and H. Cohen, “Polarimetric imaging for the diagnosis of cervical cancer,” Phys. Status Solidi C 5, 991-999 (2008).
[CrossRef]

Devlaminck, V.

Dolfi, D.

A. Bénière, F. Goudail, M. Alouini, and D. Dolfi, “Estimation precision of degree of polarization in the presence of signal-dependent and additive Poisson noises,” J. Eur. Opt. Soc. Rapid Publ. 3, 08,002 (2008).
[CrossRef]

A. Bénière, F. Goudail, M. Alouini, and D. Dolfi, “Degree of polarization estimation in the presence of nonuniform illumination and additive Gaussian noise,” J. Opt. Soc. Am. A 25, 919-929 (2008).
[CrossRef]

M. Alouini, F. Goudail, N. Roux, L. Le Hors, P. Hartemann, S. Breugnot, and D. Dolfi, “Active spectro-polarimetric imaging: signature modeling, imaging demonstrator and target detection,” Eur. Phys. J.: Appl. Phys. 42, 129-139 (2008).
[CrossRef]

M. Alouini, F. Goudail, A. Grisard, J. Bourderionnet, D. Dolfi, I. Baarstad, T. Løke, P. Kaspersen, and X. Normandin, “Active polarimetric and multispectral laboratory demonstrator: contrast enhancement for target detection,” Proc. SPIE 6396, 63960B (2006).
[CrossRef]

Dreyfuss, J.

M. Anastasiadou, A. De Martino, D. Clement, F. Lige, B. Laude-Boulesteix, N. Quang, J. Dreyfuss, B. Huynh, A. Nazac, L. Schwartz, and H. Cohen, “Polarimetric imaging for the diagnosis of cervical cancer,” Phys. Status Solidi C 5, 991-999 (2008).
[CrossRef]

Fade, J.

Gamiz, V. L.

V. L. Gamiz and J. F. Belsher, “Performance limitations of a four-channel polarimeter in the presence of detection noise,” Opt. Eng. 41, 973-980 (2002).
[CrossRef]

Germain, O.

Goudail, F.

A. Bénière, F. Goudail, M. Alouini, and D. Dolfi, “Estimation precision of degree of polarization in the presence of signal-dependent and additive Poisson noises,” J. Eur. Opt. Soc. Rapid Publ. 3, 08,002 (2008).
[CrossRef]

A. Bénière, F. Goudail, M. Alouini, and D. Dolfi, “Degree of polarization estimation in the presence of nonuniform illumination and additive Gaussian noise,” J. Opt. Soc. Am. A 25, 919-929 (2008).
[CrossRef]

M. Alouini, F. Goudail, N. Roux, L. Le Hors, P. Hartemann, S. Breugnot, and D. Dolfi, “Active spectro-polarimetric imaging: signature modeling, imaging demonstrator and target detection,” Eur. Phys. J.: Appl. Phys. 42, 129-139 (2008).
[CrossRef]

P. Réfrégier, M. Roche, and F. Goudail, “Cramer-Rao lower bound for the estimation of the degree of polarization in active coherent imagery at low photon level,” Opt. Lett. 31, 3565-3567 (2006).
[CrossRef] [PubMed]

M. Alouini, F. Goudail, A. Grisard, J. Bourderionnet, D. Dolfi, I. Baarstad, T. Løke, P. Kaspersen, and X. Normandin, “Active polarimetric and multispectral laboratory demonstrator: contrast enhancement for target detection,” Proc. SPIE 6396, 63960B (2006).
[CrossRef]

F. Goudail and P. Réfrégier, “Target segmentation in active polarimetric images by use of statistical active contours,” Appl. Opt. 41, 874-883 (2002).
[CrossRef] [PubMed]

F. Goudail and P. Réfrégier, “Statistical techniques for target detection in polarization diversity images,” Opt. Lett. 26, 644-646 (2001).
[CrossRef]

Grisard, A.

M. Alouini, F. Goudail, A. Grisard, J. Bourderionnet, D. Dolfi, I. Baarstad, T. Løke, P. Kaspersen, and X. Normandin, “Active polarimetric and multispectral laboratory demonstrator: contrast enhancement for target detection,” Proc. SPIE 6396, 63960B (2006).
[CrossRef]

Hartemann, P.

M. Alouini, F. Goudail, N. Roux, L. Le Hors, P. Hartemann, S. Breugnot, and D. Dolfi, “Active spectro-polarimetric imaging: signature modeling, imaging demonstrator and target detection,” Eur. Phys. J.: Appl. Phys. 42, 129-139 (2008).
[CrossRef]

Hunter, J.

