Abstract

Traditional spectral sensors are intentionally designed to minimize overlap among spectral response functions of different bands. In contrast, some emerging classes of spectral sensors exhibit significant band overlap. An effect introduced by such band overlap is that the photodetector noise of one band is coupled into the others in subsequent data processing steps. Because of this, the traditional band-by-band definition of signal-to-noise ratio (SNR) cannot fully describe the detector’s noise level. We devise a general definition of SNR in spectral space based on a recently developed geometrical spectral imaging model [J. Opt. Soc. Am. A 24, 2864 (2007) ]. With this model, we can find an orthogonal basis of the spectral response functions for the spectral sensor with decreasing instrument SNRs. We can also define the average instrument SNR for the whole sensor, which makes it possible to characterize quantitatively the photodetector noise of a spectral sensor with correlated bands.

© 2008 Optical Society of America

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References

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  1. Z. Wang, M. M. Hayat, and J. S. Tyo, “Data interpretation for spectral sensors with correlated bands,” J. Opt. Soc. Am. A 24, 2864-2870 (2007).
    [CrossRef]
  2. Ü. Sakoğlu, J. S. Tyo, M. M. Hayat, S. Raghavan, and S. Krishna, “Spectrally adaptive infrared photodetectors with bias-tunable quantum dots,” J. Opt. Soc. Am. B 21, 7-17 (2004).
    [CrossRef]
  3. Z. Wang, Ü. Sakoğlu, S. Annamalai, N.-R. Weisse-Bernstein, P. Dowd, J. S. Tyo, M. M. Hayat, and S. Krishna, “Real-time implementation of spectral matched filtering algorithms using adaptive focal plane array technology,” Proc. SPIE 5546, 73-83 (2004).
    [CrossRef]
  4. J. P. Kerekes and D. A. Landgrebe, “Simulation of optical remote sensing systems,” IEEE Trans. Geosci. Remote Sens. 27, 762-771 (1989).
    [CrossRef]
  5. F. A. Kruse, J. W. Boardman, and J. F. Huntington, “Comparison of airborne hyperspectral data and co-1 hyperion for mineral mapping,” IEEE Trans. Geosci. Remote Sens. 41, 1388-1400 (2003).
    [CrossRef]
  6. B. Paskaleva, M. M. Hayat, Z. Wang, and J. S. Tyo, “Canonical correlation feature selection for sensors with overlapping bands: Theory and application,” IEEE Trans. Geosci. Remote Sens. (to be published, 2008).
    [CrossRef]
  7. K. M. Harrison and H. Barrett, Foundations of Image Science, 2nd ed. (Wiley-Interscience, 2004).
  8. Z. Wang, B. S. Paskaleva, J. S. Tyo, and M. M. Hayat, “Canonical correlations analysis for assessing the performance of adaptive spectral imagers,” Proc. SPIE 5806, 23-34 (2005).
    [CrossRef]
  9. J. P. Kerekes and D. A. Landgrebe, “An analytical model of earth-observational remote sensing systems,” IEEE Trans. Syst. Man Cybern. 21, 125-133 (1991).
    [CrossRef]
  10. A. A. Green, M. Berman, P. Switzer, and M. D. Craig, “A transformation for ordering multispectral data in terms of image quality with implications for noise removal,” IEEE Trans. Geosci. Remote Sens. 26, 65-74 (1988).
    [CrossRef]
  11. G. H. Golub and C. F. V. Loan, Matrix Computations, 3rd ed. (Johns Hopkins U. Press, 1996).
  12. P. Bhattacharya, S. Krishna, J. Phillips, P. J. McCann, and K. Namjou, “Carrier dynamics in self-organized quantum dots and their application to long-wavelength sources and detectors,” J. Cryst. Growth 227, 27-35 (2001).
    [CrossRef]
  13. B. M. Ratliff, M. M. Hayat, and J. S. Tyo, “Generalized algebraic nonuniformity correction algorithm,” J. Opt. Soc. Am. A 20, 1890-1899 (2003).
    [CrossRef]

2008 (1)

B. Paskaleva, M. M. Hayat, Z. Wang, and J. S. Tyo, “Canonical correlation feature selection for sensors with overlapping bands: Theory and application,” IEEE Trans. Geosci. Remote Sens. (to be published, 2008).
[CrossRef]

