Abstract

We report on the design and exploitation of a real-field laboratory demonstrator combining active polarimetric and multispectral functions. Its building blocks, including a multiwavelength pulsed optical parametric oscillator at the emission side and a hyperspectral imager with polarimetric capability at the reception side, are described. The results obtained with this demonstrator are illustrated on some examples and discussed. In particular it is found that good detection performances rely on joint use of intensity and polarimetric images, with these images exhibiting complementary signatures in most cases.

© 2009 Optical Society of America

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2008

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]

C. R. Howle, D. J. M. Stothard, C. F. Rae, M. Ross, B. S. Truscott, C. D. Dyer, and M. H. Dunn, “Active hyperspectral imaging system for the detection of liquids,” Proc. SPIE 6954, 69540L (2008).
[CrossRef]

K. M. Twietmeyer, R. A. Chipman, A. E. Elsner, Y. Zhao, and D. VanNasdale, “Mueller matrix retinal imager with optimized polarization conditions,” Opt. Express 16, 21339-21354 (2008).
[CrossRef] [PubMed]

2006

2005

2004

B. Laude-Boulesteix, A. De Martino, B. Drévillon, and L. Schwartz, “Mueller polarimetric imaging system with liquid crystals,” Appl. Opt. 43, 2824-2832 (2004).
[CrossRef] [PubMed]

J. Bourderionnet, D. Dolfi, J. P. Huignard, B. Ea-Kim, E. Corbel, D. Purer, A. Poudoulec, and D. Gauden, “Endless fiber-to-fiber polarization controller based on ceramic programmable waveplates,” IEEE Photonics Technol. Lett. 16, 1101-1103 (2004).
[CrossRef]

M. Alouini, F. Goudail, P. Refregier, A. Grisard, E. Lallier, and D. Dolfi, “Multispectral polarimetric imaging with coherent illumination: towards higher image contrast,” Proc. SPIE 5432, 133-144 (2004).
[CrossRef]

2003

2002

C. A. Farlow, D. B. Chenault, J. L. Pezzaniti, K. D. Spradley, and M. G. Gulley, “Imaging polarimeter development and applications,” Proc. SPIE 4481, 118-125 (2002).
[CrossRef]

2001

F. A. Sadjadi and C. S. Chun, “New experiments in the use of infrared polarization in the detection of small targets,” Proc. SPIE 4379, 144-155 (2001).
[CrossRef]

L. Le Hors, P. Hartemann, D. Dolfi, and S. Breugnot, “A phenomenological model of paints for multispectral polarimetric imaging,” Proc. SPIE 4370, 94-105 (2001).
[CrossRef]

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

2000

L. Le Hors, P. Hartemann, and S. Breugnot, “Multispectral polarization active imager in the visible band,” Proc. SPIE 4035, 380-389 (2000).
[CrossRef]

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

1999

J. S. Tyo and T. S. Turner, “Imaging spectropolarimeters for use in visible and infrared remote sensing,” Proc. SPIE 3753, 214-224 (1999).
[CrossRef]

1998

J. L. Denes, M. S. Gottlieb, B. Kaminsky, and D. F. Huber, “Spectropolarimetric imaging for object recognition,” Proc. SPIE 3240, 8-18 (1998).
[CrossRef]

1996

1987

G. H. Haertling, “PLZT electrooptic materials and applications, a review,” Ferroelectrics 75, 25-55 (1987).
[CrossRef]

1931

P. Kubelka and F. Munk, “Ein Beitrag zur Optik der Farbanstriche,” Z. Tech. Phys. 12, 593-601 (1931).

Alouini, M.

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]

F. Goudail, N. Roux, I. Baarstad, T. Løke, P. Kaspersen, M. Alouini, and X. Normandin, “Some practical issues in anomaly detection and exploitation of regions of interest in hyperspectral images,” Appl. Opt. 45, 5223-5236 (2006).
[CrossRef] [PubMed]

M. Alouini, F. Goudail, P. Refregier, A. Grisard, E. Lallier, and D. Dolfi, “Multispectral polarimetric imaging with coherent illumination: towards higher image contrast,” Proc. SPIE 5432, 133-144 (2004).
[CrossRef]

Annen, K.

H. E. Scott, S. H. Jones, F. J. Iannarilli, and K. Annen, “Hyperspectral IR polarimetry with applications in demining and unexploded ordnance detection,” Proc. SPIE 3534, 300-320 (1998).

Baarstad, I.

Bourderionnet, J.

