Abstract

New classes of spectral sensors are emerging that have significant overlap in the band spectral response functions. While conventional sensors such as the Multispectral Thermal Images (MTI) or Landsat may have responses with a few percent overlap between adjacent bands, some of the emerging sensors can have more than 50% correlation among all spectral bands. The traditional geometrical models used to describe spectral data fail when such high levels of correlation exist. In this paper we present a generalized geometrical model that relies on functional analysis. We define a sensor space and a scene space that can be used to characterize the suitability of a sensor for a particular spectral sensing task. We demonstrate that classifiers based on first-order distance and angle metrics fail for sensors with highly correlated bands unless appropriate preprocessing is carried out. We further show that second-order statistical classifiers are largely immune to many of the problems introduced by the correlated band responses.

© 2007 Optical Society of America

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References

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  1. L. O. Jimenez and D. A. Landgrebe, "Supervised classification in high-dimensional space: geometrical, statistical, and asymptotical properties of multivariate data," IEEE Trans. Syst. Man. Cybern., Part C Appl. Rev. 28, 39-54 (1998).
  2. J. A. Richards and X. Jia, Remote Sensing Digital Image Analysis (Springer, 1999).
  3. M. E. Leeser, P. Belanovic, M. Estlick, M. Gokhale, J. J. Szymanski, and J. P. Theiler, "Applying reconfigurable hardware to the analysis of multispectral and hyperspectral imagery," Proc. SPIE 4480, 100-107 (2002).
  4. D. Manolakis and G. A. Shaw, "Detection algorithms for hyperspectral imaging applications," IEEE Signal Process. Mag. 19, 29-43 (2002).
    [CrossRef]
  5. S. Krishna, S. Raghavan, G. von Winckel, P. Rotella, A. Stintz, C. P. Morath, D. Le, and S. W. Kennerly, "Two color InAs/InGaAs dots-in-a-well detector with background-limited performance at 91 K," Appl. Phys. Lett. 82, 2574-2576 (2003).
    [CrossRef]
  6. U. Sakoglu, J. S. Tyo, M. M. Hayat, S. Raghavan, and S. Krishna, "Spectrally adaptive infrared photodetectors using bias-tunable quantum dots," J. Opt. Soc. Am. B 21, 7-17 (2004).
    [CrossRef]
  7. U. Sakoglu, M. M. Hayat, J. S. Tyo, P. Dowd, S. Annamalai, K. T. Posani, and S. Krishna, "Statistical adaptive sensing using detectors with spectrally overlapping bands," Appl. Opt. 45, 7224-7234 (2006).
    [CrossRef]
  8. H. H. Barrett and K. J. Myers, Foundations of Image Science (Wiley-Interscience, 2003).
  9. J. W. Boardman, "Analysis, understanding, and visualization of hyperspectral data as convex sets in n-space," Proc. SPIE 2480, 14-22 (1995).
  10. J. S. Tyo, A. Konsolakis, D. I. Diersen, and R. C. Olsen, "Principal-components-based display strategy for spectral imagery," IEEE Trans. Geosci. Remote Sens. 41, 708-720 (2003).
  11. J. W. Boardman, "Automating spectral unmixing of AVIRIS data using convex geometry concepts," in Summaries of the 4th Annual JPL Airborne Geoscience Workshop, R.O.Green, ed. (Jet Propulsion Laboratory, 1993), JPL Pub 93-26, pp. 11-26.
  12. J. C. Harsanyi and C.-I. Chang, "Hyperspectral image classification and dimensionality reduction: an orthogonal subspace projection approach," IEEE Trans. Geosci. Remote Sens. 32, 779-785 (1994).
    [CrossRef]
  13. G. Healey and D. Slater, "Models and methods for automated material identification in hyperspectral imagery acquired under unknown illumination and atmospheric conditions," IEEE Trans. Geosci. Remote Sens. 37, 2706-2717 (1999).
    [CrossRef]
  14. ASTER spectral library, http://speclib.jpl.nasa.gov/.
  15. Note that the 7-band data has nearly identical classification performance to the 50-band data for second-order classifiers. This artificially high rate is due to the fact that there is no additional noise added in the synthesis process.
  16. G. H. Golub and C. F. van Loan, Matrix Computations (Johns Hopkins U. Press, 1983).
  17. I. Daubecheis, Ten Lectures on Wavelets (SIAM, 1992), pp. 53-106.
  18. N. Keshava and J. F. Mustard, "Spectral unmixing," IEEE Signal Process. Mag. 19, 44-57 (2002).
  19. W. R. Bell, "MTI: overview," Proc. SPIE 4381, 173-183 (2001).
  20. G. Vane, R. Green, T. Chrien, H. Enmark, E. Hansen, and W. Porter, "The airborne visible infrared imaging spectrometer," Remote Sens. Environ. 44, 127-143 (1993).
    [CrossRef]
  21. J. S. Pearlman, P. S. Barry, C. C. Segal, J. Shepanski, D. Beiso, and S. L. Carman, "Hyperion, a space-based imaging spectrometer," IEEE Trans. Geosci. Remote Sens. 41, 1160-1173 (2003).