2007 (1)

2005 (1)

Z. Wang, B. S. Paskaleva, J. S. Tyo, and M. M. Hayat, “Canonical correlations analysis for assessing the performance of adaptive spectral imagers,” Proc. SPIE 5806, 23-34 (2005).
[CrossRef]

2004 (2)

Z. Wang, Ü. Sakoğlu, S. Annamalai, N.-R. Weisse-Bernstein, P. Dowd, J. S. Tyo, M. M. Hayat, and S. Krishna, “Real-time implementation of spectral matched filtering algorithms using adaptive focal plane array technology,” Proc. SPIE 5546, 73-83 (2004).
[CrossRef]

Ü. Sakoğlu, J. S. Tyo, M. M. Hayat, S. Raghavan, and S. Krishna, “Spectrally adaptive infrared photodetectors with bias-tunable quantum dots,” J. Opt. Soc. Am. B 21, 7-17 (2004).
[CrossRef]

2003 (2)

B. M. Ratliff, M. M. Hayat, and J. S. Tyo, “Generalized algebraic nonuniformity correction algorithm,” J. Opt. Soc. Am. A 20, 1890-1899 (2003).
[CrossRef]

F. A. Kruse, J. W. Boardman, and J. F. Huntington, “Comparison of airborne hyperspectral data and co-1 hyperion for mineral mapping,” IEEE Trans. Geosci. Remote Sens. 41, 1388-1400 (2003).
[CrossRef]

2001 (1)

P. Bhattacharya, S. Krishna, J. Phillips, P. J. McCann, and K. Namjou, “Carrier dynamics in self-organized quantum dots and their application to long-wavelength sources and detectors,” J. Cryst. Growth 227, 27-35 (2001).
[CrossRef]

1991 (1)

J. P. Kerekes and D. A. Landgrebe, “An analytical model of earth-observational remote sensing systems,” IEEE Trans. Syst. Man Cybern. 21, 125-133 (1991).
[CrossRef]

1989 (1)

J. P. Kerekes and D. A. Landgrebe, “Simulation of optical remote sensing systems,” IEEE Trans. Geosci. Remote Sens. 27, 762-771 (1989).
[CrossRef]

1988 (1)

A. A. Green, M. Berman, P. Switzer, and M. D. Craig, “A transformation for ordering multispectral data in terms of image quality with implications for noise removal,” IEEE Trans. Geosci. Remote Sens. 26, 65-74 (1988).
[CrossRef]

Annamalai, S.

Z. Wang, Ü. Sakoğlu, S. Annamalai, N.-R. Weisse-Bernstein, P. Dowd, J. S. Tyo, M. M. Hayat, and S. Krishna, “Real-time implementation of spectral matched filtering algorithms using adaptive focal plane array technology,” Proc. SPIE 5546, 73-83 (2004).
[CrossRef]

Barrett, H.

K. M. Harrison and H. Barrett, Foundations of Image Science, 2nd ed. (Wiley-Interscience, 2004).

Berman, M.

A. A. Green, M. Berman, P. Switzer, and M. D. Craig, “A transformation for ordering multispectral data in terms of image quality with implications for noise removal,” IEEE Trans. Geosci. Remote Sens. 26, 65-74 (1988).
[CrossRef]

Bhattacharya, P.

P. Bhattacharya, S. Krishna, J. Phillips, P. J. McCann, and K. Namjou, “Carrier dynamics in self-organized quantum dots and their application to long-wavelength sources and detectors,” J. Cryst. Growth 227, 27-35 (2001).
[CrossRef]

Boardman, J. W.

F. A. Kruse, J. W. Boardman, and J. F. Huntington, “Comparison of airborne hyperspectral data and co-1 hyperion for mineral mapping,” IEEE Trans. Geosci. Remote Sens. 41, 1388-1400 (2003).
[CrossRef]

Craig, M. D.

A. A. Green, M. Berman, P. Switzer, and M. D. Craig, “A transformation for ordering multispectral data in terms of image quality with implications for noise removal,” IEEE Trans. Geosci. Remote Sens. 26, 65-74 (1988).
[CrossRef]

Dowd, P.