J. Bourderionnet, D. Dolfi, J. P. Huignard, B. Ea-Kim, E. Corbel, D. Purer, A. Poudoulec, and D. Gauden, “Endless fiber-to-fiber polarization controller based on ceramic programmable waveplates,” IEEE Photonics Technol. Lett. 16, 1101-1103 (2004).
[CrossRef]

Brady, D. J.

A. D. Portnoy, M. E. Gehm, and D. J. Brady, “Pushbroom hyperspectral imaging with a coded aperture,” in Frontiers in Optics, OSA Technical Digest (CD) (Optical Society of America, 2006), paper FMB2.

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]

L. Le Hors, P. Hartemann, D. Dolfi, and S. Breugnot, “A phenomenological model of paints for multispectral polarimetric imaging,” Proc. SPIE 4370, 94-105 (2001).
[CrossRef]

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

L. Le Hors, P. Hartemann, and S. Breugnot, “Multispectral polarization active imager in the visible band,” Proc. SPIE 4035, 380-389 (2000).
[CrossRef]

Brosseau, C.

C. Brosseau, Polarized Light (Wiley, 1998).

Buddhiwant, P.

Cairns, B.

Chenault, D. B.

C. A. Farlow, D. B. Chenault, J. L. Pezzaniti, K. D. Spradley, and M. G. Gulley, “Imaging polarimeter development and applications,” Proc. SPIE 4481, 118-125 (2002).
[CrossRef]

Chipman, R. A.

Chowdhary, J.

Chun, C. S.

F. A. Sadjadi and C. S. Chun, “New experiments in the use of infrared polarization in the detection of small targets,” Proc. SPIE 4379, 144-155 (2001).
[CrossRef]

Clémenceau, P.

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

Collet, E.

E. Collet, Polarized Light (Marcel Deckker, 1993).

Corbel, E.

J. Bourderionnet, D. Dolfi, J. P. Huignard, B. Ea-Kim, E. Corbel, D. Purer, A. Poudoulec, and D. Gauden, “Endless fiber-to-fiber polarization controller based on ceramic programmable waveplates,” IEEE Photonics Technol. Lett. 16, 1101-1103 (2004).
[CrossRef]

De Martino, A.

Denes, J. L.

J. L. Denes, M. S. Gottlieb, B. Kaminsky, and D. F. Huber, “Spectropolarimetric imaging for object recognition,” Proc. SPIE 3240, 8-18 (1998).
[CrossRef]

Dolfi, D.

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]

J. Bourderionnet, D. Dolfi, J. P. Huignard, B. Ea-Kim, E. Corbel, D. Purer, A. Poudoulec, and D. Gauden, “Endless fiber-to-fiber polarization controller based on ceramic programmable waveplates,” IEEE Photonics Technol. Lett. 16, 1101-1103 (2004).
[CrossRef]

M. Alouini, F. Goudail, P. Refregier, A. Grisard, E. Lallier, and D. Dolfi, “Multispectral polarimetric imaging with coherent illumination: towards higher image contrast,” Proc. SPIE 5432, 133-144 (2004).
[CrossRef]

L. Le Hors, P. Hartemann, D. Dolfi, and S. Breugnot, “A phenomenological model of paints for multispectral polarimetric imaging,” Proc. SPIE 4370, 94-105 (2001).
[CrossRef]

Drévillon, B.

Duggin, M.

S. R. Loe and M. Duggin, “Hyperspectral imaging polarimeter design and calibration,” Proc. SPIE 4481, 195-205(2001).

Dunn, M. H.

C. R. Howle, D. J. M. Stothard, C. F. Rae, M. Ross, B. S. Truscott, C. D. Dyer, and M. H. Dunn, “Active hyperspectral imaging system for the detection of liquids,” Proc. SPIE 6954, 69540L (2008).
[CrossRef]

Dyer, C. D.

C. R. Howle, D. J. M. Stothard, C. F. Rae, M. Ross, B. S. Truscott, C. D. Dyer, and M. H. Dunn, “Active hyperspectral imaging system for the detection of liquids,” Proc. SPIE 6954, 69540L (2008).
[CrossRef]

Ea-Kim, B.

J. Bourderionnet, D. Dolfi, J. P. Huignard, B. Ea-Kim, E. Corbel, D. Purer, A. Poudoulec, and D. Gauden, “Endless fiber-to-fiber polarization controller based on ceramic programmable waveplates,” IEEE Photonics Technol. Lett. 16, 1101-1103 (2004).
[CrossRef]

Eliès, P.

P. Eliès, B. Le Jeune, M. Floc'h, and J. Lotrian, “Depolarization classification of metallic and dielectric targets,” Proc. SPIE 3059, 174 (1997).

Elsner, A. E.