Huynh, B.

M. Anastasiadou, A. De Martino, D. Clement, F. Lige, B. Laude-Boulesteix, N. Quang, J. Dreyfuss, B. Huynh, A. Nazac, L. Schwartz, and H. Cohen, “Polarimetric imaging for the diagnosis of cervical cancer,” Phys. Status Solidi C 5, 991-999 (2008).
[CrossRef]

Jacques, S. L.

J. C. Ramella-Roman, K. Lee, S. A. Prahl, and S. L. Jacques, “Polarized light imaging with a handheld camera,” Proc. SPIE 5068, 284-293 (2003).
[CrossRef]

Jaynes, E. T.

E. T. Jaynes, Probability Theory: The Logic of Science (Cambridge U. Press, 1995).

Joseph, J.

P. J. Wu, J. Joseph, and T. Walsh, “Stokes polarimetry imaging of rat tail tissue in a turbid medium: degree of linear polarization image maps using incident linearly polarized light,” J. Biomed. Opt. 11, 014031 (2006).
[CrossRef] [PubMed]

Kaspersen, P.

M. Alouini, F. Goudail, A. Grisard, J. Bourderionnet, D. Dolfi, I. Baarstad, T. Løke, P. Kaspersen, and X. Normandin, “Active polarimetric and multispectral laboratory demonstrator: contrast enhancement for target detection,” Proc. SPIE 6396, 63960B (2006).
[CrossRef]

Kay, S. M.

S. M. Kay, Fundamentals of Statistical Signal Processing, Vol. I, Estimation Theory (Prentice-Hall, 1993).

Kisilak, M.

Laude-Boulesteix, B.

M. Anastasiadou, A. De Martino, D. Clement, F. Lige, B. Laude-Boulesteix, N. Quang, J. Dreyfuss, B. Huynh, A. Nazac, L. Schwartz, and H. Cohen, “Polarimetric imaging for the diagnosis of cervical cancer,” Phys. Status Solidi C 5, 991-999 (2008).
[CrossRef]

Le Hors, L.

M. Alouini, F. Goudail, N. Roux, L. Le Hors, P. Hartemann, S. Breugnot, and D. Dolfi, “Active spectro-polarimetric imaging: signature modeling, imaging demonstrator and target detection,” Eur. Phys. J.: Appl. Phys. 42, 129-139 (2008).
[CrossRef]

Lee, K.

J. C. Ramella-Roman, K. Lee, S. A. Prahl, and S. L. Jacques, “Polarized light imaging with a handheld camera,” Proc. SPIE 5068, 284-293 (2003).
[CrossRef]

Lige, F.

M. Anastasiadou, A. De Martino, D. Clement, F. Lige, B. Laude-Boulesteix, N. Quang, J. Dreyfuss, B. Huynh, A. Nazac, L. Schwartz, and H. Cohen, “Polarimetric imaging for the diagnosis of cervical cancer,” Phys. Status Solidi C 5, 991-999 (2008).
[CrossRef]

Løke, T.

M. Alouini, F. Goudail, A. Grisard, J. Bourderionnet, D. Dolfi, I. Baarstad, T. Løke, P. Kaspersen, and X. Normandin, “Active polarimetric and multispectral laboratory demonstrator: contrast enhancement for target detection,” Proc. SPIE 6396, 63960B (2006).
[CrossRef]

Nazac, A.

M. Anastasiadou, A. De Martino, D. Clement, F. Lige, B. Laude-Boulesteix, N. Quang, J. Dreyfuss, B. Huynh, A. Nazac, L. Schwartz, and H. Cohen, “Polarimetric imaging for the diagnosis of cervical cancer,” Phys. Status Solidi C 5, 991-999 (2008).
[CrossRef]

Normandin, X.

M. Alouini, F. Goudail, A. Grisard, J. Bourderionnet, D. Dolfi, I. Baarstad, T. Løke, P. Kaspersen, and X. Normandin, “Active polarimetric and multispectral laboratory demonstrator: contrast enhancement for target detection,” Proc. SPIE 6396, 63960B (2006).
[CrossRef]

Nothdurft, R. E.

Papoulis, A.

A. Papoulis, Probability, Random Variables and Stochastic Processes (McGraw-Hill, 1991).

Prahl, S. A.