Z. Wang, Ü. Sakoğlu, S. Annamalai, N.-R. Weisse-Bernstein, P. Dowd, J. S. Tyo, M. M. Hayat, and S. Krishna, “Real-time implementation of spectral matched filtering algorithms using adaptive focal plane array technology,” Proc. SPIE 5546, 73-83 (2004).
[CrossRef]

Golub, G. H.

G. H. Golub and C. F. V. Loan, Matrix Computations, 3rd ed. (Johns Hopkins U. Press, 1996).

Green, A. A.

A. A. Green, M. Berman, P. Switzer, and M. D. Craig, “A transformation for ordering multispectral data in terms of image quality with implications for noise removal,” IEEE Trans. Geosci. Remote Sens. 26, 65-74 (1988).
[CrossRef]

Harrison, K. M.

K. M. Harrison and H. Barrett, Foundations of Image Science, 2nd ed. (Wiley-Interscience, 2004).

Hayat, M. M.

B. Paskaleva, M. M. Hayat, Z. Wang, and J. S. Tyo, “Canonical correlation feature selection for sensors with overlapping bands: Theory and application,” IEEE Trans. Geosci. Remote Sens. (to be published, 2008).
[CrossRef]

Z. Wang, M. M. Hayat, and J. S. Tyo, “Data interpretation for spectral sensors with correlated bands,” J. Opt. Soc. Am. A 24, 2864-2870 (2007).
[CrossRef]

Z. Wang, B. S. Paskaleva, J. S. Tyo, and M. M. Hayat, “Canonical correlations analysis for assessing the performance of adaptive spectral imagers,” Proc. SPIE 5806, 23-34 (2005).
[CrossRef]

Z. Wang, Ü. Sakoğlu, S. Annamalai, N.-R. Weisse-Bernstein, P. Dowd, J. S. Tyo, M. M. Hayat, and S. Krishna, “Real-time implementation of spectral matched filtering algorithms using adaptive focal plane array technology,” Proc. SPIE 5546, 73-83 (2004).
[CrossRef]

Ü. Sakoğlu, J. S. Tyo, M. M. Hayat, S. Raghavan, and S. Krishna, “Spectrally adaptive infrared photodetectors with bias-tunable quantum dots,” J. Opt. Soc. Am. B 21, 7-17 (2004).
[CrossRef]

B. M. Ratliff, M. M. Hayat, and J. S. Tyo, “Generalized algebraic nonuniformity correction algorithm,” J. Opt. Soc. Am. A 20, 1890-1899 (2003).
[CrossRef]

Huntington, J. F.

F. A. Kruse, J. W. Boardman, and J. F. Huntington, “Comparison of airborne hyperspectral data and co-1 hyperion for mineral mapping,” IEEE Trans. Geosci. Remote Sens. 41, 1388-1400 (2003).
[CrossRef]

Kerekes, J. P.

J. P. Kerekes and D. A. Landgrebe, “An analytical model of earth-observational remote sensing systems,” IEEE Trans. Syst. Man Cybern. 21, 125-133 (1991).
[CrossRef]

J. P. Kerekes and D. A. Landgrebe, “Simulation of optical remote sensing systems,” IEEE Trans. Geosci. Remote Sens. 27, 762-771 (1989).
[CrossRef]

Krishna, S.

Z. Wang, Ü. Sakoğlu, S. Annamalai, N.-R. Weisse-Bernstein, P. Dowd, J. S. Tyo, M. M. Hayat, and S. Krishna, “Real-time implementation of spectral matched filtering algorithms using adaptive focal plane array technology,” Proc. SPIE 5546, 73-83 (2004).
[CrossRef]

Ü. Sakoğlu, J. S. Tyo, M. M. Hayat, S. Raghavan, and S. Krishna, “Spectrally adaptive infrared photodetectors with bias-tunable quantum dots,” J. Opt. Soc. Am. B 21, 7-17 (2004).
[CrossRef]

P. Bhattacharya, S. Krishna, J. Phillips, P. J. McCann, and K. Namjou, “Carrier dynamics in self-organized quantum dots and their application to long-wavelength sources and detectors,” J. Cryst. Growth 227, 27-35 (2001).
[CrossRef]

Kruse, F. A.

