Abstract

The continued development of new and fundamentally different classes of imaging spectrometer has increased both the scope and the complexity of comparisons of their relative signal-to-noise ratios. Although the throughput and multiplex advantages of Fourier-transform spectrometers were established in the early 1950s, the application of this terminology to imaging spectrometers is often ambiguous and has led to some confusion and debate. For comparisons of signal-collection abilities to be useful to a system designer, they must be based on identical requirements and constraints. We present unambiguous definitions of terminology for application to imaging spectrometers and comparisons of signal-collection abilities and signal-to-noise-ratios on a basis that is useful to a systems designer and inclusive of six fundamentally different classes (both traditional and novel) of imaging spectrometers.

© 2005 Optical Society of America

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References

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  1. R. G. Sellar, G. D. Boreman, “Classification of imaging spectrometers for remote sensing applications,” Opt. Eng. (to be published).
  2. P. Jacquinot, C. Dufour, “Condition optique d’emploi des cellules photo-électriques dans les spectrographes et les interférometrès,” J. Rech. Centre Nat. Rech. Sci. Lab. Bellevue 6, 91–103 (1948).
  3. P. Jacquinot, “The etendue advantage,” presented at the Seventeenth XVII Meeting of Congrès du Groupment avancement des méthodes d’analyse spectrométriques (Paris, 1954).
  4. P. Jacquinot, “Caractères communs aux nouvelles méthodes de spectroscopie interférentielle; facteur de mérite,” J. Phys. Radium 19, 223–229 (1958).
    [CrossRef]
  5. P. B. Fellgett, “The multiplex advantage,” Ph.D. dissertation (University of Cambridge, Cambridge, UK, 1951).
  6. P. B. Fellgett, “Multi-channel spectrometry,” J. Opt. Soc. Am. 42, 872 (1952).
  7. M. R. Descour, “Throughput advantage in imaging Fourier-transform spectrometers,” in Imaging Spectrometry II, M. R. Descour, J. M. Mooney, eds., Proc. SPIE2819, 285–290 (1996).
    [CrossRef]
  8. L. W. Schumann, T. S. Lomheim, “Infrared hyperspectral imaging Fourier transform and dispersive spectrometers: comparison of signal-to-noise based performance,” in Imaging Spectrometry VII, M. R. Descour, S. S. Shen, eds., Proc. SPIE4480, 1–14 (2002).
    [CrossRef]
  9. E. L. Dereniak, G. D. Boreman, Infrared Detectors and Systems (Wiley, New York, 1996), pp. 45–48.
  10. J. C. Demro, R. Hartshorne, L. M. Woody, “Design of a multispectral, wedge filter, remote sensing instrument incorporating a multi-port, thinned, CCD area array,” in Imaging Spectroscopy, M. R. Descour, J. M. Mooney, D. L. Perry, L. R. Illing, eds., Proc. SPIE2480, 280–294 (1995).
  11. R. F. Horton, “Optical design for a high-etendue imaging Fourier-transform spectrometer,” in Imaging Spectrometry II, M. R. Descour, J. M. Mooney, eds., Proc. SPIE2819, 300–315 (1996).
    [CrossRef]
  12. Ref. 9, p. 165.
  13. F. D. Kahn, “The signal–noise ratio of a suggested spectral analyzer,” Astrophys. J. 129, 518–521 (1959).
    [CrossRef]
  14. S. P. Davis, M. C. Abrams, J. W. Brault, Fourier Transform Spectrometry (Academic, New York, 2001).
  15. P. J. Miller, A. R. Harvey, “Signal to noise analysis of various imaging systems,” in Biomarkers and Biological Spectral Imaging, G. H. Bearman, D. J. Bornhop, R. M. Levenson, eds., Proc. SPIE4259, 16–21 (2001).
    [CrossRef]

1959 (1)

F. D. Kahn, “The signal–noise ratio of a suggested spectral analyzer,” Astrophys. J. 129, 518–521 (1959).
[CrossRef]

1958 (1)

P. Jacquinot, “Caractères communs aux nouvelles méthodes de spectroscopie interférentielle; facteur de mérite,” J. Phys. Radium 19, 223–229 (1958).
[CrossRef]

1952 (1)

P. B. Fellgett, “Multi-channel spectrometry,” J. Opt. Soc. Am. 42, 872 (1952).

1948 (1)

P. Jacquinot, C. Dufour, “Condition optique d’emploi des cellules photo-électriques dans les spectrographes et les interférometrès,” J. Rech. Centre Nat. Rech. Sci. Lab. Bellevue 6, 91–103 (1948).

