Abstract

We consider target detection in images perturbed with additive noise. We determine the conditions in which polarimetric imaging, which consists of analyzing of the polarization of the light scattered by the scene before forming the image, yields better performance than classical intensity imaging. These results give important information to assess the interest of polarimetric imaging in a given application.

© 2011 Optical Society of America

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References

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  1. L. B. Wolff, “Polarization vision: a new sensory approach to image understanding,” Image Vis. Comput. 15, 81–93 (1997).
    [CrossRef]
  2. J. S. Tyo, D. L. Goldstein, D. B. Chenault, and J. A. Shaw, “Review of passive imaging polarimetry for remote sensing applications,” Appl. Opt. 45, 5453–5469 (2006).
    [CrossRef] [PubMed]
  3. 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]
  4. 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]
  5. 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]
  6. J. M. Bueno, J. Hunter, C. Cookson, M. Kisilak, and M. Campbell, “Improved scanning laser fundus imaging using polarimetry,” J. Opt. Soc. Am. A 24, 1337–1348 (2007).
    [CrossRef]
  7. O. Morel, C. Stolz, F. Meriaudeau, and P. Gorria, “Active lighting applied to three-dimensional reconstruction of specular metallic surfaces by polarization imaging,” Appl. Opt. 45, 4062–4068(2006).
    [CrossRef] [PubMed]
  8. A. Ambirajan and D. C. Look, “Optimium angles for a polarimeter: part I,” Opt. Eng. 34, 1651–1658 (1995).
    [CrossRef]
  9. D. S. Sabatke, M. R. Descour, E. L. Dereniak, W. C. Sweatt, S. A. Kemme, and G. S. Phipps, “Optimization of retardance for a complete Stokes polarimeter,” Opt. Lett. 25, 802–804(2000).
    [CrossRef]
  10. J. S. Tyo, “Noise equalization in Stokes parameter images obtained by use of variable-retardance polarimeters,” Opt. Lett. 25, 1198–1200 (2000).
    [CrossRef]
  11. J. S. Tyo, “Design of optimal polarimeters: maximization of the signal-to-noise ratio and minimization of systematic error,” Appl. Opt. 41, 619–630 (2002).
    [CrossRef] [PubMed]
  12. S. N. Savenkov, “Optimization and structuring of the instrument matrix for polarimetric measurements,” Opt. Eng. 41, 965–972 (2002).
    [CrossRef]
  13. A. A. Swartz, H. A. Yueh, J. A. Kong, L. M. Novak, and R. T. Shin, “Optimal polarizations for achieving maximal contrast in radar images,” J. Geophys. Res. 93, 15252–15260 (1988).
    [CrossRef]
  14. M. Floc’h, G. Le Brun, C. Kieleck, J. Cariou, and J. Lotrian, “Polarimetric considerations to optimize lidar detection of immersed targets,” Pure Appl. Opt. 7, 1327–1340 (1998).
    [CrossRef]
  15. F. Goudail, “Optimization of the contrast in active Stokes images,” Opt. Lett. 34, 121–123 (2009).
    [CrossRef] [PubMed]
  16. J. S. Tyo, S. J. Johnson, Z. Wang, and B. G. Hoover, “Designing partial Mueller matrix polarimeters,” Proc. SPIE 7461, 74610V (2009).
    [CrossRef]
  17. A. B. Kostinski and W. M. Boerner, “On the polarimetric contrast optimization,” IEEE Trans. Antennas Propag. 35, 988–991(1987).
    [CrossRef]
  18. F. Goudail and A. Bénière, “Optimization of the contrast in polarimetric scalar images,” Opt. Lett. 34, 1471–1473 (2009).
    [CrossRef] [PubMed]
  19. F. Goudail, “Comparison of the maximal achievable contrast in scalar, Stokes and Mueller images,” Opt. Lett. 35, 2600–2602(2010).
    [CrossRef] [PubMed]
  20. D. Goldstein, Polarized Light, 2nd ed. (Marcel Dekker, 2003).
    [CrossRef]
  21. P. Réfrégier and F. Goudail, “Invariant polarimetric contrast parameters for coherent light,” J. Opt. Soc. Am. A 19, 1223–1233 (2002).
    [CrossRef]
  22. F. Goudail, P. Réfrégier, and G. Delyon, “Bhattacharyya distance as a contrast parameter for statistical processing of noisy optical images,” J. Opt. Soc. Am. A 21, 1231–1240 (2004).
    [CrossRef]
  23. A. Bénière, F. Goudail, M. Alouini, and D. Dolfi, “Design and experimental validation of a snapshot polarization contrast imager,” Appl. Opt. 48, 5764–5773 (2009).
    [CrossRef] [PubMed]
  24. M. H. Smith, J. B. Woodruff, and J. D. Howe, “Beam wander considerations in imaging polarimetry,” Proc. SPIE 3754, 50–54 (1999).
    [CrossRef]

