Abstract

An equation-based triangle orientation discrimination (TOD) performance model was first developed to focus on staring thermal imagers. Specifically, the spatial spectra distribution of a standard triangle pattern is determined. The modulation effects of the overall components of the system on a nonperiodic standard triangle pattern are analyzed and modeled. The matched filter idea is used to characterize quantitatively the spatiotemporal integration of the eye and brain to signal, aliasing, and various noise components over a triangle pattern area, and the perceived signal-to-noise ratio of the staring thermal imager is derived. Further, the TOD performance theoretical model is established. Comparison with the experimental results shows that this theoretical model gives a reasonable prediction of the TOD performance curve for staring thermal imagers. Although more tests and modifications are required, these preliminary results suggest that this model can be developed into a model that predicts the TOD for a wide range of sensors.

© 2005 Optical Society of America

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References

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  1. P. Bijl, J. M. Valeton, “TOD, a new method to characterize electro-optical system performance,” in Infrared Imaging Systems: Design, Analysis, Modeling, and Testing IX, G. C. Holst, ed., Proc. SPIE3377, 182–193 (1998).
  2. P. Bijl, J. M. Valeton, “Triangle orientation discrimination: the alternative to minimum resolvable temperature difference and minimum resolvable contrast,” Opt. Eng. 37, 1976–1983 (1998).
    [CrossRef]
  3. R. G. Driggers, R. Vollmerhausen, C. E. Halford, “Sampled imaging systems,” Opt. Eng. 38, 740–741 (1999).
    [CrossRef]
  4. R. H. Vollmerhausen, R. G. Driggers, Analysis of Sampled Imaging Systems, Vol. TT-39 of SPIE Tutorial Texts (SPIE Press, Bellingham, Wash., 2000), Chap. 4.
    [CrossRef]
  5. P. Bijl, J. M. Valeton, A. N. de Jong, “TOD predicts target acquisition performance for staring and scanning thermal imagers,” in Infrared Imaging Systems: Design, Analysis, Modeling, and Testing XI, G. C. Holst, ed., Proc. SPIE4030, 96–103 (2000).
  6. M. A. Hogervorst, P. Bijl, J. M. Valeton, “Capturing the sampling effects: a TOD sensor performance model,” in Infrared Imaging Systems: Design, Analysis, Modeling, and Testing XII, G. C. Holst, ed., Proc. SPIE4372, 62–73 (2001).
  7. R. G. Driggers, R. H. Vollmerhausen, B. L. O’Kane, “Sampled imaging sensor design using the MTF squeeze model to characterize spurious response,” in Infrared Imaging Systems: Design, Analysis, Modeling, and Testing X, G. C. Holst, ed., Proc. SPIE3701, 61–73 (1999).
  8. W. Wittenstein, “Thermal range model TRM3,” in Infrared Technology and Applications XXIV, B. F. Andresen, M. Strojnik, eds., Proc. SPIE3436, 413–424 (1998).
    [CrossRef]
  9. C. M. Webb, C. E. Halford, “Dynamic minimum resolvable temperature testing for staring array imagers,” Opt. Eng. 38, 845–851 (1999).
    [CrossRef]
  10. C. S. Bendall, “Automated objective minimum resolvable temperature difference,” in Infrared Imaging Systems: Design, Analysis, Modeling, and Testing XI, G. C. Holst, ed., Proc. SPIE4030, 50–59 (2000).
  11. P. Bijl, M. A. Hogervorst, J. M. Valeton, “TOD, NVTherm, and TRM3 model calculations: a comparison,” in Infrared and Passive Millimeter-Wave Imaging Systems: Design, Analysis, Modeling, and Testing, R. Appleby, G. C. Holst, D. W. Wikner, eds., Proc. SPIE4719, 51–62 (2002).
    [CrossRef]
  12. P. Bijl, J. M. Valeton, “Guidelines for accurate TOD measurement,” in Infrared Imaging Systems: Design, Analysis, Modeling, and Testing X, G. C. Holst, ed., Proc. SPIE3701, 14–25 (1999).
  13. P. Bijl, J. M. Valeton, “Validation of the new triangle orientation discrimination method and ACQUIRE model predictions using observer performance data for ship targets,” Opt. Eng. 37, 1984–1994 (1998).
    [CrossRef]
  14. G. C. Holst, “Electro-Optical Imaging System Performance, 2nd ed., Vol. PM-121 of SPIE Press Monographs (SPIE Press, Bellingham, Wash., 2002).
  15. R. G. Driggers, R. H. Vollmerhausen, T. C. Edwards, “Target identification performance of infrared imager models as a function of blur and sampling,” in Infrared Imaging Systems: Design, Analysis, Modeling, and Testing X, G. C. Holst, ed., Proc. SPIE3701, 26–34 (1999).
  16. R. Vollmerhausen, “Incorporating display limitations into right vision performance models,” IRIS Passive Sensors 2, 11–31 (1995).
  17. S. K. Park, R. Hazra, “Aliasing as noise: a quantitative and qualitative assessment,” in Infrared Imaging Systems: Design, Analysis, Modeling, and Testing IV, G. C. Holst, ed., Proc. SPIE1969, 54–65 (1993).
  18. X.-H. Peng, Y.-C. Wu, Y.-L. Chen, “Performance evaluation of sampled imaging system,” J. Infrared Millim. Waves (China) 20, 477–480 (2001).
  19. M. Wittenstein, “Minimum temperature difference perceived—a new approach to assess undersampled thermal imagers,” Opt. Eng. 38, 773–781 (1999).
    [CrossRef]
  20. J. A. D’Agostino, C. M. Webb, “Three-dimensional analysis framework and measurement methodology for imaging system noise,” in Infrared Imaging Systems: Design, Analysis, Modeling, and Testing II, G. C. Holst, ed., Proc. SPIE1488, 110–121 (1991).
  21. G. C. Holst, Testing and Evaluation of Infrared Imaging Systems, 2nd ed., Vol. PM-58 of SPIE Press Monographs (SPIE Press, Bellingham, Wash., 1998), pp. 371–374.

