Abstract

A framework is proposed for the selection of wavelength bands for multispectral sensors by use of hyperspectral reference data. Using the results from the detection theory we derive a cost function that is minimized by a set of spectral bands optimal in terms of detection performance for discrimination between a class of small rare targets and clutter with known spectral distribution. The method may be used, e.g., in the design of multispectral infrared search and track and electro-optical missile warning sensors, where a low false-alarm rate and a high-detection probability for detection of small targets against a clutter background are of critical importance, but the required high frame rate prevents the use of hyperspectral sensors.

© 2002 Optical Society of America

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References

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  1. J. C. Price, “Band selection procedure for multispectral scanners,” Appl. Opt. 33, 3281–3288 (1994).
    [CrossRef] [PubMed]
  2. A. Kanodia, R. C. Hardie, R. O. Johnson, “Band selection and performance analysis for multispectral target detectors using truthed Bomem spectrometer data,” in Proc. IEEE National Aerospace Electronics Conference (NAECON), B. Moore, ed., 1, 33–40 (1996).
  3. R. C. Hardie, M. Vaidyanathan, P. F. McManamon, “Spectral band selection and classifier design for a multispectral imaging laser radar,” Opt. Eng. 37, 752–762 (1998).
    [CrossRef]
  4. K. Fukunaga, Introduction to Statistical Pattern Recognition, 2nd ed. (Academic, San Diego, Calif., 1990).
  5. S. Mallat, A Wavelet Tour of Signal Processing (Academic, San Diego, Calif., 1998).
  6. J. C. Harsanyi, C.-I. Chang, “Hyperspectral image classification and dimensionality reduction: an orthogonal subspace projection approach,” IEEE Trans. Geosci. Remote Sens. 32, 779–785 (1994).
    [CrossRef]
  7. B. S. Everitt, An Introduction to Latent Variable Models (Chapman and Hall, London, 1984), ISBN 0-412-25310-0.
  8. S. M. Kay, Fundamentals of Statistical Signal Processing: Detection Theory (Prentice-Hall, Englewood Cliffs, N.J., 1998), ISBN 0-13-504135-X.
  9. D. Manolakis, G. Shaw, N. Keshava, “Comparative analysis of hyperspectral adaptive matched filter detectors,” in Algorithms for Multispectral, Hyperspectral, and Ultraspectral Imagery VI, S. S. Chen, M. R. Descour, eds., Proc. SPIE4049, 2–17 (2000).
  10. D. Manolakis, G. Shaw, “Detection algorithms for hyperspectral imaging applications,” IEEE Signal Process. Mag. 19, 29–43 (2002).
    [CrossRef]
  11. G. M. Davis, S. Mallat, M. Avellaneda, “Greedy adaptive approximations,” J. Const. Appr. 13, 57–98 (1997).
  12. S. S. Chen, D. L. Donoho, M. A. Saunders, “Atomic decomposition by basis pursuit,” SIAM J. Sci. Comput. 20, 33–61 (1998).
    [CrossRef]
  13. S. Mallat, Z. Zhang, “Matching pursuits with time-frequency dictionaries,” IEEE Trans. Signal Process. 41, 3397–3415 (1993).
    [CrossRef]
  14. G. M. Davis, S. Mallat, Z. Zhang, “Adaptive time-frequency approximations,” Opt. Eng. 33, 2183–2191 (1994).
    [CrossRef]
  15. G. W. Stewart, On the Early History of the Singular Value Decomposition, Technical Report TR-92-31, Dept. of Computer Science, University of Maryland, College Park, 1992, ftp://thales.cs.umd.edu/pub/reports/ehsvd.ps .
  16. A. Berk, L. S. Bernstein, D. C. Robertson, “modtran, a moderate resolution model for lowtran 7,” Technical Report GL-TR-89-0122, Air Force Geophysics Laboratory, Hanscom Air Force Base, Mass., (1989).
  17. S. F. Cotter, B. D. Rao, K. Kreutz-Delgado, “Forward sequential algorithms for best basis selection,” IEE Proc. Vision Image Signal Process. 146, 235–244 (1999).
    [CrossRef]
  18. B. K. Natarajan, “Sparse approximate solutions to linear systems,” SIAM J. Comput. 24, 227–234 (1995).
    [CrossRef]
  19. D. Goldberg, Genetic Algorithms in Search, Optimization, and Machine Learning (Addison-Wesley, Reading, Mass., 1989).