2010

2009

2008

2007

2006

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]

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]

O. Morel, C. Stolz, F. Meriaudeau, and P. Gorria, “Active lighting applied to three-dimensional reconstruction of specular metallic surfaces by polarization imaging,” Appl. Opt. 45, 4062–4068(2006).
[CrossRef] [PubMed]

J. S. Tyo, D. L. Goldstein, D. B. Chenault, and J. A. Shaw, “Review of passive imaging polarimetry for remote sensing applications,” Appl. Opt. 45, 5453–5469 (2006).
[CrossRef] [PubMed]

2004

2002

2000

1999

M. H. Smith, J. B. Woodruff, and J. D. Howe, “Beam wander considerations in imaging polarimetry,” Proc. SPIE 3754, 50–54 (1999).
[CrossRef]

1998

M. Floc’h, G. Le Brun, C. Kieleck, J. Cariou, and J. Lotrian, “Polarimetric considerations to optimize lidar detection of immersed targets,” Pure Appl. Opt. 7, 1327–1340 (1998).
[CrossRef]

1997

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

1995

A. Ambirajan and D. C. Look, “Optimium angles for a polarimeter: part I,” Opt. Eng. 34, 1651–1658 (1995).
[CrossRef]

1988

A. A. Swartz, H. A. Yueh, J. A. Kong, L. M. Novak, and R. T. Shin, “Optimal polarizations for achieving maximal contrast in radar images,” J. Geophys. Res. 93, 15252–15260 (1988).
[CrossRef]

1987

A. B. Kostinski and W. M. Boerner, “On the polarimetric contrast optimization,” IEEE Trans. Antennas Propag. 35, 988–991(1987).
[CrossRef]

Alouini, M.

A. Bénière, F. Goudail, M. Alouini, and D. Dolfi, “Design and experimental validation of a snapshot polarization contrast imager,” Appl. Opt. 48, 5764–5773 (2009).
[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]

Ambirajan, A.

A. Ambirajan and D. C. Look, “Optimium angles for a polarimeter: part I,” Opt. Eng. 34, 1651–1658 (1995).
[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]

Bénière, A.

Boerner, W. M.

A. B. Kostinski and W. M. Boerner, “On the polarimetric contrast optimization,” IEEE Trans. Antennas Propag. 35, 988–991(1987).
[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]

Bueno, J. M.

Campbell, M.

Cariou, J.

M. Floc’h, G. Le Brun, C. Kieleck, J. Cariou, and J. Lotrian, “Polarimetric considerations to optimize lidar detection of immersed targets,” Pure Appl. Opt. 7, 1327–1340 (1998).
[CrossRef]

Charbois, J. M.

Chenault, D. B.

Cookson, C.

Delyon, G.

Dereniak, E. L.

Descour, M. R.

Devlaminck, V.

Dolfi, D.

A. Bénière, F. Goudail, M. Alouini, and D. Dolfi, “Design and experimental validation of a snapshot polarization contrast imager,” Appl. Opt. 48, 5764–5773 (2009).
[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]

Floc’h, M.

M. Floc’h, G. Le Brun, C. Kieleck, J. Cariou, and J. Lotrian, “Polarimetric considerations to optimize lidar detection of immersed targets,” Pure Appl. Opt. 7, 1327–1340 (1998).
[CrossRef]

Goldstein, D.

D. Goldstein, Polarized Light, 2nd ed. (Marcel Dekker, 2003).
[CrossRef]

Goldstein, D. L.

Gorria, P.

Goudail, F.

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]

Hoover, B. G.

J. S. Tyo, S. J. Johnson, Z. Wang, and B. G. Hoover, “Designing partial Mueller matrix polarimeters,” Proc. SPIE 7461, 74610V (2009).
[CrossRef]

Howe, J. D.