Farlow, C. A.

C. A. Farlow, D. B. Chenault, J. L. Pezzaniti, K. D. Spradley, and M. G. Gulley, “Imaging polarimeter development and applications,” Proc. SPIE 4481, 118-125 (2002).
[CrossRef]

Floc'h, M.

P. Eliès, B. Le Jeune, M. Floc'h, and J. Lotrian, “Depolarization classification of metallic and dielectric targets,” Proc. SPIE 3059, 174 (1997).

Gauden, D.

J. Bourderionnet, D. Dolfi, J. P. Huignard, B. Ea-Kim, E. Corbel, D. Purer, A. Poudoulec, and D. Gauden, “Endless fiber-to-fiber polarization controller based on ceramic programmable waveplates,” IEEE Photonics Technol. Lett. 16, 1101-1103 (2004).
[CrossRef]

Gehm, M. E.

A. D. Portnoy, M. E. Gehm, and D. J. Brady, “Pushbroom hyperspectral imaging with a coded aperture,” in Frontiers in Optics, OSA Technical Digest (CD) (Optical Society of America, 2006), paper FMB2.

Ghosh, N.

Gottlieb, M. S.

J. L. Denes, M. S. Gottlieb, B. Kaminsky, and D. F. Huber, “Spectropolarimetric imaging for object recognition,” Proc. SPIE 3240, 8-18 (1998).
[CrossRef]

Goudail, F.

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]

F. Goudail, N. Roux, I. Baarstad, T. Løke, P. Kaspersen, M. Alouini, and X. Normandin, “Some practical issues in anomaly detection and exploitation of regions of interest in hyperspectral images,” Appl. Opt. 45, 5223-5236 (2006).
[CrossRef] [PubMed]

M. Alouini, F. Goudail, P. Refregier, A. Grisard, E. Lallier, and D. Dolfi, “Multispectral polarimetric imaging with coherent illumination: towards higher image contrast,” Proc. SPIE 5432, 133-144 (2004).
[CrossRef]

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, P. Refregier, A. Grisard, E. Lallier, and D. Dolfi, “Multispectral polarimetric imaging with coherent illumination: towards higher image contrast,” Proc. SPIE 5432, 133-144 (2004).
[CrossRef]

Gulley, M. G.

C. A. Farlow, D. B. Chenault, J. L. Pezzaniti, K. D. Spradley, and M. G. Gulley, “Imaging polarimeter development and applications,” Proc. SPIE 4481, 118-125 (2002).
[CrossRef]

Gupta, P. K.

Haertling, G. H.

G. H. Haertling, “PLZT electrooptic materials and applications, a review,” Ferroelectrics 75, 25-55 (1987).
[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]

L. Le Hors, P. Hartemann, D. Dolfi, and S. Breugnot, “A phenomenological model of paints for multispectral polarimetric imaging,” Proc. SPIE 4370, 94-105 (2001).
[CrossRef]

L. Le Hors, P. Hartemann, and S. Breugnot, “Multispectral polarization active imager in the visible band,” Proc. SPIE 4035, 380-389 (2000).
[CrossRef]

Hong, S. H.

Howle, C. R.

C. R. Howle, D. J. M. Stothard, C. F. Rae, M. Ross, B. S. Truscott, C. D. Dyer, and M. H. Dunn, “Active hyperspectral imaging system for the detection of liquids,” Proc. SPIE 6954, 69540L (2008).
[CrossRef]

Huard, S.

S. Huard, Polarization of Light (Wiley, 1997).

Huber, D. F.

J. L. Denes, M. S. Gottlieb, B. Kaminsky, and D. F. Huber, “Spectropolarimetric imaging for object recognition,” Proc. SPIE 3240, 8-18 (1998).
[CrossRef]

Huignard, J. P.

J. Bourderionnet, D. Dolfi, J. P. Huignard, B. Ea-Kim, E. Corbel, D. Purer, A. Poudoulec, and D. Gauden, “Endless fiber-to-fiber polarization controller based on ceramic programmable waveplates,” IEEE Photonics Technol. Lett. 16, 1101-1103 (2004).
[CrossRef]

Iannarilli, F. J.

H. E. Scott, S. H. Jones, F. J. Iannarilli, and K. Annen, “Hyperspectral IR polarimetry with applications in demining and unexploded ordnance detection,” Proc. SPIE 3534, 300-320 (1998).

Illing, R.

Javidi, B.

Jones, S. H.

H. E. Scott, S. H. Jones, F. J. Iannarilli, and K. Annen, “Hyperspectral IR polarimetry with applications in demining and unexploded ordnance detection,” Proc. SPIE 3534, 300-320 (1998).