2006 (1)

2004 (1)

2003 (3)

J. S. Pearlman, P. S. Barry, C. C. Segal, J. Shepanski, D. Beiso, and S. L. Carman, "Hyperion, a space-based imaging spectrometer," IEEE Trans. Geosci. Remote Sens. 41, 1160-1173 (2003).

S. Krishna, S. Raghavan, G. von Winckel, P. Rotella, A. Stintz, C. P. Morath, D. Le, and S. W. Kennerly, "Two color InAs/InGaAs dots-in-a-well detector with background-limited performance at 91 K," Appl. Phys. Lett. 82, 2574-2576 (2003).
[CrossRef]

J. S. Tyo, A. Konsolakis, D. I. Diersen, and R. C. Olsen, "Principal-components-based display strategy for spectral imagery," IEEE Trans. Geosci. Remote Sens. 41, 708-720 (2003).

2002 (3)

M. E. Leeser, P. Belanovic, M. Estlick, M. Gokhale, J. J. Szymanski, and J. P. Theiler, "Applying reconfigurable hardware to the analysis of multispectral and hyperspectral imagery," Proc. SPIE 4480, 100-107 (2002).

D. Manolakis and G. A. Shaw, "Detection algorithms for hyperspectral imaging applications," IEEE Signal Process. Mag. 19, 29-43 (2002).
[CrossRef]

N. Keshava and J. F. Mustard, "Spectral unmixing," IEEE Signal Process. Mag. 19, 44-57 (2002).

1999 (1)

G. Healey and D. Slater, "Models and methods for automated material identification in hyperspectral imagery acquired under unknown illumination and atmospheric conditions," IEEE Trans. Geosci. Remote Sens. 37, 2706-2717 (1999).
[CrossRef]

1998 (1)

L. O. Jimenez and D. A. Landgrebe, "Supervised classification in high-dimensional space: geometrical, statistical, and asymptotical properties of multivariate data," IEEE Trans. Syst. Man. Cybern., Part C Appl. Rev. 28, 39-54 (1998).

1995 (1)

J. W. Boardman, "Analysis, understanding, and visualization of hyperspectral data as convex sets in n-space," Proc. SPIE 2480, 14-22 (1995).

1994 (1)

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

1993 (1)

G. Vane, R. Green, T. Chrien, H. Enmark, E. Hansen, and W. Porter, "The airborne visible infrared imaging spectrometer," Remote Sens. Environ. 44, 127-143 (1993).
[CrossRef]

Annamalai, S.

Barrett, H. H.

H. H. Barrett and K. J. Myers, Foundations of Image Science (Wiley-Interscience, 2003).

Barry, P. S.

J. S. Pearlman, P. S. Barry, C. C. Segal, J. Shepanski, D. Beiso, and S. L. Carman, "Hyperion, a space-based imaging spectrometer," IEEE Trans. Geosci. Remote Sens. 41, 1160-1173 (2003).

Beiso, D.

J. S. Pearlman, P. S. Barry, C. C. Segal, J. Shepanski, D. Beiso, and S. L. Carman, "Hyperion, a space-based imaging spectrometer," IEEE Trans. Geosci. Remote Sens. 41, 1160-1173 (2003).

Belanovic, P.

M. E. Leeser, P. Belanovic, M. Estlick, M. Gokhale, J. J. Szymanski, and J. P. Theiler, "Applying reconfigurable hardware to the analysis of multispectral and hyperspectral imagery," Proc. SPIE 4480, 100-107 (2002).

Bell, W. R.