J. C. Ramella-Roman, K. Lee, S. A. Prahl, and S. L. Jacques, “Polarized light imaging with a handheld camera,” Proc. SPIE 5068, 284-293 (2003).
[CrossRef]

Quang, N.

M. Anastasiadou, A. De Martino, D. Clement, F. Lige, B. Laude-Boulesteix, N. Quang, J. Dreyfuss, B. Huynh, A. Nazac, L. Schwartz, and H. Cohen, “Polarimetric imaging for the diagnosis of cervical cancer,” Phys. Status Solidi C 5, 991-999 (2008).
[CrossRef]

Ramella-Roman, J. C.

J. C. Ramella-Roman, K. Lee, S. A. Prahl, and S. L. Jacques, “Polarized light imaging with a handheld camera,” Proc. SPIE 5068, 284-293 (2003).
[CrossRef]

Réfrégier, P.

Roche, M.

Roux, N.

M. Alouini, F. Goudail, N. Roux, L. Le Hors, P. Hartemann, S. Breugnot, and D. Dolfi, “Active spectro-polarimetric imaging: signature modeling, imaging demonstrator and target detection,” Eur. Phys. J.: Appl. Phys. 42, 129-139 (2008).
[CrossRef]

Schwartz, L.

M. Anastasiadou, A. De Martino, D. Clement, F. Lige, B. Laude-Boulesteix, N. Quang, J. Dreyfuss, B. Huynh, A. Nazac, L. Schwartz, and H. Cohen, “Polarimetric imaging for the diagnosis of cervical cancer,” Phys. Status Solidi C 5, 991-999 (2008).
[CrossRef]

Terrier, P.

Walsh, T.

P. J. Wu, J. Joseph, and T. Walsh, “Stokes polarimetry imaging of rat tail tissue in a turbid medium: degree of linear polarization image maps using incident linearly polarized light,” J. Biomed. Opt. 11, 014031 (2006).
[CrossRef] [PubMed]

Wolff, L. B.

L. B. Wolff, “Polarization vision: a new sensory approach to image understanding,” Image Vis. Comput. 15, 81-93 (1997).
[CrossRef]

Wu, P. J.

P. J. Wu, J. Joseph, and T. Walsh, “Stokes polarimetry imaging of rat tail tissue in a turbid medium: degree of linear polarization image maps using incident linearly polarized light,” J. Biomed. Opt. 11, 014031 (2006).
[CrossRef] [PubMed]

Yao, G.

Appl. Opt. (3)

Eur. Phys. J.: Appl. Phys. (1)

M. Alouini, F. Goudail, N. Roux, L. Le Hors, P. Hartemann, S. Breugnot, and D. Dolfi, “Active spectro-polarimetric imaging: signature modeling, imaging demonstrator and target detection,” Eur. Phys. J.: Appl. Phys. 42, 129-139 (2008).
[CrossRef]

Image Vis. Comput. (1)

L. B. Wolff, “Polarization vision: a new sensory approach to image understanding,” Image Vis. Comput. 15, 81-93 (1997).
[CrossRef]

J. Biomed. Opt. (1)

P. J. Wu, J. Joseph, and T. Walsh, “Stokes polarimetry imaging of rat tail tissue in a turbid medium: degree of linear polarization image maps using incident linearly polarized light,” J. Biomed. Opt. 11, 014031 (2006).
[CrossRef] [PubMed]

J. Eur. Opt. Soc. Rapid Publ. (1)

A. Bénière, F. Goudail, M. Alouini, and D. Dolfi, “Estimation precision of degree of polarization in the presence of signal-dependent and additive Poisson noises,” J. Eur. Opt. Soc. Rapid Publ. 3, 08,002 (2008).
[CrossRef]

J. Opt. Soc. Am. A (4)

Opt. Eng. (2)

V. L. Gamiz and J. F. Belsher, “Performance limitations of a four-channel polarimeter in the presence of detection noise,” Opt. Eng. 41, 973-980 (2002).
[CrossRef]

S. Breugnot and P. Clémenceau, “Modeling and performances of a polarization active imager at λ=806 nm,” Opt. Eng. 39, 2681-2688 (2000).
[CrossRef]

Opt. Lett. (3)

Phys. Status Solidi C (1)

M. Anastasiadou, A. De Martino, D. Clement, F. Lige, B. Laude-Boulesteix, N. Quang, J. Dreyfuss, B. Huynh, A. Nazac, L. Schwartz, and H. Cohen, “Polarimetric imaging for the diagnosis of cervical cancer,” Phys. Status Solidi C 5, 991-999 (2008).
[CrossRef]