Kaminsky, B.

J. L. Denes, M. S. Gottlieb, B. Kaminsky, and D. F. Huber, “Spectropolarimetric imaging for object recognition,” Proc. SPIE 3240, 8-18 (1998).
[CrossRef]

Kaspersen, P.

Kay, S. M.

S. M. Kay, Fundamentals of Statistical Signal Processing: Detection Theory, Vol. 2 (Prentice-Hall, 1998).

Kim, D.

Kubelka, P.

P. Kubelka and F. Munk, “Ein Beitrag zur Optik der Farbanstriche,” Z. Tech. Phys. 12, 593-601 (1931).

Lallier, E.

M. Alouini, F. Goudail, P. Refregier, A. Grisard, E. Lallier, and D. Dolfi, “Multispectral polarimetric imaging with coherent illumination: towards higher image contrast,” Proc. SPIE 5432, 133-144 (2004).
[CrossRef]

Laude-Boulesteix, B.

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]

L. Le Hors, P. Hartemann, D. Dolfi, and S. Breugnot, “A phenomenological model of paints for multispectral polarimetric imaging,” Proc. SPIE 4370, 94-105 (2001).
[CrossRef]

L. Le Hors, P. Hartemann, and S. Breugnot, “Multispectral polarization active imager in the visible band,” Proc. SPIE 4035, 380-389 (2000).
[CrossRef]

Le Jeune, B.

P. Eliès, B. Le Jeune, M. Floc'h, and J. Lotrian, “Depolarization classification of metallic and dielectric targets,” Proc. SPIE 3059, 174 (1997).

Loe, S. R.

S. R. Loe and M. Duggin, “Hyperspectral imaging polarimeter design and calibration,” Proc. SPIE 4481, 195-205(2001).

Løke, T.

Lotrian, J.

P. Eliès, B. Le Jeune, M. Floc'h, and J. Lotrian, “Depolarization classification of metallic and dielectric targets,” Proc. SPIE 3059, 174 (1997).

Lu, S. Y.

Manhas, S.

Matoba, O.

Miecznik, G.

Munk, F.

P. Kubelka and F. Munk, “Ein Beitrag zur Optik der Farbanstriche,” Z. Tech. Phys. 12, 593-601 (1931).

Normandin, X.

Petroy, S.

Pezzaniti, J. L.

C. A. Farlow, D. B. Chenault, J. L. Pezzaniti, K. D. Spradley, and M. G. Gulley, “Imaging polarimeter development and applications,” Proc. SPIE 4481, 118-125 (2002).
[CrossRef]

Portnoy, A. D.

A. D. Portnoy, M. E. Gehm, and D. J. Brady, “Pushbroom hyperspectral imaging with a coded aperture,” in Frontiers in Optics, OSA Technical Digest (CD) (Optical Society of America, 2006), paper FMB2.

Poudoulec, A.

J. Bourderionnet, D. Dolfi, J. P. Huignard, B. Ea-Kim, E. Corbel, D. Purer, A. Poudoulec, and D. Gauden, “Endless fiber-to-fiber polarization controller based on ceramic programmable waveplates,” IEEE Photonics Technol. Lett. 16, 1101-1103 (2004).
[CrossRef]

Purer, D.

J. Bourderionnet, D. Dolfi, J. P. Huignard, B. Ea-Kim, E. Corbel, D. Purer, A. Poudoulec, and D. Gauden, “Endless fiber-to-fiber polarization controller based on ceramic programmable waveplates,” IEEE Photonics Technol. Lett. 16, 1101-1103 (2004).
[CrossRef]

Rae, C. F.

C. R. Howle, D. J. M. Stothard, C. F. Rae, M. Ross, B. S. Truscott, C. D. Dyer, and M. H. Dunn, “Active hyperspectral imaging system for the detection of liquids,” Proc. SPIE 6954, 69540L (2008).
[CrossRef]

Refregier, P.

M. Alouini, F. Goudail, P. Refregier, A. Grisard, E. Lallier, and D. Dolfi, “Multispectral polarimetric imaging with coherent illumination: towards higher image contrast,” Proc. SPIE 5432, 133-144 (2004).
[CrossRef]

Réfrégier, P.

Ross, M.

C. R. Howle, D. J. M. Stothard, C. F. Rae, M. Ross, B. S. Truscott, C. D. Dyer, and M. H. Dunn, “Active hyperspectral imaging system for the detection of liquids,” Proc. SPIE 6954, 69540L (2008).
[CrossRef]

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]

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Figures (13)

Fig. 1
Fig. 1

Building blocks of the active polarimetric and multispectral imager.