W. R. Bell, "MTI: overview," Proc. SPIE 4381, 173-183 (2001).

Boardman, J. W.

J. W. Boardman, "Analysis, understanding, and visualization of hyperspectral data as convex sets in n-space," Proc. SPIE 2480, 14-22 (1995).

J. W. Boardman, "Automating spectral unmixing of AVIRIS data using convex geometry concepts," in Summaries of the 4th Annual JPL Airborne Geoscience Workshop, R.O.Green, ed. (Jet Propulsion Laboratory, 1993), JPL Pub 93-26, pp. 11-26.

Carman, S. L.

J. S. Pearlman, P. S. Barry, C. C. Segal, J. Shepanski, D. Beiso, and S. L. Carman, "Hyperion, a space-based imaging spectrometer," IEEE Trans. Geosci. Remote Sens. 41, 1160-1173 (2003).

Chang, C.-I.

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

Chrien, T.

G. Vane, R. Green, T. Chrien, H. Enmark, E. Hansen, and W. Porter, "The airborne visible infrared imaging spectrometer," Remote Sens. Environ. 44, 127-143 (1993).
[CrossRef]

Daubecheis, I.

I. Daubecheis, Ten Lectures on Wavelets (SIAM, 1992), pp. 53-106.

Diersen, D. I.

J. S. Tyo, A. Konsolakis, D. I. Diersen, and R. C. Olsen, "Principal-components-based display strategy for spectral imagery," IEEE Trans. Geosci. Remote Sens. 41, 708-720 (2003).

Dowd, P.

Enmark, H.

G. Vane, R. Green, T. Chrien, H. Enmark, E. Hansen, and W. Porter, "The airborne visible infrared imaging spectrometer," Remote Sens. Environ. 44, 127-143 (1993).
[CrossRef]

Estlick, M.

M. E. Leeser, P. Belanovic, M. Estlick, M. Gokhale, J. J. Szymanski, and J. P. Theiler, "Applying reconfigurable hardware to the analysis of multispectral and hyperspectral imagery," Proc. SPIE 4480, 100-107 (2002).

Gokhale, M.

M. E. Leeser, P. Belanovic, M. Estlick, M. Gokhale, J. J. Szymanski, and J. P. Theiler, "Applying reconfigurable hardware to the analysis of multispectral and hyperspectral imagery," Proc. SPIE 4480, 100-107 (2002).

Golub, G. H.

G. H. Golub and C. F. van Loan, Matrix Computations (Johns Hopkins U. Press, 1983).

Green, R.

G. Vane, R. Green, T. Chrien, H. Enmark, E. Hansen, and W. Porter, "The airborne visible infrared imaging spectrometer," Remote Sens. Environ. 44, 127-143 (1993).
[CrossRef]

Hansen, E.

G. Vane, R. Green, T. Chrien, H. Enmark, E. Hansen, and W. Porter, "The airborne visible infrared imaging spectrometer," Remote Sens. Environ. 44, 127-143 (1993).
[CrossRef]

Harsanyi, J. C.

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

Hayat, M. M.

Healey, G.

G. Healey and D. Slater, "Models and methods for automated material identification in hyperspectral imagery acquired under unknown illumination and atmospheric conditions," IEEE Trans. Geosci. Remote Sens. 37, 2706-2717 (1999).
[CrossRef]

Jia, X.

J. A. Richards and X. Jia, Remote Sensing Digital Image Analysis (Springer, 1999).

Jimenez, L. O.

L. O. Jimenez and D. A. Landgrebe, "Supervised classification in high-dimensional space: geometrical, statistical, and asymptotical properties of multivariate data," IEEE Trans. Syst. Man. Cybern., Part C Appl. Rev. 28, 39-54 (1998).

Kennerly, S. W.

S. Krishna, S. Raghavan, G. von Winckel, P. Rotella, A. Stintz, C. P. Morath, D. Le, and S. W. Kennerly, "Two color InAs/InGaAs dots-in-a-well detector with background-limited performance at 91 K," Appl. Phys. Lett. 82, 2574-2576 (2003).
[CrossRef]

Keshava, N.

N. Keshava and J. F. Mustard, "Spectral unmixing," IEEE Signal Process. Mag. 19, 44-57 (2002).

Konsolakis, A.

J. S. Tyo, A. Konsolakis, D. I. Diersen, and R. C. Olsen, "Principal-components-based display strategy for spectral imagery," IEEE Trans. Geosci. Remote Sens. 41, 708-720 (2003).