2002 (1)

D. Manolakis, G. Shaw, “Detection algorithms for hyperspectral imaging applications,” IEEE Signal Process. Mag. 19, 29–43 (2002).
[CrossRef]

1999 (1)

S. F. Cotter, B. D. Rao, K. Kreutz-Delgado, “Forward sequential algorithms for best basis selection,” IEE Proc. Vision Image Signal Process. 146, 235–244 (1999).
[CrossRef]

1998 (2)

S. S. Chen, D. L. Donoho, M. A. Saunders, “Atomic decomposition by basis pursuit,” SIAM J. Sci. Comput. 20, 33–61 (1998).
[CrossRef]

R. C. Hardie, M. Vaidyanathan, P. F. McManamon, “Spectral band selection and classifier design for a multispectral imaging laser radar,” Opt. Eng. 37, 752–762 (1998).
[CrossRef]

1997 (1)

G. M. Davis, S. Mallat, M. Avellaneda, “Greedy adaptive approximations,” J. Const. Appr. 13, 57–98 (1997).

1995 (1)

B. K. Natarajan, “Sparse approximate solutions to linear systems,” SIAM J. Comput. 24, 227–234 (1995).
[CrossRef]

1994 (3)

J. C. Price, “Band selection procedure for multispectral scanners,” Appl. Opt. 33, 3281–3288 (1994).
[CrossRef] [PubMed]

J. C. Harsanyi, C.-I. Chang, “Hyperspectral image classification and dimensionality reduction: an orthogonal subspace projection approach,” IEEE Trans. Geosci. Remote Sens. 32, 779–785 (1994).
[CrossRef]

G. M. Davis, S. Mallat, Z. Zhang, “Adaptive time-frequency approximations,” Opt. Eng. 33, 2183–2191 (1994).
[CrossRef]

1993 (1)

S. Mallat, Z. Zhang, “Matching pursuits with time-frequency dictionaries,” IEEE Trans. Signal Process. 41, 3397–3415 (1993).
[CrossRef]

Avellaneda, M.

G. M. Davis, S. Mallat, M. Avellaneda, “Greedy adaptive approximations,” J. Const. Appr. 13, 57–98 (1997).

Berk, A.

A. Berk, L. S. Bernstein, D. C. Robertson, “modtran, a moderate resolution model for lowtran 7,” Technical Report GL-TR-89-0122, Air Force Geophysics Laboratory, Hanscom Air Force Base, Mass., (1989).

Bernstein, L. S.

A. Berk, L. S. Bernstein, D. C. Robertson, “modtran, a moderate resolution model for lowtran 7,” Technical Report GL-TR-89-0122, Air Force Geophysics Laboratory, Hanscom Air Force Base, Mass., (1989).

Chang, C.-I.

J. C. Harsanyi, C.-I. Chang, “Hyperspectral image classification and dimensionality reduction: an orthogonal subspace projection approach,” IEEE Trans. Geosci. Remote Sens. 32, 779–785 (1994).
[CrossRef]

Chen, S. S.

S. S. Chen, D. L. Donoho, M. A. Saunders, “Atomic decomposition by basis pursuit,” SIAM J. Sci. Comput. 20, 33–61 (1998).
[CrossRef]

Cotter, S. F.

S. F. Cotter, B. D. Rao, K. Kreutz-Delgado, “Forward sequential algorithms for best basis selection,” IEE Proc. Vision Image Signal Process. 146, 235–244 (1999).
[CrossRef]

Davis, G. M.

G. M. Davis, S. Mallat, M. Avellaneda, “Greedy adaptive approximations,” J. Const. Appr. 13, 57–98 (1997).

G. M. Davis, S. Mallat, Z. Zhang, “Adaptive time-frequency approximations,” Opt. Eng. 33, 2183–2191 (1994).
[CrossRef]

Donoho, D. L.

S. S. Chen, D. L. Donoho, M. A. Saunders, “Atomic decomposition by basis pursuit,” SIAM J. Sci. Comput. 20, 33–61 (1998).
[CrossRef]

Everitt, B. S.

B. S. Everitt, An Introduction to Latent Variable Models (Chapman and Hall, London, 1984), ISBN 0-412-25310-0.

Fukunaga, K.

K. Fukunaga, Introduction to Statistical Pattern Recognition, 2nd ed. (Academic, San Diego, Calif., 1990).

Goldberg, D.