F. A. Kruse, J. W. Boardman, and J. F. Huntington, “Comparison of airborne hyperspectral data and co-1 hyperion for mineral mapping,” IEEE Trans. Geosci. Remote Sens. 41, 1388-1400 (2003).
[CrossRef]

Landgrebe, D. A.

J. P. Kerekes and D. A. Landgrebe, “An analytical model of earth-observational remote sensing systems,” IEEE Trans. Syst. Man Cybern. 21, 125-133 (1991).
[CrossRef]

J. P. Kerekes and D. A. Landgrebe, “Simulation of optical remote sensing systems,” IEEE Trans. Geosci. Remote Sens. 27, 762-771 (1989).
[CrossRef]

Loan, C. F. V.

G. H. Golub and C. F. V. Loan, Matrix Computations, 3rd ed. (Johns Hopkins U. Press, 1996).

McCann, P. J.

P. Bhattacharya, S. Krishna, J. Phillips, P. J. McCann, and K. Namjou, “Carrier dynamics in self-organized quantum dots and their application to long-wavelength sources and detectors,” J. Cryst. Growth 227, 27-35 (2001).
[CrossRef]

Namjou, K.

P. Bhattacharya, S. Krishna, J. Phillips, P. J. McCann, and K. Namjou, “Carrier dynamics in self-organized quantum dots and their application to long-wavelength sources and detectors,” J. Cryst. Growth 227, 27-35 (2001).
[CrossRef]

Paskaleva, B.

B. Paskaleva, M. M. Hayat, Z. Wang, and J. S. Tyo, “Canonical correlation feature selection for sensors with overlapping bands: Theory and application,” IEEE Trans. Geosci. Remote Sens. (to be published, 2008).
[CrossRef]

Paskaleva, B. S.

Z. Wang, B. S. Paskaleva, J. S. Tyo, and M. M. Hayat, “Canonical correlations analysis for assessing the performance of adaptive spectral imagers,” Proc. SPIE 5806, 23-34 (2005).
[CrossRef]

Phillips, J.

P. Bhattacharya, S. Krishna, J. Phillips, P. J. McCann, and K. Namjou, “Carrier dynamics in self-organized quantum dots and their application to long-wavelength sources and detectors,” J. Cryst. Growth 227, 27-35 (2001).
[CrossRef]

Raghavan, S.

Ratliff, B. M.

Sakoglu, Ü.

Ü. Sakoğlu, J. S. Tyo, M. M. Hayat, S. Raghavan, and S. Krishna, “Spectrally adaptive infrared photodetectors with bias-tunable quantum dots,” J. Opt. Soc. Am. B 21, 7-17 (2004).
[CrossRef]

Z. Wang, Ü. Sakoğlu, S. Annamalai, N.-R. Weisse-Bernstein, P. Dowd, J. S. Tyo, M. M. Hayat, and S. Krishna, “Real-time implementation of spectral matched filtering algorithms using adaptive focal plane array technology,” Proc. SPIE 5546, 73-83 (2004).
[CrossRef]

Switzer, P.

A. A. Green, M. Berman, P. Switzer, and M. D. Craig, “A transformation for ordering multispectral data in terms of image quality with implications for noise removal,” IEEE Trans. Geosci. Remote Sens. 26, 65-74 (1988).
[CrossRef]

Tyo, J. S.

B. Paskaleva, M. M. Hayat, Z. Wang, and J. S. Tyo, “Canonical correlation feature selection for sensors with overlapping bands: Theory and application,” IEEE Trans. Geosci. Remote Sens. (to be published, 2008).
[CrossRef]

Z. Wang, M. M. Hayat, and J. S. Tyo, “Data interpretation for spectral sensors with correlated bands,” J. Opt. Soc. Am. A 24, 2864-2870 (2007).
[CrossRef]

Z. Wang, B. S. Paskaleva, J. S. Tyo, and M. M. Hayat, “Canonical correlations analysis for assessing the performance of adaptive spectral imagers,” Proc. SPIE 5806, 23-34 (2005).
[CrossRef]