2001

X.-H. Peng, Y.-C. Wu, Y.-L. Chen, “Performance evaluation of sampled imaging system,” J. Infrared Millim. Waves (China) 20, 477–480 (2001).

1999

M. Wittenstein, “Minimum temperature difference perceived—a new approach to assess undersampled thermal imagers,” Opt. Eng. 38, 773–781 (1999).
[CrossRef]

R. G. Driggers, R. Vollmerhausen, C. E. Halford, “Sampled imaging systems,” Opt. Eng. 38, 740–741 (1999).
[CrossRef]

C. M. Webb, C. E. Halford, “Dynamic minimum resolvable temperature testing for staring array imagers,” Opt. Eng. 38, 845–851 (1999).
[CrossRef]

1998

P. Bijl, J. M. Valeton, “Validation of the new triangle orientation discrimination method and ACQUIRE model predictions using observer performance data for ship targets,” Opt. Eng. 37, 1984–1994 (1998).
[CrossRef]

P. Bijl, J. M. Valeton, “Triangle orientation discrimination: the alternative to minimum resolvable temperature difference and minimum resolvable contrast,” Opt. Eng. 37, 1976–1983 (1998).
[CrossRef]

1995

R. Vollmerhausen, “Incorporating display limitations into right vision performance models,” IRIS Passive Sensors 2, 11–31 (1995).

Bendall, C. S.

C. S. Bendall, “Automated objective minimum resolvable temperature difference,” in Infrared Imaging Systems: Design, Analysis, Modeling, and Testing XI, G. C. Holst, ed., Proc. SPIE4030, 50–59 (2000).

Bijl, P.