M. H. Smith, J. B. Woodruff, and J. D. Howe, “Beam wander considerations in imaging polarimetry,” Proc. SPIE 3754, 50–54 (1999).
[CrossRef]

Hunter, J.

Johnson, S. J.

J. S. Tyo, S. J. Johnson, Z. Wang, and B. G. Hoover, “Designing partial Mueller matrix polarimeters,” Proc. SPIE 7461, 74610V (2009).
[CrossRef]

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]

Kemme, S. A.

Kieleck, C.

M. Floc’h, G. Le Brun, C. Kieleck, J. Cariou, and J. Lotrian, “Polarimetric considerations to optimize lidar detection of immersed targets,” Pure Appl. Opt. 7, 1327–1340 (1998).
[CrossRef]

Kisilak, M.

Kong, J. A.

A. A. Swartz, H. A. Yueh, J. A. Kong, L. M. Novak, and R. T. Shin, “Optimal polarizations for achieving maximal contrast in radar images,” J. Geophys. Res. 93, 15252–15260 (1988).
[CrossRef]

Kostinski, A. B.

A. B. Kostinski and W. M. Boerner, “On the polarimetric contrast optimization,” IEEE Trans. Antennas Propag. 35, 988–991(1987).
[CrossRef]

Le Brun, G.

M. Floc’h, G. Le Brun, C. Kieleck, J. Cariou, and J. Lotrian, “Polarimetric considerations to optimize lidar detection of immersed targets,” Pure Appl. Opt. 7, 1327–1340 (1998).
[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]

Look, D. C.

A. Ambirajan and D. C. Look, “Optimium angles for a polarimeter: part I,” Opt. Eng. 34, 1651–1658 (1995).
[CrossRef]

Lotrian, J.

M. Floc’h, G. Le Brun, C. Kieleck, J. Cariou, and J. Lotrian, “Polarimetric considerations to optimize lidar detection of immersed targets,” Pure Appl. Opt. 7, 1327–1340 (1998).
[CrossRef]

Meriaudeau, F.

Morel, O.

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]

Novak, L. M.

A. A. Swartz, H. A. Yueh, J. A. Kong, L. M. Novak, and R. T. Shin, “Optimal polarizations for achieving maximal contrast in radar images,” J. Geophys. Res. 93, 15252–15260 (1988).
[CrossRef]

Phipps, G. S.

Réfrégier, P.

Sabatke, D. S.

Savenkov, S. N.

S. N. Savenkov, “Optimization and structuring of the instrument matrix for polarimetric measurements,” Opt. Eng. 41, 965–972 (2002).
[CrossRef]

Shaw, J. A.

Shin, R. T.

A. A. Swartz, H. A. Yueh, J. A. Kong, L. M. Novak, and R. T. Shin, “Optimal polarizations for achieving maximal contrast in radar images,” J. Geophys. Res. 93, 15252–15260 (1988).
[CrossRef]

Smith, M. H.

M. H. Smith, J. B. Woodruff, and J. D. Howe, “Beam wander considerations in imaging polarimetry,” Proc. SPIE 3754, 50–54 (1999).
[CrossRef]

Stolz, C.

Swartz, A. A.

A. A. Swartz, H. A. Yueh, J. A. Kong, L. M. Novak, and R. T. Shin, “Optimal polarizations for achieving maximal contrast in radar images,” J. Geophys. Res. 93, 15252–15260 (1988).
[CrossRef]

Sweatt, W. C.

Terrier, P.

Tyo, J. S.

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]

Wang, Z.

J. S. Tyo, S. J. Johnson, Z. Wang, and B. G. Hoover, “Designing partial Mueller matrix polarimeters,” Proc. SPIE 7461, 74610V (2009).
[CrossRef]

Wolff, L. B.

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

Woodruff, J. B.

M. H. Smith, J. B. Woodruff, and J. D. Howe, “Beam wander considerations in imaging polarimetry,” Proc. SPIE 3754, 50–54 (1999).
[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]

Yueh, H. A.

A. A. Swartz, H. A. Yueh, J. A. Kong, L. M. Novak, and R. T. Shin, “Optimal polarizations for achieving maximal contrast in radar images,” J. Geophys. Res. 93, 15252–15260 (1988).
[CrossRef]

Appl. Opt.