Proc. SPIE (2)

J. C. Ramella-Roman, K. Lee, S. A. Prahl, and S. L. Jacques, “Polarized light imaging with a handheld camera,” Proc. SPIE 5068, 284-293 (2003).
[CrossRef]

M. Alouini, F. Goudail, A. Grisard, J. Bourderionnet, D. Dolfi, I. Baarstad, T. Løke, P. Kaspersen, and X. Normandin, “Active polarimetric and multispectral laboratory demonstrator: contrast enhancement for target detection,” Proc. SPIE 6396, 63960B (2006).
[CrossRef]

Other (3)

S. M. Kay, Fundamentals of Statistical Signal Processing, Vol. I, Estimation Theory (Prentice-Hall, 1993).

A. Papoulis, Probability, Random Variables and Stochastic Processes (McGraw-Hill, 1991).

E. T. Jaynes, Probability Theory: The Logic of Science (Cambridge U. Press, 1995).

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Figures (6)

Fig. 1
Fig. 1

ROC curves of the detectors: (a) R Fknown (solid), R Funknown (dashed). (b) R Fknown (solid) R uni (dashed) with three different types of nonuniform illumination in scenario 1. The contrast C Fknown is set to 9, and F ¯ a F ¯ b = 1 (○), F ¯ a F ¯ b = 1.4 (△), and F ¯ a F ¯ b = 4.7 (◻). Other parameters are ( N a , I a , P a ) = ( 10 , 100 , 0.8 ) , ( N b , I b , P b ) = ( 10 , 100 , 0.4 ) , and σ = 15 .

Fig. 2
Fig. 2

Same as in Fig. 1 in scenario 2 which corresponds to ( N a , I a , P a ) = ( 10 , 110 , 0.7 ) , ( N b , I b , P b ) = ( 10 , 90 , 0.5 ) .

Fig. 3
Fig. 3

Location of the targets on the 256 × 256 image, and mask M.

Fig. 4
Fig. 4

Illumination pattern F for l c = 1 , 10, and 100. The values F ¯ a and F ¯ b are computed for each case in the mask superimposed on the images.

Fig. 5
Fig. 5

Detection in scenario 1 corresponding to OSC contrast only. Horizontally: illumination pattern F, intensity image, OSCI, the results of R uni and R Funknown detection. Vertically: those images are computed for l c = 1 , 10, and 100.

Fig. 6
Fig. 6

Detection in scenario 2 corresponding to contrast in OSC and intensity. Horizontally: illumination pattern F, intensity image, OSCI, the results of R uni and R Funknown detection. Vertically: those images are computed for l c = 1 , 10, and 100.

Equations (54)