Abrams, M. C.

S. P. Davis, M. C. Abrams, J. W. Brault, Fourier Transform Spectrometry (Academic, New York, 2001).

Boreman, G. D.

R. G. Sellar, G. D. Boreman, “Classification of imaging spectrometers for remote sensing applications,” Opt. Eng. (to be published).

E. L. Dereniak, G. D. Boreman, Infrared Detectors and Systems (Wiley, New York, 1996), pp. 45–48.

Brault, J. W.

S. P. Davis, M. C. Abrams, J. W. Brault, Fourier Transform Spectrometry (Academic, New York, 2001).

Davis, S. P.

S. P. Davis, M. C. Abrams, J. W. Brault, Fourier Transform Spectrometry (Academic, New York, 2001).

Demro, J. C.

J. C. Demro, R. Hartshorne, L. M. Woody, “Design of a multispectral, wedge filter, remote sensing instrument incorporating a multi-port, thinned, CCD area array,” in Imaging Spectroscopy, M. R. Descour, J. M. Mooney, D. L. Perry, L. R. Illing, eds., Proc. SPIE2480, 280–294 (1995).

Dereniak, E. L.

E. L. Dereniak, G. D. Boreman, Infrared Detectors and Systems (Wiley, New York, 1996), pp. 45–48.

Descour, M. R.

M. R. Descour, “Throughput advantage in imaging Fourier-transform spectrometers,” in Imaging Spectrometry II, M. R. Descour, J. M. Mooney, eds., Proc. SPIE2819, 285–290 (1996).
[CrossRef]

Dufour, C.

P. Jacquinot, C. Dufour, “Condition optique d’emploi des cellules photo-électriques dans les spectrographes et les interférometrès,” J. Rech. Centre Nat. Rech. Sci. Lab. Bellevue 6, 91–103 (1948).

Fellgett, P. B.

P. B. Fellgett, “Multi-channel spectrometry,” J. Opt. Soc. Am. 42, 872 (1952).

P. B. Fellgett, “The multiplex advantage,” Ph.D. dissertation (University of Cambridge, Cambridge, UK, 1951).

Hartshorne, R.

J. C. Demro, R. Hartshorne, L. M. Woody, “Design of a multispectral, wedge filter, remote sensing instrument incorporating a multi-port, thinned, CCD area array,” in Imaging Spectroscopy, M. R. Descour, J. M. Mooney, D. L. Perry, L. R. Illing, eds., Proc. SPIE2480, 280–294 (1995).

Harvey, A. R.

P. J. Miller, A. R. Harvey, “Signal to noise analysis of various imaging systems,” in Biomarkers and Biological Spectral Imaging, G. H. Bearman, D. J. Bornhop, R. M. Levenson, eds., Proc. SPIE4259, 16–21 (2001).
[CrossRef]

Horton, R. F.

R. F. Horton, “Optical design for a high-etendue imaging Fourier-transform spectrometer,” in Imaging Spectrometry II, M. R. Descour, J. M. Mooney, eds., Proc. SPIE2819, 300–315 (1996).
[CrossRef]

Jacquinot, P.

P. Jacquinot, “Caractères communs aux nouvelles méthodes de spectroscopie interférentielle; facteur de mérite,” J. Phys. Radium 19, 223–229 (1958).
[CrossRef]

P. Jacquinot, C. Dufour, “Condition optique d’emploi des cellules photo-électriques dans les spectrographes et les interférometrès,” J. Rech. Centre Nat. Rech. Sci. Lab. Bellevue 6, 91–103 (1948).

P. Jacquinot, “The etendue advantage,” presented at the Seventeenth XVII Meeting of Congrès du Groupment avancement des méthodes d’analyse spectrométriques (Paris, 1954).

Kahn, F. D.