Krishna, S.

Landgrebe, D. A.

L. O. Jimenez and D. A. Landgrebe, "Supervised classification in high-dimensional space: geometrical, statistical, and asymptotical properties of multivariate data," IEEE Trans. Syst. Man. Cybern., Part C Appl. Rev. 28, 39-54 (1998).

Le, D.

S. Krishna, S. Raghavan, G. von Winckel, P. Rotella, A. Stintz, C. P. Morath, D. Le, and S. W. Kennerly, "Two color InAs/InGaAs dots-in-a-well detector with background-limited performance at 91 K," Appl. Phys. Lett. 82, 2574-2576 (2003).
[CrossRef]

Leeser, M. E.

M. E. Leeser, P. Belanovic, M. Estlick, M. Gokhale, J. J. Szymanski, and J. P. Theiler, "Applying reconfigurable hardware to the analysis of multispectral and hyperspectral imagery," Proc. SPIE 4480, 100-107 (2002).

Manolakis, D.

D. Manolakis and G. A. Shaw, "Detection algorithms for hyperspectral imaging applications," IEEE Signal Process. Mag. 19, 29-43 (2002).
[CrossRef]

Morath, C. P.

S. Krishna, S. Raghavan, G. von Winckel, P. Rotella, A. Stintz, C. P. Morath, D. Le, and S. W. Kennerly, "Two color InAs/InGaAs dots-in-a-well detector with background-limited performance at 91 K," Appl. Phys. Lett. 82, 2574-2576 (2003).
[CrossRef]

Mustard, J. F.

N. Keshava and J. F. Mustard, "Spectral unmixing," IEEE Signal Process. Mag. 19, 44-57 (2002).

Myers, K. J.

H. H. Barrett and K. J. Myers, Foundations of Image Science (Wiley-Interscience, 2003).

Olsen, R. C.

J. S. Tyo, A. Konsolakis, D. I. Diersen, and R. C. Olsen, "Principal-components-based display strategy for spectral imagery," IEEE Trans. Geosci. Remote Sens. 41, 708-720 (2003).

Pearlman, J. S.

J. S. Pearlman, P. S. Barry, C. C. Segal, J. Shepanski, D. Beiso, and S. L. Carman, "Hyperion, a space-based imaging spectrometer," IEEE Trans. Geosci. Remote Sens. 41, 1160-1173 (2003).

Porter, W.

G. Vane, R. Green, T. Chrien, H. Enmark, E. Hansen, and W. Porter, "The airborne visible infrared imaging spectrometer," Remote Sens. Environ. 44, 127-143 (1993).
[CrossRef]

Posani, K. T.

Raghavan, S.

U. Sakoglu, J. S. Tyo, M. M. Hayat, S. Raghavan, and S. Krishna, "Spectrally adaptive infrared photodetectors using bias-tunable quantum dots," J. Opt. Soc. Am. B 21, 7-17 (2004).
[CrossRef]

S. Krishna, S. Raghavan, G. von Winckel, P. Rotella, A. Stintz, C. P. Morath, D. Le, and S. W. Kennerly, "Two color InAs/InGaAs dots-in-a-well detector with background-limited performance at 91 K," Appl. Phys. Lett. 82, 2574-2576 (2003).
[CrossRef]

Richards, J. A.

J. A. Richards and X. Jia, Remote Sensing Digital Image Analysis (Springer, 1999).

Rotella, P.

S. Krishna, S. Raghavan, G. von Winckel, P. Rotella, A. Stintz, C. P. Morath, D. Le, and S. W. Kennerly, "Two color InAs/InGaAs dots-in-a-well detector with background-limited performance at 91 K," Appl. Phys. Lett. 82, 2574-2576 (2003).
[CrossRef]

Sakoglu, U.

Segal, C. C.

J. S. Pearlman, P. S. Barry, C. C. Segal, J. Shepanski, D. Beiso, and S. L. Carman, "Hyperion, a space-based imaging spectrometer," IEEE Trans. Geosci. Remote Sens. 41, 1160-1173 (2003).

Shaw, G. A.

D. Manolakis and G. A. Shaw, "Detection algorithms for hyperspectral imaging applications," IEEE Signal Process. Mag. 19, 29-43 (2002).
[CrossRef]

Shepanski, J.