D. Goldberg, Genetic Algorithms in Search, Optimization, and Machine Learning (Addison-Wesley, Reading, Mass., 1989).

Hardie, R. C.

R. C. Hardie, M. Vaidyanathan, P. F. McManamon, “Spectral band selection and classifier design for a multispectral imaging laser radar,” Opt. Eng. 37, 752–762 (1998).
[CrossRef]

A. Kanodia, R. C. Hardie, R. O. Johnson, “Band selection and performance analysis for multispectral target detectors using truthed Bomem spectrometer data,” in Proc. IEEE National Aerospace Electronics Conference (NAECON), B. Moore, ed., 1, 33–40 (1996).

Harsanyi, J. C.

J. C. Harsanyi, C.-I. Chang, “Hyperspectral image classification and dimensionality reduction: an orthogonal subspace projection approach,” IEEE Trans. Geosci. Remote Sens. 32, 779–785 (1994).
[CrossRef]

Johnson, R. O.

A. Kanodia, R. C. Hardie, R. O. Johnson, “Band selection and performance analysis for multispectral target detectors using truthed Bomem spectrometer data,” in Proc. IEEE National Aerospace Electronics Conference (NAECON), B. Moore, ed., 1, 33–40 (1996).

Kanodia, A.

A. Kanodia, R. C. Hardie, R. O. Johnson, “Band selection and performance analysis for multispectral target detectors using truthed Bomem spectrometer data,” in Proc. IEEE National Aerospace Electronics Conference (NAECON), B. Moore, ed., 1, 33–40 (1996).

Kay, S. M.

S. M. Kay, Fundamentals of Statistical Signal Processing: Detection Theory (Prentice-Hall, Englewood Cliffs, N.J., 1998), ISBN 0-13-504135-X.

Keshava, N.

D. Manolakis, G. Shaw, N. Keshava, “Comparative analysis of hyperspectral adaptive matched filter detectors,” in Algorithms for Multispectral, Hyperspectral, and Ultraspectral Imagery VI, S. S. Chen, M. R. Descour, eds., Proc. SPIE4049, 2–17 (2000).

Kreutz-Delgado, K.

S. F. Cotter, B. D. Rao, K. Kreutz-Delgado, “Forward sequential algorithms for best basis selection,” IEE Proc. Vision Image Signal Process. 146, 235–244 (1999).
[CrossRef]

Mallat, S.

G. M. Davis, S. Mallat, M. Avellaneda, “Greedy adaptive approximations,” J. Const. Appr. 13, 57–98 (1997).

G. M. Davis, S. Mallat, Z. Zhang, “Adaptive time-frequency approximations,” Opt. Eng. 33, 2183–2191 (1994).
[CrossRef]

S. Mallat, Z. Zhang, “Matching pursuits with time-frequency dictionaries,” IEEE Trans. Signal Process. 41, 3397–3415 (1993).
[CrossRef]

S. Mallat, A Wavelet Tour of Signal Processing (Academic, San Diego, Calif., 1998).

Manolakis, D.

D. Manolakis, G. Shaw, “Detection algorithms for hyperspectral imaging applications,” IEEE Signal Process. Mag. 19, 29–43 (2002).
[CrossRef]

D. Manolakis, G. Shaw, N. Keshava, “Comparative analysis of hyperspectral adaptive matched filter detectors,” in Algorithms for Multispectral, Hyperspectral, and Ultraspectral Imagery VI, S. S. Chen, M. R. Descour, eds., Proc. SPIE4049, 2–17 (2000).

McManamon, P. F.

R. C. Hardie, M. Vaidyanathan, P. F. McManamon, “Spectral band selection and classifier design for a multispectral imaging laser radar,” Opt. Eng. 37, 752–762 (1998).
[CrossRef]

Natarajan, B. K.

B. K. Natarajan, “Sparse approximate solutions to linear systems,” SIAM J. Comput. 24, 227–234 (1995).
[CrossRef]

Price, J. C.

Rao, B. D.

S. F. Cotter, B. D. Rao, K. Kreutz-Delgado, “Forward sequential algorithms for best basis selection,” IEE Proc. Vision Image Signal Process. 146, 235–244 (1999).
[CrossRef]

Robertson, D. C.

A. Berk, L. S. Bernstein, D. C. Robertson, “modtran, a moderate resolution model for lowtran 7,” Technical Report GL-TR-89-0122, Air Force Geophysics Laboratory, Hanscom Air Force Base, Mass., (1989).

Saunders, M. A.