P. Bijl, J. M. Valeton, “Validation of the new triangle orientation discrimination method and ACQUIRE model predictions using observer performance data for ship targets,” Opt. Eng. 37, 1984–1994 (1998).
[CrossRef]

P. Bijl, J. M. Valeton, “Triangle orientation discrimination: the alternative to minimum resolvable temperature difference and minimum resolvable contrast,” Opt. Eng. 37, 1976–1983 (1998).
[CrossRef]

P. Bijl, J. M. Valeton, “TOD, a new method to characterize electro-optical system performance,” in Infrared Imaging Systems: Design, Analysis, Modeling, and Testing IX, G. C. Holst, ed., Proc. SPIE3377, 182–193 (1998).

P. Bijl, J. M. Valeton, A. N. de Jong, “TOD predicts target acquisition performance for staring and scanning thermal imagers,” in Infrared Imaging Systems: Design, Analysis, Modeling, and Testing XI, G. C. Holst, ed., Proc. SPIE4030, 96–103 (2000).

M. A. Hogervorst, P. Bijl, J. M. Valeton, “Capturing the sampling effects: a TOD sensor performance model,” in Infrared Imaging Systems: Design, Analysis, Modeling, and Testing XII, G. C. Holst, ed., Proc. SPIE4372, 62–73 (2001).

P. Bijl, J. M. Valeton, “Guidelines for accurate TOD measurement,” in Infrared Imaging Systems: Design, Analysis, Modeling, and Testing X, G. C. Holst, ed., Proc. SPIE3701, 14–25 (1999).

P. Bijl, M. A. Hogervorst, J. M. Valeton, “TOD, NVTherm, and TRM3 model calculations: a comparison,” in Infrared and Passive Millimeter-Wave Imaging Systems: Design, Analysis, Modeling, and Testing, R. Appleby, G. C. Holst, D. W. Wikner, eds., Proc. SPIE4719, 51–62 (2002).
[CrossRef]

Chen, Y.-L.

X.-H. Peng, Y.-C. Wu, Y.-L. Chen, “Performance evaluation of sampled imaging system,” J. Infrared Millim. Waves (China) 20, 477–480 (2001).

D’Agostino, J. A.

J. A. D’Agostino, C. M. Webb, “Three-dimensional analysis framework and measurement methodology for imaging system noise,” in Infrared Imaging Systems: Design, Analysis, Modeling, and Testing II, G. C. Holst, ed., Proc. SPIE1488, 110–121 (1991).

de Jong, A. N.

P. Bijl, J. M. Valeton, A. N. de Jong, “TOD predicts target acquisition performance for staring and scanning thermal imagers,” in Infrared Imaging Systems: Design, Analysis, Modeling, and Testing XI, G. C. Holst, ed., Proc. SPIE4030, 96–103 (2000).

Driggers, R. G.

R. G. Driggers, R. Vollmerhausen, C. E. Halford, “Sampled imaging systems,” Opt. Eng. 38, 740–741 (1999).
[CrossRef]

R. G. Driggers, R. H. Vollmerhausen, B. L. O’Kane, “Sampled imaging sensor design using the MTF squeeze model to characterize spurious response,” in Infrared Imaging Systems: Design, Analysis, Modeling, and Testing X, G. C. Holst, ed., Proc. SPIE3701, 61–73 (1999).

R. H. Vollmerhausen, R. G. Driggers, Analysis of Sampled Imaging Systems, Vol. TT-39 of SPIE Tutorial Texts (SPIE Press, Bellingham, Wash., 2000), Chap. 4.
[CrossRef]

R. G. Driggers, R. H. Vollmerhausen, T. C. Edwards, “Target identification performance of infrared imager models as a function of blur and sampling,” in Infrared Imaging Systems: Design, Analysis, Modeling, and Testing X, G. C. Holst, ed., Proc. SPIE3701, 26–34 (1999).

Edwards, T. C.