Z. Wang, Ü. Sakoğlu, S. Annamalai, N.-R. Weisse-Bernstein, P. Dowd, J. S. Tyo, M. M. Hayat, and S. Krishna, “Real-time implementation of spectral matched filtering algorithms using adaptive focal plane array technology,” Proc. SPIE 5546, 73-83 (2004).
[CrossRef]

Ü. Sakoğlu, J. S. Tyo, M. M. Hayat, S. Raghavan, and S. Krishna, “Spectrally adaptive infrared photodetectors with bias-tunable quantum dots,” J. Opt. Soc. Am. B 21, 7-17 (2004).
[CrossRef]

B. M. Ratliff, M. M. Hayat, and J. S. Tyo, “Generalized algebraic nonuniformity correction algorithm,” J. Opt. Soc. Am. A 20, 1890-1899 (2003).
[CrossRef]

Wang, Z.

B. Paskaleva, M. M. Hayat, Z. Wang, and J. S. Tyo, “Canonical correlation feature selection for sensors with overlapping bands: Theory and application,” IEEE Trans. Geosci. Remote Sens. (to be published, 2008).
[CrossRef]

Z. Wang, M. M. Hayat, and J. S. Tyo, “Data interpretation for spectral sensors with correlated bands,” J. Opt. Soc. Am. A 24, 2864-2870 (2007).
[CrossRef]

Z. Wang, B. S. Paskaleva, J. S. Tyo, and M. M. Hayat, “Canonical correlations analysis for assessing the performance of adaptive spectral imagers,” Proc. SPIE 5806, 23-34 (2005).
[CrossRef]

Z. Wang, Ü. Sakoğlu, S. Annamalai, N.-R. Weisse-Bernstein, P. Dowd, J. S. Tyo, M. M. Hayat, and S. Krishna, “Real-time implementation of spectral matched filtering algorithms using adaptive focal plane array technology,” Proc. SPIE 5546, 73-83 (2004).
[CrossRef]

Weisse-Bernstein, N.-R.

Z. Wang, Ü. Sakoğlu, S. Annamalai, N.-R. Weisse-Bernstein, P. Dowd, J. S. Tyo, M. M. Hayat, and S. Krishna, “Real-time implementation of spectral matched filtering algorithms using adaptive focal plane array technology,” Proc. SPIE 5546, 73-83 (2004).
[CrossRef]

IEEE Trans. Geosci. Remote Sens. (4)

J. P. Kerekes and D. A. Landgrebe, “Simulation of optical remote sensing systems,” IEEE Trans. Geosci. Remote Sens. 27, 762-771 (1989).
[CrossRef]

F. A. Kruse, J. W. Boardman, and J. F. Huntington, “Comparison of airborne hyperspectral data and co-1 hyperion for mineral mapping,” IEEE Trans. Geosci. Remote Sens. 41, 1388-1400 (2003).
[CrossRef]

B. Paskaleva, M. M. Hayat, Z. Wang, and J. S. Tyo, “Canonical correlation feature selection for sensors with overlapping bands: Theory and application,” IEEE Trans. Geosci. Remote Sens. (to be published, 2008).
[CrossRef]

A. A. Green, M. Berman, P. Switzer, and M. D. Craig, “A transformation for ordering multispectral data in terms of image quality with implications for noise removal,” IEEE Trans. Geosci. Remote Sens. 26, 65-74 (1988).
[CrossRef]

IEEE Trans. Syst. Man Cybern. (1)

J. P. Kerekes and D. A. Landgrebe, “An analytical model of earth-observational remote sensing systems,” IEEE Trans. Syst. Man Cybern. 21, 125-133 (1991).
[CrossRef]

J. Cryst. Growth (1)

P. Bhattacharya, S. Krishna, J. Phillips, P. J. McCann, and K. Namjou, “Carrier dynamics in self-organized quantum dots and their application to long-wavelength sources and detectors,” J. Cryst. Growth 227, 27-35 (2001).
[CrossRef]

J. Opt. Soc. Am. A (2)

J. Opt. Soc. Am. B (1)

Proc. SPIE (2)

Z. Wang, Ü. Sakoğlu, S. Annamalai, N.-R. Weisse-Bernstein, P. Dowd, J. S. Tyo, M. M. Hayat, and S. Krishna, “Real-time implementation of spectral matched filtering algorithms using adaptive focal plane array technology,” Proc. SPIE 5546, 73-83 (2004).
[CrossRef]