J. S. Pearlman, P. S. Barry, C. C. Segal, J. Shepanski, D. Beiso, and S. L. Carman, "Hyperion, a space-based imaging spectrometer," IEEE Trans. Geosci. Remote Sens. 41, 1160-1173 (2003).

Slater, D.

G. Healey and D. Slater, "Models and methods for automated material identification in hyperspectral imagery acquired under unknown illumination and atmospheric conditions," IEEE Trans. Geosci. Remote Sens. 37, 2706-2717 (1999).
[CrossRef]

Stintz, A.

S. Krishna, S. Raghavan, G. von Winckel, P. Rotella, A. Stintz, C. P. Morath, D. Le, and S. W. Kennerly, "Two color InAs/InGaAs dots-in-a-well detector with background-limited performance at 91 K," Appl. Phys. Lett. 82, 2574-2576 (2003).
[CrossRef]

Szymanski, J. J.

M. E. Leeser, P. Belanovic, M. Estlick, M. Gokhale, J. J. Szymanski, and J. P. Theiler, "Applying reconfigurable hardware to the analysis of multispectral and hyperspectral imagery," Proc. SPIE 4480, 100-107 (2002).

Theiler, J. P.

M. E. Leeser, P. Belanovic, M. Estlick, M. Gokhale, J. J. Szymanski, and J. P. Theiler, "Applying reconfigurable hardware to the analysis of multispectral and hyperspectral imagery," Proc. SPIE 4480, 100-107 (2002).

Tyo, J. S.

van Loan, C. F.

G. H. Golub and C. F. van Loan, Matrix Computations (Johns Hopkins U. Press, 1983).

Vane, G.

G. Vane, R. Green, T. Chrien, H. Enmark, E. Hansen, and W. Porter, "The airborne visible infrared imaging spectrometer," Remote Sens. Environ. 44, 127-143 (1993).
[CrossRef]

von Winckel, G.

S. Krishna, S. Raghavan, G. von Winckel, P. Rotella, A. Stintz, C. P. Morath, D. Le, and S. W. Kennerly, "Two color InAs/InGaAs dots-in-a-well detector with background-limited performance at 91 K," Appl. Phys. Lett. 82, 2574-2576 (2003).
[CrossRef]

Appl. Opt. (1)

Appl. Phys. Lett. (1)

S. Krishna, S. Raghavan, G. von Winckel, P. Rotella, A. Stintz, C. P. Morath, D. Le, and S. W. Kennerly, "Two color InAs/InGaAs dots-in-a-well detector with background-limited performance at 91 K," Appl. Phys. Lett. 82, 2574-2576 (2003).
[CrossRef]

IEEE Signal Process. Mag. (2)

N. Keshava and J. F. Mustard, "Spectral unmixing," IEEE Signal Process. Mag. 19, 44-57 (2002).

D. Manolakis and G. A. Shaw, "Detection algorithms for hyperspectral imaging applications," IEEE Signal Process. Mag. 19, 29-43 (2002).
[CrossRef]

IEEE Trans. Geosci. Remote Sens. (4)

J. S. Pearlman, P. S. Barry, C. C. Segal, J. Shepanski, D. Beiso, and S. L. Carman, "Hyperion, a space-based imaging spectrometer," IEEE Trans. Geosci. Remote Sens. 41, 1160-1173 (2003).

J. S. Tyo, A. Konsolakis, D. I. Diersen, and R. C. Olsen, "Principal-components-based display strategy for spectral imagery," IEEE Trans. Geosci. Remote Sens. 41, 708-720 (2003).

J. C. Harsanyi and 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. Healey and D. Slater, "Models and methods for automated material identification in hyperspectral imagery acquired under unknown illumination and atmospheric conditions," IEEE Trans. Geosci. Remote Sens. 37, 2706-2717 (1999).
[CrossRef]

IEEE Trans. Syst. Man. Cybern., Part C Appl. Rev. (1)

L. O. Jimenez and D. A. Landgrebe, "Supervised classification in high-dimensional space: geometrical, statistical, and asymptotical properties of multivariate data," IEEE Trans. Syst. Man. Cybern., Part C Appl. Rev. 28, 39-54 (1998).

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

Proc. SPIE (2)

M. E. Leeser, P. Belanovic, M. Estlick, M. Gokhale, J. J. Szymanski, and J. P. Theiler, "Applying reconfigurable hardware to the analysis of multispectral and hyperspectral imagery," Proc. SPIE 4480, 100-107 (2002).