R. G. Driggers, R. H. Vollmerhausen, T. C. Edwards, “Target identification performance of infrared imager models as a function of blur and sampling,” in Infrared Imaging Systems: Design, Analysis, Modeling, and Testing X, G. C. Holst, ed., Proc. SPIE3701, 26–34 (1999).

Halford, C. E.

C. M. Webb, C. E. Halford, “Dynamic minimum resolvable temperature testing for staring array imagers,” Opt. Eng. 38, 845–851 (1999).
[CrossRef]

R. G. Driggers, R. Vollmerhausen, C. E. Halford, “Sampled imaging systems,” Opt. Eng. 38, 740–741 (1999).
[CrossRef]

Hazra, R.

S. K. Park, R. Hazra, “Aliasing as noise: a quantitative and qualitative assessment,” in Infrared Imaging Systems: Design, Analysis, Modeling, and Testing IV, G. C. Holst, ed., Proc. SPIE1969, 54–65 (1993).

Hogervorst, M. A.

M. A. Hogervorst, P. Bijl, J. M. Valeton, “Capturing the sampling effects: a TOD sensor performance model,” in Infrared Imaging Systems: Design, Analysis, Modeling, and Testing XII, G. C. Holst, ed., Proc. SPIE4372, 62–73 (2001).

P. Bijl, M. A. Hogervorst, J. M. Valeton, “TOD, NVTherm, and TRM3 model calculations: a comparison,” in Infrared and Passive Millimeter-Wave Imaging Systems: Design, Analysis, Modeling, and Testing, R. Appleby, G. C. Holst, D. W. Wikner, eds., Proc. SPIE4719, 51–62 (2002).
[CrossRef]

Holst, G. C.

G. C. Holst, “Electro-Optical Imaging System Performance, 2nd ed., Vol. PM-121 of SPIE Press Monographs (SPIE Press, Bellingham, Wash., 2002).

G. C. Holst, Testing and Evaluation of Infrared Imaging Systems, 2nd ed., Vol. PM-58 of SPIE Press Monographs (SPIE Press, Bellingham, Wash., 1998), pp. 371–374.

O’Kane, B. L.

R. G. Driggers, R. H. Vollmerhausen, B. L. O’Kane, “Sampled imaging sensor design using the MTF squeeze model to characterize spurious response,” in Infrared Imaging Systems: Design, Analysis, Modeling, and Testing X, G. C. Holst, ed., Proc. SPIE3701, 61–73 (1999).

Park, S. K.

S. K. Park, R. Hazra, “Aliasing as noise: a quantitative and qualitative assessment,” in Infrared Imaging Systems: Design, Analysis, Modeling, and Testing IV, G. C. Holst, ed., Proc. SPIE1969, 54–65 (1993).

Peng, X.-H.

X.-H. Peng, Y.-C. Wu, Y.-L. Chen, “Performance evaluation of sampled imaging system,” J. Infrared Millim. Waves (China) 20, 477–480 (2001).

Valeton, J. M.

P. Bijl, J. M. Valeton, “Validation of the new triangle orientation discrimination method and ACQUIRE model predictions using observer performance data for ship targets,” Opt. Eng. 37, 1984–1994 (1998).
[CrossRef]

P. Bijl, J. M. Valeton, “Triangle orientation discrimination: the alternative to minimum resolvable temperature difference and minimum resolvable contrast,” Opt. Eng. 37, 1976–1983 (1998).
[CrossRef]

P. Bijl, J. M. Valeton, “TOD, a new method to characterize electro-optical system performance,” in Infrared Imaging Systems: Design, Analysis, Modeling, and Testing IX, G. C. Holst, ed., Proc. SPIE3377, 182–193 (1998).

P. Bijl, J. M. Valeton, A. N. de Jong, “TOD predicts target acquisition performance for staring and scanning thermal imagers,” in Infrared Imaging Systems: Design, Analysis, Modeling, and Testing XI, G. C. Holst, ed., Proc. SPIE4030, 96–103 (2000).