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X i = F i m X + n i x ,
Y i = F i m Y + n i y ,
P = m X m Y m X + m Y .
m X = I 1 + P 2 , m Y = I 1 P 2 .
R Fknown = 1 2 σ 2 Φ a Φ b Φ a + Φ b [ ( m ̂ X a m ̂ X b ) 2 + ( m ̂ Y a m ̂ Y b ) 2 ] ,
Φ v = i Ω v F i 2 , m ̂ U v = 1 Φ v i Ω v F i U i ,
R uni = 1 2 σ 2 N a N b N a + N b [ ( i Ω a X i N a i Ω b X i N b ) 2 + ( i Ω a Y i N a i Ω b Y i N b ) 2 ] .
C Fknown = Φ a Φ b ( Φ a + Φ b ) [ ( SNR a SNR b ) 2 + ( SNR a P a SNR b P b ) 2 ] ,
C Fknown = Φ a Φ b ( Φ a + Φ b ) SNR 2 ( P a P b ) 2 .
R Funknown = 1 4 σ 2 { D a 2 + W a 2 + D b 2 + W b 2 ( D a + D b ) 2 + ( W a + W b ) 2 } ,
D v = i Ω v ( X i 2 Y i 2 ) , W v = 2 i Ω v X i Y i ,
X i σ = 0 = F i m X ,
Y i σ = 0 = F i m Y .
R Funknown σ = 0 = SNR F a SNR F b 4 × Q [ 1 1 ( 2 Δ P Q ) 2 ] ,
SNR F u = ( Φ u I u ) ( 2 σ ) ,
Δ P = P a P b ,
Q = SNR F a SNR F b ( 1 + P a 2 ) + SNR F b SNR F a ( 1 + P b 2 ) .
C = ( SNR a SNR b ) 2 + ( SNR a P a SNR b P b ) 2 ,
SNR v = N a N b N F ¯ v I v 2 σ , F ¯ v = 1 N v i Ω v F i ,
C = I 2 ( 1 + P 2 ) 2 σ 2 N a N b N ( F ¯ a F ¯ b ) 2 .
C g g ( m , n ) = σ F 2 exp ( m 2 + n 2 l c ) .
l ( χ F ) = arg max m X , m Y [ L ( χ F , m X , m Y ) ] .
l ( χ F , m X , m Y ) = 2 N ln [ 2 π ] 1 2 σ 2 i = 1 N ( X i F i m X ) 2 1 2 σ 2 i = 1 N ( Y i F i m Y ) 2 .
l ( χ F ) = 2 N ln [ 2 π ] 1 2 σ 2 { i = 1 N X i 2 ( i = 1 N F i X i ) 2 i = 1 N F i 2 } 1 2 σ 2 { i = 1 N Y i 2 ( i = 1 N F i Y i ) 2 i = 1 N F i 2 } .
m ̂ U a = m U 0 + n U a Φ a ,
m ̂ U b = m U 0 + n U b Φ b ,
R Fknown 0 = ( Φ b n a X Φ a n b X ) 2 + ( Φ b n a Y Φ a n b Y ) 2 2 σ 2 Φ a Φ b ( Φ a + Φ b ) = b X 2 2 + b Y 2 2 ,
b U = Φ b n a U Φ a n b U σ Φ a Φ b ( Φ a + Φ b )
m ̂ U a = m a + n U a Φ a ,
m ̂ U b = m b + n U b Φ b ,
R Fknown 1 = 1 2 ( C X + b X ) 2 + 1 2 ( C Y + b Y ) 2 ,
C U = Φ a Φ b ( Φ a + Φ b ) m U a m U b σ
C Fknown = C X 2 + C Y 2
= 1 σ 2 Φ a Φ b ( Φ a + Φ b ) × [ ( m X a m X b ) 2 + ( m Y a m Y b ) 2 ] .
l ( χ ) = arg max m X , m Y , F [ l ( χ F , m X , m Y ) ] .
F i ̂ = m X X i + m Y Y i m X 2 + m Y 2 .
l ( χ m X , m Y ) = 1 2 σ 2 i = 1 N ( m y X i m X Y i ) 2 m X 2 + m Y 2 ,
l ( χ ρ ) = 1 2 σ 2 T X W ρ + T Y ρ 2 1 + ρ 2 ,
l ρ = 1 σ 2 W ρ 2 2 D ρ W 2 ( 1 + ρ 2 ) 2 ,
W ρ 2 2 D ρ W = 0 .
ρ = D ± Δ W ,
Δ = D 2 + W 2 .
2 l ρ 2 = 1 2 σ 2 2 W ρ 3 + 6 D ρ 2 + 6 W ρ 2 D ( 1 + ρ 2 ) 4 .
2 l ρ 2 = 2 σ 2 ρ W ( D 2 + W 2 ) ( 1 + ρ 2 ) 4 .
D ± D 2 + W 2 W 2 0 .
ρ ̂ = D + D 2 + W 2 W .
ρ ̂ = r + sgn ( W ) 1 + r 2 ,
l ( χ ρ ̂ ) = 1 4 σ 2 ( 2 T Y Δ 1 + D 2 W 2 + D Δ W 2 ) ,
Δ 1 + D 2 W 2 + D Δ W 2 = W 2 Δ W 2 + D 2 + D Δ .
W 2 Δ W 2 + D 2 + D Δ = Δ D .
l ( χ ρ ̂ ) = 1 4 σ 2 ( T X + T Y ( T X T Y ) 2 + W 2 ) .
l ( χ ρ ̂ ) = 1 4 σ 2 { i = 1 N ( X i 2 + Y i 2 ) ( i = 1 N ( X i 2 Y i 2 ) ) 2 + 4 ( i = 1 N X i Y i ) 2 } .
l 0 ( χ ) = 1 4 σ 2 { ( T X + T Y ) ( T X T Y ) 2 + W 2 } ,
l 1 ( χ ) = l ( χ a ) + l ( χ b ) = 1 4 σ 2 { ( T X + T Y ) ( T X a T Y a ) 2 + W a 2 ( T X b T Y b ) 2 + W b 2 } ,

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