F. D. Kahn, “The signal–noise ratio of a suggested spectral analyzer,” Astrophys. J. 129, 518–521 (1959).
[CrossRef]

Lomheim, T. S.

L. W. Schumann, T. S. Lomheim, “Infrared hyperspectral imaging Fourier transform and dispersive spectrometers: comparison of signal-to-noise based performance,” in Imaging Spectrometry VII, M. R. Descour, S. S. Shen, eds., Proc. SPIE4480, 1–14 (2002).
[CrossRef]

Miller, P. J.

P. J. Miller, A. R. Harvey, “Signal to noise analysis of various imaging systems,” in Biomarkers and Biological Spectral Imaging, G. H. Bearman, D. J. Bornhop, R. M. Levenson, eds., Proc. SPIE4259, 16–21 (2001).
[CrossRef]

Schumann, L. W.

L. W. Schumann, T. S. Lomheim, “Infrared hyperspectral imaging Fourier transform and dispersive spectrometers: comparison of signal-to-noise based performance,” in Imaging Spectrometry VII, M. R. Descour, S. S. Shen, eds., Proc. SPIE4480, 1–14 (2002).
[CrossRef]

Sellar, R. G.

R. G. Sellar, G. D. Boreman, “Classification of imaging spectrometers for remote sensing applications,” Opt. Eng. (to be published).

Woody, L. M.

J. C. Demro, R. Hartshorne, L. M. Woody, “Design of a multispectral, wedge filter, remote sensing instrument incorporating a multi-port, thinned, CCD area array,” in Imaging Spectroscopy, M. R. Descour, J. M. Mooney, D. L. Perry, L. R. Illing, eds., Proc. SPIE2480, 280–294 (1995).

Astrophys. J. (1)

F. D. Kahn, “The signal–noise ratio of a suggested spectral analyzer,” Astrophys. J. 129, 518–521 (1959).
[CrossRef]

J. Opt. Soc. Am. (1)

P. B. Fellgett, “Multi-channel spectrometry,” J. Opt. Soc. Am. 42, 872 (1952).

J. Phys. Radium (1)

P. Jacquinot, “Caractères communs aux nouvelles méthodes de spectroscopie interférentielle; facteur de mérite,” J. Phys. Radium 19, 223–229 (1958).
[CrossRef]

J. Rech. Centre Nat. Rech. Sci. Lab. Bellevue (1)

P. Jacquinot, C. Dufour, “Condition optique d’emploi des cellules photo-électriques dans les spectrographes et les interférometrès,” J. Rech. Centre Nat. Rech. Sci. Lab. Bellevue 6, 91–103 (1948).

Other (11)

P. Jacquinot, “The etendue advantage,” presented at the Seventeenth XVII Meeting of Congrès du Groupment avancement des méthodes d’analyse spectrométriques (Paris, 1954).

P. B. Fellgett, “The multiplex advantage,” Ph.D. dissertation (University of Cambridge, Cambridge, UK, 1951).

M. R. Descour, “Throughput advantage in imaging Fourier-transform spectrometers,” in Imaging Spectrometry II, M. R. Descour, J. M. Mooney, eds., Proc. SPIE2819, 285–290 (1996).
[CrossRef]

L. W. Schumann, T. S. Lomheim, “Infrared hyperspectral imaging Fourier transform and dispersive spectrometers: comparison of signal-to-noise based performance,” in Imaging Spectrometry VII, M. R. Descour, S. S. Shen, eds., Proc. SPIE4480, 1–14 (2002).
[CrossRef]

E. L. Dereniak, G. D. Boreman, Infrared Detectors and Systems (Wiley, New York, 1996), pp. 45–48.

J. C. Demro, R. Hartshorne, L. M. Woody, “Design of a multispectral, wedge filter, remote sensing instrument incorporating a multi-port, thinned, CCD area array,” in Imaging Spectroscopy, M. R. Descour, J. M. Mooney, D. L. Perry, L. R. Illing, eds., Proc. SPIE2480, 280–294 (1995).

R. F. Horton, “Optical design for a high-etendue imaging Fourier-transform spectrometer,” in Imaging Spectrometry II, M. R. Descour, J. M. Mooney, eds., Proc. SPIE2819, 300–315 (1996).
[CrossRef]

Ref. 9, p. 165.