M. A. Hogervorst, P. Bijl, J. M. Valeton, “Capturing the sampling effects: a TOD sensor performance model,” in Infrared Imaging Systems: Design, Analysis, Modeling, and Testing XII, G. C. Holst, ed., Proc. SPIE4372, 62–73 (2001).

P. Bijl, J. M. Valeton, “Guidelines for accurate TOD measurement,” in Infrared Imaging Systems: Design, Analysis, Modeling, and Testing X, G. C. Holst, ed., Proc. SPIE3701, 14–25 (1999).

P. Bijl, M. A. Hogervorst, J. M. Valeton, “TOD, NVTherm, and TRM3 model calculations: a comparison,” in Infrared and Passive Millimeter-Wave Imaging Systems: Design, Analysis, Modeling, and Testing, R. Appleby, G. C. Holst, D. W. Wikner, eds., Proc. SPIE4719, 51–62 (2002).
[CrossRef]

Vollmerhausen, R.

R. G. Driggers, R. Vollmerhausen, C. E. Halford, “Sampled imaging systems,” Opt. Eng. 38, 740–741 (1999).
[CrossRef]

R. Vollmerhausen, “Incorporating display limitations into right vision performance models,” IRIS Passive Sensors 2, 11–31 (1995).

Vollmerhausen, R. H.

R. H. Vollmerhausen, R. G. Driggers, Analysis of Sampled Imaging Systems, Vol. TT-39 of SPIE Tutorial Texts (SPIE Press, Bellingham, Wash., 2000), Chap. 4.
[CrossRef]

R. G. Driggers, R. H. Vollmerhausen, B. L. O’Kane, “Sampled imaging sensor design using the MTF squeeze model to characterize spurious response,” in Infrared Imaging Systems: Design, Analysis, Modeling, and Testing X, G. C. Holst, ed., Proc. SPIE3701, 61–73 (1999).

R. G. Driggers, R. H. Vollmerhausen, T. C. Edwards, “Target identification performance of infrared imager models as a function of blur and sampling,” in Infrared Imaging Systems: Design, Analysis, Modeling, and Testing X, G. C. Holst, ed., Proc. SPIE3701, 26–34 (1999).

Webb, C. M.

C. M. Webb, C. E. Halford, “Dynamic minimum resolvable temperature testing for staring array imagers,” Opt. Eng. 38, 845–851 (1999).
[CrossRef]

J. A. D’Agostino, C. M. Webb, “Three-dimensional analysis framework and measurement methodology for imaging system noise,” in Infrared Imaging Systems: Design, Analysis, Modeling, and Testing II, G. C. Holst, ed., Proc. SPIE1488, 110–121 (1991).

Wittenstein, M.

M. Wittenstein, “Minimum temperature difference perceived—a new approach to assess undersampled thermal imagers,” Opt. Eng. 38, 773–781 (1999).
[CrossRef]

Wittenstein, W.

W. Wittenstein, “Thermal range model TRM3,” in Infrared Technology and Applications XXIV, B. F. Andresen, M. Strojnik, eds., Proc. SPIE3436, 413–424 (1998).
[CrossRef]

Wu, Y.-C.

X.-H. Peng, Y.-C. Wu, Y.-L. Chen, “Performance evaluation of sampled imaging system,” J. Infrared Millim. Waves (China) 20, 477–480 (2001).

IRIS Passive Sensors

R. Vollmerhausen, “Incorporating display limitations into right vision performance models,” IRIS Passive Sensors 2, 11–31 (1995).

J. Infrared Millim. Waves (China)

X.-H. Peng, Y.-C. Wu, Y.-L. Chen, “Performance evaluation of sampled imaging system,” J. Infrared Millim. Waves (China) 20, 477–480 (2001).

Opt. Eng.