S. S. Chen, D. L. Donoho, M. A. Saunders, “Atomic decomposition by basis pursuit,” SIAM J. Sci. Comput. 20, 33–61 (1998).
[CrossRef]

Shaw, G.

D. Manolakis, G. Shaw, “Detection algorithms for hyperspectral imaging applications,” IEEE Signal Process. Mag. 19, 29–43 (2002).
[CrossRef]

D. Manolakis, G. Shaw, N. Keshava, “Comparative analysis of hyperspectral adaptive matched filter detectors,” in Algorithms for Multispectral, Hyperspectral, and Ultraspectral Imagery VI, S. S. Chen, M. R. Descour, eds., Proc. SPIE4049, 2–17 (2000).

Stewart, G. W.

G. W. Stewart, On the Early History of the Singular Value Decomposition, Technical Report TR-92-31, Dept. of Computer Science, University of Maryland, College Park, 1992, ftp://thales.cs.umd.edu/pub/reports/ehsvd.ps .

Vaidyanathan, M.

R. C. Hardie, M. Vaidyanathan, P. F. McManamon, “Spectral band selection and classifier design for a multispectral imaging laser radar,” Opt. Eng. 37, 752–762 (1998).
[CrossRef]

Zhang, Z.

G. M. Davis, S. Mallat, Z. Zhang, “Adaptive time-frequency approximations,” Opt. Eng. 33, 2183–2191 (1994).
[CrossRef]

S. Mallat, Z. Zhang, “Matching pursuits with time-frequency dictionaries,” IEEE Trans. Signal Process. 41, 3397–3415 (1993).
[CrossRef]

Appl. Opt. (1)

IEE Proc. Vision Image Signal Process. (1)

S. F. Cotter, B. D. Rao, K. Kreutz-Delgado, “Forward sequential algorithms for best basis selection,” IEE Proc. Vision Image Signal Process. 146, 235–244 (1999).
[CrossRef]

IEEE Signal Process. Mag. (1)

D. Manolakis, G. Shaw, “Detection algorithms for hyperspectral imaging applications,” IEEE Signal Process. Mag. 19, 29–43 (2002).
[CrossRef]

IEEE Trans. Geosci. Remote Sens. (1)

J. C. Harsanyi, C.-I. Chang, “Hyperspectral image classification and dimensionality reduction: an orthogonal subspace projection approach,” IEEE Trans. Geosci. Remote Sens. 32, 779–785 (1994).
[CrossRef]

IEEE Trans. Signal Process. (1)

S. Mallat, Z. Zhang, “Matching pursuits with time-frequency dictionaries,” IEEE Trans. Signal Process. 41, 3397–3415 (1993).
[CrossRef]

J. Const. Appr. (1)

G. M. Davis, S. Mallat, M. Avellaneda, “Greedy adaptive approximations,” J. Const. Appr. 13, 57–98 (1997).

Opt. Eng. (2)

R. C. Hardie, M. Vaidyanathan, P. F. McManamon, “Spectral band selection and classifier design for a multispectral imaging laser radar,” Opt. Eng. 37, 752–762 (1998).
[CrossRef]

G. M. Davis, S. Mallat, Z. Zhang, “Adaptive time-frequency approximations,” Opt. Eng. 33, 2183–2191 (1994).
[CrossRef]

SIAM J. Comput. (1)

B. K. Natarajan, “Sparse approximate solutions to linear systems,” SIAM J. Comput. 24, 227–234 (1995).
[CrossRef]

SIAM J. Sci. Comput. (1)

S. S. Chen, D. L. Donoho, M. A. Saunders, “Atomic decomposition by basis pursuit,” SIAM J. Sci. Comput. 20, 33–61 (1998).
[CrossRef]

Other (9)

K. Fukunaga, Introduction to Statistical Pattern Recognition, 2nd ed. (Academic, San Diego, Calif., 1990).

S. Mallat, A Wavelet Tour of Signal Processing (Academic, San Diego, Calif., 1998).

B. S. Everitt, An Introduction to Latent Variable Models (Chapman and Hall, London, 1984), ISBN 0-412-25310-0.

S. M. Kay, Fundamentals of Statistical Signal Processing: Detection Theory (Prentice-Hall, Englewood Cliffs, N.J., 1998), ISBN 0-13-504135-X.