S. P. Davis, M. C. Abrams, J. W. Brault, Fourier Transform Spectrometry (Academic, New York, 2001).

P. J. Miller, A. R. Harvey, “Signal to noise analysis of various imaging systems,” in Biomarkers and Biological Spectral Imaging, G. H. Bearman, D. J. Bornhop, R. M. Levenson, eds., Proc. SPIE4259, 16–21 (2001).
[CrossRef]

R. G. Sellar, G. D. Boreman, “Classification of imaging spectrometers for remote sensing applications,” Opt. Eng. (to be published).

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Figures (3)

Fig. 1
Fig. 1

Determination of efficiency η and temporal extent Δτ for an object voxel.

Fig. 2
Fig. 2

Example SNR factors for an object voxel as a function of noise-condition parameter Q for My = 4000, Mλ = 200, Mj = 1000, and S(λ)/S = 1.

Fig. 3
Fig. 3

Relative SNRs—normalized to the SNR for the dispersive pushbroom class—for an object voxel as a function of noise-condition parameter Q for the example My = 4000, Mλ = 200, Mj = 1000, and S(λ)/S = 1.

Tables (10)

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Table 1 Classification of Imaging Spectrometers

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Table 2 Common Set of Requirements

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Table 3 Detector Array Constraints in Common

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Table 4 Other Constraints in Common

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Table 5 Signal Factors and Example Relative Signals for an Object Voxel and for the Entire Object

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Table 6 Spectral Extent Δλi for a Raw Data Element, Number of Exposures Mk, and Temporal Extent Δτi for a Raw Data Element

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Table 7 Signal Factors and Example Relative Signals for a Raw Data Element (Image Space)

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Table 8 Number of Data Elements and Signal Factors for the Sum of all Data Elements (Image Space)

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Table 9 SNR Factors for an Object Voxel

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Table 10 Example Relative SNRs for an Object Voxel

Equations (15)

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S = L λ η A Ω ( Δ λ ) ( Δ τ ) ,
S = L λ ( λ ) η ( λ ) A Ω ( Δ τ ) d λ .
s o = L λ η ( Δ x o ) ( Δ y o ) Ω o ( Δ λ o ) ( Δ τ o ) .
S o = L λ η ( Δ x ) ( Δ y ) Ω ( Δ λ ) ( Δ τ ) M = L λ η ( Δ x o ) ( Δ y o ) Ω o ( Δ λ ) ( Δ τ ) M x M y M λ = L λ η ( Δ x o M x ) ( Δ y o M y ) Ω o ( Δ λ M λ ) ( Δ τ ) = L λ η X o Y o Ω o Λ ( Δ τ ) ,
s i = L λ η ( Δ x i ) ( Δ y i ) Ω i ( Δ λ i ) ( Δ τ i ) .
M = M i M j M k .
M = M i M j M k - M i M j M j .
S i = L λ η ( Δ x ) ( Δ y ) Ω ( Δ λ ) ( Δ τ ) M = L λ η ( Δ x i ) ( Δ y i ) Ω i ( Δ λ i ) ( Δ τ i ) M i M j M k = L λ η ( Δ x i M i ) ( Δ y i M j ) Ω i ( Δ λ i ) ( Δ τ i M k ) = L λ η X i Y i Ω i ( Δ λ i ) τ ,
S i = L λ η ( Δ x ) ( Δ y ) Ω ( Δ λ ) ( Δ τ ) M = L λ η ( Δ x i ) ( Δ y i ) Ω i ( Δ λ i ) ( Δ τ i ) ( M i M j M k - M i M j M j ) = L λ η ( Δ x i M i ) ( Δ y i M j ) Ω i ( Δ λ i ) ( Δ τ i ) ( M k - M j ) = L λ η X i Y i Ω i ( Δ λ i ) ( τ - Δ τ M j ) ,
N p = S .
N = N p + N d .
SNR = S N = S N P + N d .
SNR s ( λ ) = SNR i { [ S ( λ ) S ¯ ] 1 M λ } ,
N d = Q [ L λ A o Ω o ( Δ λ o ) τ ] 1 / 2 .
SNR fac ( Q ) = S fac Q + S fac ,

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