M. Wittenstein, “Minimum temperature difference perceived—a new approach to assess undersampled thermal imagers,” Opt. Eng. 38, 773–781 (1999).
[CrossRef]

P. Bijl, J. M. Valeton, “Validation of the new triangle orientation discrimination method and ACQUIRE model predictions using observer performance data for ship targets,” Opt. Eng. 37, 1984–1994 (1998).
[CrossRef]

P. Bijl, J. M. Valeton, “Triangle orientation discrimination: the alternative to minimum resolvable temperature difference and minimum resolvable contrast,” Opt. Eng. 37, 1976–1983 (1998).
[CrossRef]

R. G. Driggers, R. Vollmerhausen, C. E. Halford, “Sampled imaging systems,” Opt. Eng. 38, 740–741 (1999).
[CrossRef]

C. M. Webb, C. E. Halford, “Dynamic minimum resolvable temperature testing for staring array imagers,” Opt. Eng. 38, 845–851 (1999).
[CrossRef]

Other

C. S. Bendall, “Automated objective minimum resolvable temperature difference,” in Infrared Imaging Systems: Design, Analysis, Modeling, and Testing XI, G. C. Holst, ed., Proc. SPIE4030, 50–59 (2000).

P. Bijl, M. A. Hogervorst, J. M. Valeton, “TOD, NVTherm, and TRM3 model calculations: a comparison,” in Infrared and Passive Millimeter-Wave Imaging Systems: Design, Analysis, Modeling, and Testing, R. Appleby, G. C. Holst, D. W. Wikner, eds., Proc. SPIE4719, 51–62 (2002).
[CrossRef]

P. Bijl, J. M. Valeton, “Guidelines for accurate TOD measurement,” in Infrared Imaging Systems: Design, Analysis, Modeling, and Testing X, G. C. Holst, ed., Proc. SPIE3701, 14–25 (1999).

R. H. Vollmerhausen, R. G. Driggers, Analysis of Sampled Imaging Systems, Vol. TT-39 of SPIE Tutorial Texts (SPIE Press, Bellingham, Wash., 2000), Chap. 4.
[CrossRef]

P. Bijl, J. M. Valeton, A. N. de Jong, “TOD predicts target acquisition performance for staring and scanning thermal imagers,” in Infrared Imaging Systems: Design, Analysis, Modeling, and Testing XI, G. C. Holst, ed., Proc. SPIE4030, 96–103 (2000).

M. A. Hogervorst, P. Bijl, J. M. Valeton, “Capturing the sampling effects: a TOD sensor performance model,” in Infrared Imaging Systems: Design, Analysis, Modeling, and Testing XII, G. C. Holst, ed., Proc. SPIE4372, 62–73 (2001).

R. G. Driggers, R. H. Vollmerhausen, B. L. O’Kane, “Sampled imaging sensor design using the MTF squeeze model to characterize spurious response,” in Infrared Imaging Systems: Design, Analysis, Modeling, and Testing X, G. C. Holst, ed., Proc. SPIE3701, 61–73 (1999).

W. Wittenstein, “Thermal range model TRM3,” in Infrared Technology and Applications XXIV, B. F. Andresen, M. Strojnik, eds., Proc. SPIE3436, 413–424 (1998).
[CrossRef]

G. C. Holst, “Electro-Optical Imaging System Performance, 2nd ed., Vol. PM-121 of SPIE Press Monographs (SPIE Press, Bellingham, Wash., 2002).

R. G. Driggers, R. H. Vollmerhausen, T. C. Edwards, “Target identification performance of infrared imager models as a function of blur and sampling,” in Infrared Imaging Systems: Design, Analysis, Modeling, and Testing X, G. C. Holst, ed., Proc. SPIE3701, 26–34 (1999).

S. K. Park, R. Hazra, “Aliasing as noise: a quantitative and qualitative assessment,” in Infrared Imaging Systems: Design, Analysis, Modeling, and Testing IV, G. C. Holst, ed., Proc. SPIE1969, 54–65 (1993).