Fig. 2
Fig. 2

Left-hand side: hyperspectral imager operating principle. Parallel and crossed polarization analysis is provided by a PLZT-based polarization rotator followed by a linear polarizer. Right-hand side: illustration of single shot image (on the two two-dimensional detector array) when the scene is uniform. This image corresponds, along the x axis, to one spatial dimension of the scene (a line) imaged at different wavelengths along the y axis. The final image is reconstructed after scanning the scene.

Fig. 3
Fig. 3

Example of INT passive image acquired with the HySpex hyperspectral imager. The RGB image has been obtained by merging three spectral channels: 600 nm (red), 550 nm (green), and 450 nm (blue). The spectrum related to each pixel is contained in the third dimension of the final image.

Fig. 4
Fig. 4

Linear absorption and electro-optic coefficient of the PLZT ceramic, used for polarization switching, as a function of wavelength.

Fig. 5
Fig. 5

(a) Top and (b) side microscope views of the processed PLZT polarization rotator showing the parallel grooves receiving the electrodes. (c), (d) Drawing and photograph, respectively, of the PLZT polarization rotator plate with its contacts.

Fig. 6
Fig. 6

(a) Drawing of the multiwavelength OPO used as illumination source. M1 and M2 are the cavity mirrors. (b) Typical spectral shape of one wavelength channel. The large spectral width of each channel enables us to obtain active images free of speckle.

Fig. 7
Fig. 7

Overview of the active polarimetric and multispectral imager compacted on 60 cm × 60 cm breadboard. The pump laser in gray appears on the right (1). After beam reshaping, this laser feeds a multiwavelength OPO appearing on the left of the view (2). The OPO output beam is polarized and carries different discrete wavelengths. It is reshaped underneath (3) in order to shine the scene with a vertical straight line matching the FOV of the hyperspectral imager. The hyperspectral imager appears in the middle of the view with a black housing (3). The wire entering the imager controls the PLZT-based polarimetric module. The scanning mirror labeled (4) acts for both emission and reception.

Fig. 8
Fig. 8

Representation of the regions T (target) and B (background) involved in the definition of the PSR, which is defined in Eq. (7).

Fig. 9
Fig. 9

(a) View (photograph in the visible domain) of a green painted panel positioned on a foliage background. The panel is difficult to see in the visible domain. (b) Demonstrator acquisition: RGB-coded INT and polarimetric (OSC) images in the near-infrared obtained by merging channels λ 1 S , λ 2 S , and λ 3 S . Here the target appears on both the INT and the OSC images. The OSC image offers a uniform background more suitable for anomaly detection.

Fig. 10
Fig. 10

Same scene as in Fig. 9 observed with the active system in three wavelengths and the corresponding GLRT detection planes. First row: INT image; second row, GLRT detection plane obtained from the INT images; third row, OSC image; and fourth row, GLRT detection plane obtained from the OSC image.

Fig. 11
Fig. 11

Values of the PSR for the three considered wavelengths ( λ 1 S , λ 2 S , and λ 3 S ) in the INT and OSC detection planes.

Fig. 12
Fig. 12

Detection of two identical green plates appearing on a grass background. The first plate (on the left) is lying on the grass, while the second one (on the right) is positioned perpendicular to the floor. The detection algorithm applied to the INT image allows only the second plate to be detected. When the algorithm is applied to the OSC image, the two plates are detected.

Fig. 13
Fig. 13

Decamouflage of a metallic panel hidden by a camouflage net.

Equations (8)

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M = [ 1 0 0 0 0 a 0 0 0 0 a 0 0 0 0 b ] ,
OSC = ( I I ) / ( I + I ) .
I / / ( x , λ ) = I act ( x , λ ) I pas ( x , λ ) , I ( x , λ ) = I act ( x , λ ) I pas ( x , λ ) .
INT ( x , λ ) = I ( x , λ ) + I ( x , λ ) ,
OSC ( x , λ ) = I ( x , λ ) I ( x , λ ) I ( x , λ ) + I ( x , λ ) .
R ( i , j ) = N w N w ¯ N w + N w ¯ ( m ^ w k m ^ w ¯ k ) 2 σ k 2 ,
PSR = 10 log 10 [ ( m ^ T m ^ B ) 2 σ ^ B 2 ] ,
m ^ T = 1 N T ( i , j ) T R ( i , j ) , m ^ B = 1 N B ( i , j ) B R ( i , j ) , σ ^ B 2 = 1 N B ( i , j ) T [ R ( i , j ) m ^ B ] 2 ,

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