J. W. Boardman, "Analysis, understanding, and visualization of hyperspectral data as convex sets in n-space," Proc. SPIE 2480, 14-22 (1995).

Remote Sens. Environ. (1)

G. Vane, R. Green, T. Chrien, H. Enmark, E. Hansen, and W. Porter, "The airborne visible infrared imaging spectrometer," Remote Sens. Environ. 44, 127-143 (1993).
[CrossRef]

Other (8)

J. A. Richards and X. Jia, Remote Sensing Digital Image Analysis (Springer, 1999).

W. R. Bell, "MTI: overview," Proc. SPIE 4381, 173-183 (2001).

H. H. Barrett and K. J. Myers, Foundations of Image Science (Wiley-Interscience, 2003).

J. W. Boardman, "Automating spectral unmixing of AVIRIS data using convex geometry concepts," in Summaries of the 4th Annual JPL Airborne Geoscience Workshop, R.O.Green, ed. (Jet Propulsion Laboratory, 1993), JPL Pub 93-26, pp. 11-26.

ASTER spectral library, http://speclib.jpl.nasa.gov/.

Note that the 7-band data has nearly identical classification performance to the 50-band data for second-order classifiers. This artificially high rate is due to the fact that there is no additional noise added in the synthesis process.

G. H. Golub and C. F. van Loan, Matrix Computations (Johns Hopkins U. Press, 1983).

I. Daubecheis, Ten Lectures on Wavelets (SIAM, 1992), pp. 53-106.

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Figures (6)

Fig. 1
Fig. 1

QDIP spectral responses measured from a 10-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

(a) ETM + image of a site near Canon City, Colorado (band 1, image of band 2 looks similar). This image can be found in data CD of ENVI 4.0 [filename]. (b) Spectral responses of ETM + for Landsat 7, bands 1 and 2. (c) 2D-scatter plot of data of TM bands 1 and 2. (d) Biorthogonal representation of the Landsat and simulated responses. (e) Simulated spectral bands created by superposition of Landsat bands 1 and 2. (f) 2D scatterplot of output images of simulated sensor in Fig. 2e.

Fig. 3
Fig. 3

Seven bias-dependent QDIP responses used for the simulation in Sections 2, 4.

Fig. 4
Fig. 4

Process flow for creating simulated hyperspectral images in the 3–5 and 8 12 μ m spectral ranges. We start with AVIRIS images in the 1.9 2.4 μ m range, then unmix the scene using in-scene endmembers. These endmembers are identified, and the spectra are extrapolated to the midwave IR and long-wave IR by using signatures in the ASTER database. The extrapolated spectra are used to remix the hyperspectral images.

Fig. 5
Fig. 5

Minimum Euclidean distance classification results for the (a) AVIRIS scene and (b) simulated sensor data.

Fig. 6
Fig. 6

Minimum distance classification results after preprocessing as indicated in Eq. (14). Results are compared with Fig. 5.

Tables (1)

Tables Icon

Table 1 Description of the Band Layout of Popular Spectral Sensors

Equations (18)

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

r 1 = r TM 1 + 2 r TM 2 ,
r 2 = 2 r TM 1 + 1.5 r TM 2 ,
R i ( λ ) = R 0 r i ( λ ) .
ρ i j = r i ( λ ) r j ( λ ) d λ r i 2 d λ r j 2 d λ cos θ i j .
T ( λ ) = T 0 t ( λ ) .
f ( λ ) , g ( λ ) = λ f ( λ ) g ( λ ) d λ .
x = [ T ( λ ) , R 1 ( λ ) T ( λ ) , R p ( λ ) ] T ,
T ( λ ) , R i ( λ ) = A T 0 R 0 r i T t = A T 0 R 0 t ( λ ) r i ( λ ) d λ .
x = R T t ,
R = [ r 1 r p ] .
t = Z x [ R ( R T R ) 1 ] R T t .
Z = [ z 1 z p ] ,
z i T r j = δ i j
t ( λ ) = i r i ( λ ) , t ( λ ) z i ( λ ) .
t = Z x = U ( S 1 V T x ) = U x .
x n = R T t + n ,
x n = P R t + P n
M i = ( x μ ¯ i ) T Σ i 1 ( x μ ¯ i ) ,

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