IEEE Trans. Antennas Propag.

A. B. Kostinski and W. M. Boerner, “On the polarimetric contrast optimization,” IEEE Trans. Antennas Propag. 35, 988–991(1987).
[CrossRef]

Image Vis. Comput.

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

J Biomed. Opt.

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. Geophys. Res.

A. A. Swartz, H. A. Yueh, J. A. Kong, L. M. Novak, and R. T. Shin, “Optimal polarizations for achieving maximal contrast in radar images,” J. Geophys. Res. 93, 15252–15260 (1988).
[CrossRef]

J. Opt. Soc. Am. A

Opt. Eng.

A. Ambirajan and D. C. Look, “Optimium angles for a polarimeter: part I,” Opt. Eng. 34, 1651–1658 (1995).
[CrossRef]

S. N. Savenkov, “Optimization and structuring of the instrument matrix for polarimetric measurements,” Opt. Eng. 41, 965–972 (2002).
[CrossRef]

Opt. Lett.

Proc. SPIE

J. S. Tyo, S. J. Johnson, Z. Wang, and B. G. Hoover, “Designing partial Mueller matrix polarimeters,” Proc. SPIE 7461, 74610V (2009).
[CrossRef]

M. H. Smith, J. B. Woodruff, and J. D. Howe, “Beam wander considerations in imaging polarimetry,” Proc. SPIE 3754, 50–54 (1999).
[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]

Pure Appl. Opt.

M. Floc’h, G. Le Brun, C. Kieleck, J. Cariou, and J. Lotrian, “Polarimetric considerations to optimize lidar detection of immersed targets,” Pure Appl. Opt. 7, 1327–1340 (1998).
[CrossRef]

Other

D. Goldstein, Polarized Light, 2nd ed. (Marcel Dekker, 2003).
[CrossRef]

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Figures (3)

Fig. 1
Fig. 1

Polarimetric imaging setups. (a) Single intensity measurement. (b) Two intensity measurements.

Fig. 2
Fig. 2

Variation of the maximal achievable contrast as a function of β when M b = diag ( 1 , 0.3 , 0.3 , 0.3 ) and M a = diag ( 0.95 , β , β , 0.3 ) , for the six configurations considered in Table 1 (a) in the presence of detector noise ( SNR D = 1 ), (b) in the presence of background noise ( SNR B = 1 ).

Fig. 3
Fig. 3

Variation of the maximal achievable contrast as a function of β when M b = diag ( 1 , 0.3 , 0.3 , 0.3 ) and M a is defined in Eq. (45), for the six configurations considered in Table 1(a) in the presence of detector noise ( SNR D = 1 ), (b) in the presence of background noise ( SNR B = 1 ).

Tables (2)

Tables Icon

Table 1 Summary of the Maximal Achievable Contrast for Different Acquisition Setups and Different Types of Noise for the Case of the Unpolarized Light Source

Tables Icon

Table 2 Summary of the Maximal Achievable Contrast for Different Acquisition Setups and Different Types of Noise for the Case of the Polarized Light Source

Equations (45)