Z. Wang, B. S. Paskaleva, J. S. Tyo, and M. M. Hayat, “Canonical correlations analysis for assessing the performance of adaptive spectral imagers,” Proc. SPIE 5806, 23-34 (2005).
[CrossRef]

Other (2)

G. H. Golub and C. F. V. Loan, Matrix Computations, 3rd ed. (Johns Hopkins U. Press, 1996).

K. M. Harrison and H. Barrett, Foundations of Image Science, 2nd ed. (Wiley-Interscience, 2004).

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Figures (4)

Fig. 1
Fig. 1

QDIP spectral responses measured from a ten-layer In As In 0.15 Ga 0.85 As 100 μ m -aperture structure at 38 K under selected bias voltages. Each response is normalized by its peak responsivity to highlight the spectral shift among them.

Fig. 2
Fig. 2

The noise distribution in sensor space for two 2-band sensors. The measured SNRs are assumed to be the same for these sensors. (a) Sensor I spectral response functions are highly overlapping. (b) Sensor II (traditional) spectral response functions are nonoverlapping, thus orthogonal.

Fig. 3
Fig. 3

Spectral response functions of a QDIP with ten-layer In As In 0.15 Ga 0.85 As 100 μ m -aperture structure at 38 K .

Fig. 4
Fig. 4

The left singular column vectors { u i } multiplied by their corresponding singular values s i .

Tables (2)

Tables Icon

Table 1 Description of the Band Layout of Popular Spectral sensors

Tables Icon

Table 2 SNRs of the Transformed Bands for QDIPs with Selected Bias Voltage Combinations

Equations (28)

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

x i = R 0 r i ( λ ) T 0 t ( λ ) d λ = R 0 T 0 r i , t .
X = [ x 1 , x 2 , , x p ] = R 0 T 0 [ r 1 , t , r 2 , t , , r p , t ] = H t ,
t = H X ,
X = R 0 T 0 R T t ,
t = 1 R 0 T 0 ( R T ) X = ( 1 R 0 T 0 ) R ( R T R ) 1 X = ( 1 R 0 T 0 ) Z X ,
R = [ r 1 , r 2 , , r p ]
Z = [ z 1 , z 2 , , z p ]
X = R 0 T 0 R T t + n = X + n ,
SNR i = r i , t σ i = x i σ i .
t n = 1 R 0 T 0 R ( R T R ) 1 n = 1 R 0 T 0 Z n .
Γ N P = 1 R 0 2 T 0 2 Z Γ N Z T .
t ̂ = r r .
SNR i = x σ noise = r i T t ̂ σ i R 0 T 0 = r i σ i R 0 T 0 .
SNR inst = 1 t ̂ T Σ N P t ̂ .
SNR i = R 0 2 T 0 2 ( r i r i ) T R ( R T R ) 1 Σ N ( R ( R T R ) 1 ) T ( r i r i ) ,
= r i 2 σ i 2 R 0 2 T 0 2
R ( R T R ) 1 r i = [ 0 , , 0 , 1 , 0 , , 0 ] .
SNR general = t 2 t ̂ T Σ N P t ̂ = t 2 SNR ,
SNR general = t 2 t ̂ T Σ N P t ̂ = t 2 t ̂ T Σ N P t ̂ cos 2 θ = t 2 SNR cos 2 θ .
R = U S V T ,
Z = U S V T [ ( U S V T ) T U S V T ] 1 = U S V T ( V S 2 V T ) 1 = U S 1 V T .
SNR = 1 u ̂ i T Σ N P u i ̂ = 1 u i T Z Σ N Z T u i = s i 2 v i T Σ N v i .
Z T u i = ( U S 1 V T ) T u i = V [ 0 , , 0 , s i 1 , 0 , , 0 ] T .
r i = r i σ i
SNR i = 200 r i r 1 ,
SNR average = p i = 1 p 1 SNR i .
SNR i = 1 l i T Σ N P l i .
i = 1 p 1 SNR i = i = 1 p l i T Σ N P l i = tr ( L T Σ N P L ) = tr ( U T Σ N P U ) = i = 1 p 1 SNR i ,

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