D. Manolakis, G. Shaw, N. Keshava, “Comparative analysis of hyperspectral adaptive matched filter detectors,” in Algorithms for Multispectral, Hyperspectral, and Ultraspectral Imagery VI, S. S. Chen, M. R. Descour, eds., Proc. SPIE4049, 2–17 (2000).

D. Goldberg, Genetic Algorithms in Search, Optimization, and Machine Learning (Addison-Wesley, Reading, Mass., 1989).

A. Kanodia, R. C. Hardie, R. O. Johnson, “Band selection and performance analysis for multispectral target detectors using truthed Bomem spectrometer data,” in Proc. IEEE National Aerospace Electronics Conference (NAECON), B. Moore, ed., 1, 33–40 (1996).

G. W. Stewart, On the Early History of the Singular Value Decomposition, Technical Report TR-92-31, Dept. of Computer Science, University of Maryland, College Park, 1992, ftp://thales.cs.umd.edu/pub/reports/ehsvd.ps .

A. Berk, L. S. Bernstein, D. C. Robertson, “modtran, a moderate resolution model for lowtran 7,” Technical Report GL-TR-89-0122, Air Force Geophysics Laboratory, Hanscom Air Force Base, Mass., (1989).

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Figures (10)

Fig. 1
Fig. 1

Geometry of the hyperspectral detector.

Fig. 2
Fig. 2

Spectral blackbody radiance L λBB at three temperatures.

Fig. 3
Fig. 3

Atmospheric transmission over a range of 5 km excluding aerosol contributions.

Fig. 4
Fig. 4

Relative spectral irradiance s b at the sensor from the thermal background.

Fig. 5
Fig. 5

Relative target intensity, s t , as observed through a 5 km atmosphere at ground level.

Fig. 6
Fig. 6

Relative intensity of solar glints, s sun at ground level from a range of 5 km.

Fig. 7
Fig. 7

Ideal projection function for suppression of thermal background, and approximation by use of the first four filters selected by the matching pursuit algorithm.

Fig. 8
Fig. 8

Ideal projection function for suppression of solar glint, and approximation with the first four filters selected by the matching pursuit algorithm.

Fig. 9
Fig. 9

Ideal projection function for suppression of thermal background and solar glint, and approximation using the first four filters selected by the matching pursuit algorithm.

Fig. 10
Fig. 10

First ten filters, in order of selection, for suppression of thermal background and solar glints.

Tables (1)

Tables Icon

Table 1 Projections ‖Pzŝ‖ = |T ŝ| of unit-norm signals ŝ

Equations (26)

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

yi= wiωxωdω, i=1,  , B,
x=k=1Maksk+n=Sa+n,
x=k=1Paksk+k=P+1Maksk+n=defStat+Sbab+n.
H0 : x=Sbab+ntarget absent,H1 : x=Stat+Sbab+ntarget present.
GLRTx=px; â1, σˆ12px; â0, σˆ02,
px; a, σ2=2πσ2-N/2×exp-12σ2x-SaTx-Sa,
GLRTx=xTPbxxTPsxN/2,
Pb=I-SbSbTSb-1SbT=I-SbSb+
Tx=N-MPxTPb-PsxxTPsx=Pzx2/PPzPbx2/N-M,
1N-MPzPbx2=σˆ12
PzPbx2N-Mσ2χN-M2Pzx2Pσ2χP2under H0,
i=1LPzxi2LPσ2χLP2  under H0
T1x=Pzx21Li=1LPzxi2,
EW=minaz-Wa2=minaz-i=1Baiwi2.
z=wγ0wγ0Tz+Rz,γ0=arg minγΓz-wγTzwγ.
Rlz=wγlwγlTRlz+Rl+1z
z=k=0lwγkTRkzwγk+Rl+1z,
EW=minAZ-WAF2=minAm,nZmn-k=1Bwmkakn2=minAvecZ-Ip  WvecA2,
M  N=m11Nm12Nm1NNm21Nm22NmM1NmM2NmMNN,
Ip  W=W000W00W.
Rlzi=wγlwγlTRlzi+Rl+1zi.
Leffλ=τλRNλLλW m-2μm-1Sr-1,
RNλ=λλpeak,λλpeak0,otherwise.
zb=Pbst=I-Pbst=st-sbTsb-1sbTstsb=st-sˆbTstsˆb
zsun=Psunst=I-Psunst =st-ssunTssun-1ssunTstssun=st-sˆsunTstsˆsun
ztot=Ptotst=I-Ptotst=st-sbssunsbssunTsbssun-1sbssunTst

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