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i p = η 4 [ μ T + μ T ] T M p [ λ S + λ S ] + n p ,
C D ( θ ) = λ 2 μ 2 η 2 16 σ 2 × [ ( T + β T ) T D ( S + α S ) ] 2 ,
D = M a M b = [ D 00 m T n D ˜ ] ,
C D ( θ ) = λ 2 μ 2 η 2 I 0 2 16 σ 2 [ ( 1 + α ) ( 1 + β ) D 00 + ( 1 α ) ( 1 + β ) s T m + ( 1 + α ) ( 1 β ) t T n + ( 1 α ) ( 1 β ) t T D ˜ s ] 2 .
SNR D = η 2 I 0 2 σ 2 ,
C D ( θ ) = SNR D 16 × [ p + α q ] 2 ,
p = ( 1 + β ) ( D 00 + s T m ) + ( 1 β ) t T ( n + D ˜ s ) q = ( 1 + β ) ( D 00 s T m ) + ( 1 β ) t T ( n D ˜ s ) .
C D ( θ ) = SNR D 16 × [ p + β q ] 2 ,
p = ( 1 + α ) ( D 00 + t T n ) + ( 1 α ) ( s T m + t T D ˜ s ) , q = ( 1 + α ) ( D 00 t T n ) + ( 1 α ) ( s T m t T D ˜ s ) .
C D ( s , t ) = SNR D 16 × [ D 00 + s T m + t T ( n + D ˜ s ) ] 2 .
max t [ C D ( s , t ) ] = SNR D 16 × [ | D 00 + s T m | + n + D ˜ s ] 2 .
C pp , D max = SNR D 16 × max s [ F ( s ) ] ,
F ( s ) = [ | D 00 + s T m | + n + D ˜ s ] 2 .
C D ( s , t ) = SNR D 4 × [ D 00 + s T m ] 2 .
C pi , D max = SNR D 4 × [ | D 00 | + m ] 2 .
C D ( s , t ) = SNR D 4 × [ D 00 + t T n ] 2 .
C up , D max = SNR D 4 × [ | D 00 | + n ] 2 .
C ui , D max = SNR D × [ D 00 ] 2 .
C D max = max { C pp , D max , C pi , D max , C up , D max , C ui , D max } ,
i p q = μ q η I 0 4 × [ T q ] T M p ( λ S + λ S ) + n p q ,
C 2 , D ( θ ) = ( i a i b ) 2 σ 2 = λ 2 μ 2 η 2 I 0 2 16 σ 2 [ p + q ] 2 C 2 , D ( θ ) = ( i a i b ) 2 σ 2 = λ 2 μ 2 η 2 I 0 2 16 σ 2 [ p q ] 2 ,
p = ( 1 + α ) D 00 + ( 1 α ) s T m q = ( 1 + α ) t T n + ( 1 α ) t T D ˜ s ,
C 2 , D ( θ ) = C 2 , D ( θ ) + C 2 , D ( θ ) .
C 2 , D ( θ ) = SNR D 8 × ( [ p ] 2 + [ q ] 2 ) ,
C 2 , D ( s , t ) = SNR D 8 × [ | D 00 + s T m | 2 + | t T ( n + D ˜ s ) | 2 ] .
max t [ C 2 , D ( s , t ) ] = SNR D 8 × [ | D 00 + s T m | 2 + n + D ˜ s 2 ] .
C p 2 , D max = SNR D 8 × max s [ G ( s ) ] ,
G ( s ) = | D 00 + s T m | 2 + n + D ˜ s 2 .
C 2 , D ( t ) = SNR D 2 × ( | D 00 | 2 + | t T n | 2 ) .
C u 2 , D max = SNR D 2 × [ | D 00 | 2 + n 2 ] .
σ 2 = ( η I n ) / 2 × ( μ + μ ) .
C B ( θ ) = μ η λ 2 I 0 2 8 I n ( 1 + β ) [ ( 1 + α ) ( 1 + β ) D 00 + ( 1 α ) ( 1 + β ) s T m + ( 1 + α ) ( 1 β ) t T n + ( 1 α ) ( 1 β ) t T D ˜ s ] 2 .
C B ( θ ) = SNR B 8 ( 1 + β ) × [ p + β q ] 2 ,
SNR B = η I 0 2 I n .
C pp , B max = SNR B 8 × max s [ F ( s ) ] ,
C pi , B max = SNR B 4 × [ | D 00 | + m ] 2 .
C up , B max = SNR B 2 × [ | D 00 | + n ] 2 .
C ui , B max = SNR B × [ D 00 ] 2 .
C 2 , B ( θ ) = ( i a i b ) 2 σ 2 = λ 2 μ η 2 I 0 2 8 I n [ p + q ] 2 C 2 , B ( θ ) = ( i a i b ) 2 σ 2 = λ 2 μ η 2 I 0 2 8 I n [ p q ] 2 ,
C 2 , B ( s , t ) = SNR B 4 × [ | D 00 + s T m | 2 + | t T ( n + D ˜ s ) | 2 ] .
C p 2 , B max = SNR D 4 × max s [ G ( s ) ] ,
C u 2 , D max = SNR B × [ | D 00 | 2 + n 2 ] .
F ( s ) = ( | D 00 + m T s | + n + D ˜ s ) 2 ,
G ( s ) = | D 00 + m T s | 2 + n + D ˜ s 2 .
M a = [ 0.95 β / 5 0 0 β / 6 β 0 0 0 0 β 0 0 0 0 0.3 ] .

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