J. A. D’Agostino, C. M. Webb, “Three-dimensional analysis framework and measurement methodology for imaging system noise,” in Infrared Imaging Systems: Design, Analysis, Modeling, and Testing II, G. C. Holst, ed., Proc. SPIE1488, 110–121 (1991).

G. C. Holst, Testing and Evaluation of Infrared Imaging Systems, 2nd ed., Vol. PM-58 of SPIE Press Monographs (SPIE Press, Bellingham, Wash., 1998), pp. 371–374.

P. Bijl, J. M. Valeton, “TOD, a new method to characterize electro-optical system performance,” in Infrared Imaging Systems: Design, Analysis, Modeling, and Testing IX, G. C. Holst, ed., Proc. SPIE3377, 182–193 (1998).

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Figures (4)

Fig. 1
Fig. 1

Spatial distribution of an equilateral triangle pattern.

Fig. 2
Fig. 2

Basic block diagram of a staring thermal imager.

Fig. 3
Fig. 3

TOD predictive curve for a typical staring infrared imaging system.

Fig. 4
Fig. 4

TOD measured curve for a typical staring infrared imaging system.

Tables (1)

Tables Icon

Table 1 Fundamental Parameters of a Staring Imaging System for TOD Calculation

Equations (36)

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

o ( x , y ) = [ 1 ( - w 2 x 0 - 3 6 w y 3 x + 3 3 w ) ( 0 x w 2 - 3 6 w y - 3 x + 3 w 3 ) 0 otherwise ] .
O ( ξ , η ) = - - o ( x , y ) exp [ - j 2 π ( x ξ + y η ) ] d x d y .
O ( ξ , η ) = A - B - C D ,
f ( x , y ) = PSF ( x , y ) * o ( x , y ) ,
f i ( x , y ) = f ( x , y ) * d ( x , y ) ,
i ( x , y ) = f i ( x , y ) samp Δ x , Δ y ( x , y ) ,
samp Δ x , Δ y ( x , y ) = m = - n = - δ ( x - m Δ x , y - n Δ y ) ,
g h ( x , y ) = i ( x , y ) * h s p ( x , y ) ,
g c ( x , y ) = g h ( x , y ) * h r e ( x , y ) .
g m ( x , y ) = g c ( x , y ) * h m ( x , y ) ,
g m ( x , y ) = [ o ( x , y ) * PSF ( x , y ) * d ( x , y ) ] samp Δ x , Δ y ( x , y ) * h sp ( x , y ) * h r e ( x , y ) * h m ( x , y )
G m ( ξ , η ) = [ O ( ξ , η ) OTF ( ξ , η ) H d ( ξ , η ) ] * SAMP ξ 1 S , ξ 2 S ( ξ , η ) H s p ( ξ , η ) H r e ( ξ , η ) H m ( ξ , η )
SAMP ξ S , η S ( ξ , η ) = J { samp ( x , y ) } = ξ S η S m = - m = - δ ( ξ - m ξ S , η - n η S ) = ξ S η n [ δ ( ξ , η ) + m 0 , n 0 δ ( ξ - m ξ S , η - n η S ) ] .
MTF pre ( ξ , η ) = OTF ( ξ , η ) H d ( ξ , η ) , MTF post ( ξ , η ) = H s p ( ξ , η ) H r e ( ξ , η ) H m ( ξ , η ) , H sys ( ξ , η ) = OTF ( ξ , η ) H d ( ξ , η ) H s p ( ξ , η ) × H r e ( ξ , η ) H m ( ξ , η ) ,
G m ( ξ , η ) = [ O ( ξ , η ) MTF pre ( ξ , η ) ] MTF post ( ξ , η ) + MTF post ( ξ , η ) m 0 , n 0 MTF pre ( ξ - m ξ S , η - n η S ) O ( ξ - m ξ S , η - n η S ) ,
N a ( ξ , η ) = MTF pout ( ξ , η ) m 0 , n 0 MTF pre ( ξ - m ξ s , η - n η S ) O ( ξ - m ξ S , η - n η S ) .
G m ( ξ , η ) = O ( ξ , η ) MTF pre ( ξ , η ) MTF post ( ξ , η ) + N a ( ξ , η ) .
M monitor ( ξ , η ) = G Δ T O ( ξ , η ) × H sys ( ξ , η ) Δ x Δ y ,
M eye ( ξ , η ) = G Δ T O ( ξ , η ) H sys ( ξ , η ) H eye ( ξ , η ) H mat ( ξ , η ) Δ x Δ y ,
H mat ( ξ , η ) = { O ( ξ , η ) H sys ( ξ , η ) H eye ( ξ , η ) w m 4 mrad O ( ξ , η ) H sys ( ξ , η ) H eye ( ξ , η ) w m = 4 mrad w m > 4 mrad .
( signal ) p = ( F r t eye ) - - M eye ( ξ , η ) d ξ d η = G Δ T ( F r t eye ) Δ x Δ y - - O ( ξ , η ) × H sys ( ξ , η ) H eye ( ξ , η ) H mat ( ξ , η ) d ξ d η ,
N monitor - alia ( ξ , η ) = G Δ T N a ( ξ , η ) Δ x Δ y ,
N eye - alia ( ξ , η ) = k G Δ T N a ( ξ , η ) H eye ( ξ , η ) H tmata ( ξ , η ) Δ x Δ y × ( t eye F ) ,
H mata ( ξ , η ) = N a ( ξ , η ) H eye ( ξ , η ) .
σ a p = k G Δ T ( F r t eye ) Δ x Δ y - - N a 2 ( ξ , η ) H eye 2 ( ξ , η ) d ξ d η .
σ t v h = 4 F 2 Δ f noise τ 0 A d λ 1 λ 2 D * ( λ ) L ( λ , T ) T ,
Δ f x , y = 1 ( Δ x Δ y ) 1 / 2 ,
Δ f x , y = ( Δ x Δ y ) 1 / 2 ( Δ x Δ y ) 1 / 2 - H NF 2 ( ξ , η ) H mat 2 ( ξ , η ) d ξ d η
σ t v h Δ f x , y / Δ f x , y .
σ t v h ( F r t eye ) Δ f x , y / Δ f x y ,
σ sys = ( σ t v h 2 E t E v h + σ v h 2 E v h ) 1 / 2 .
σ n p = [ G 2 σ sys 2 / ( Δ x Δ y ) ] 1 / 2 .
σ totalp = σ up + σ n p = ( G 2 σ sys 2 Δ x Δ y ) 1 / 2 + k G Δ T ( F r t eye ) Δ x Δ y + - - N a 2 ( ξ , η ) H eye 2 ( ξ , η ) d ξ d η .
SNR p = ( signal ) p σ totalp = G Δ T ( F r t eye ) ( Δ x Δ y ) - - O ( ξ , η ) H sys ( ξ , η ) H eye ( ξ , η ) H mat ( ξ , η ) d ξ d η k G Δ T ( F r t eye ) Δ x Δ y - - N a 2 ( ξ , η ) H eye 2 ( ξ , η ) d ξ d η + G σ t v h 2 E t E v h + σ v h 2 E v h ( Δ x Δ y ) 1 / 2 .
TOD ( w ) = SNR t h ( Δ x Δ y ) 1 / 2 ( σ t v h 2 E t E v h + σ v h 2 E v h ) 1 / 2 ( E t ) [ - - O ( ξ , η ) H sys ( ξ , η ) H eye ( ξ , η ) H mat ( ξ , η ) d ξ d η - SNR t h k - - N a 2 ( ξ , η ) H eye 2 ( ξ , η ) d ξ d η ] ,
S R = 2 3 4 1 w .

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