Abstract

Improvements to an algorithm for performing spectral unmixing of hyperspectral imagery based on the stochastic mixing model (SMM) are presented. The SMM provides a method for characterizing both subpixel mixing of the pure image constituents, or endmembers, and statistical variation in the endmember spectra that is due, for example, to sensor noise and natural variability of the pure constituents. Modifications of the iterative, expectation maximization approach to deriving the SMM parameter estimates are proposed, and their effects on unmixing performance are characterized. These modifications specifically concern algorithm initialization, random class assignment, and mixture constraints. The results show that the enhanced stochastic mixing model provides a better statistical representation of hyperspectral imagery from the perspective of achieving greater endmember class separation.

© 2004 Optical Society of America

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  1. N. Keshava, J. F. Mustard, “Spectral unmixing,” IEEE Signal Process. Mag. 19, 44–57 (2002).
    [Crossref]
  2. R. C. Hardie, M. T. Eismann, “MAP estimation for hyperspectral image resolution enhancement using an auxiliary sensor,” IEEE Trans. Image Process. 13, 1174–1184 (2004).
    [Crossref] [PubMed]
  3. R. L. Sundberg, J. H. Gruninger, R. Haren, “Extraction of hyperspectral scene statistics and scene realization,” in Algorithms and Technologies for Multispectral, Hyperspectral, and Ultraspectral Imagery VIII, S. S. Shen, P. E. Lewis, eds., Proc. SPIE4725, 184–194 (2002).
    [Crossref]
  4. K. Fukunaga, Introduction to Statistical Pattern Recognition, 2nd ed. (Academic, San Diego, Calif., 1990), Chap. 11.
  5. A. D. Stocker, A. P. Schaum, “Application of stochastic mixing models to hyperspectral detection problems,” in Algorithms and Technologies for Multispectral, Hyperspectral, and Ultraspectral Imagery III, A. E. Iverson, S. S. Shen, eds., Proc. SPIE3071, 47–60 (1997).
  6. R. A. Redner, H. F. Walker, “Mixture densities, maximum likelihood, and the EM algorithm,” SIAM Rev. 26, 195–239 (1984).
    [Crossref]
  7. D. W. Stein, “Stochastic compositional models applied to subpixel analysis of hyperspectral imagery,” in Imaging Spectrometry VII, M. R. Descour, S. S. Shen, eds., Proc. SPIE4480, 49–56 (2002).
    [Crossref]
  8. P. Masson, W. Pieczynski, “SEM algorithm and unsupervised statistical segmentation of satellite images,” IEEE Trans. Geosci. Remote Sens. 31, 618–633 (1993).
    [Crossref]
  9. R. A. Schowengerdt, Remote Sensing: Models and Methods for Image Processing, 2nd ed. (Academic, San Diego, Calif., 1997).
  10. M. E. Winter, “Fast autonomous endmember determination in hyperspectral data,” in Proceedings of the 13th International Conference on Applied Geological Remote Sensing (Environmental Research Institute of Michigan, Ann Arbor, Mich., 1999), Vol. II, pp. 337–344.
  11. H. Stark, J. W. Woods, Probability and Random Processes with Applications to Signal Processing (Prentice-Hall, Upper Saddle River, N.J., 2002), pp. 28–30.
  12. Ref. 4, p. 446.
  13. J. T. Tou, R. C. Gonzalez, Pattern Recognition Principles (Addison-Wesley, Reading, Mass., 1974), p. 87.
  14. J. R. Schott, R. Raqueno, C. Salvaggio, “Incorporation of time-dependent thermodynamic model and radiation propagation model into infrared three-dimensional synthetic image generation,” Opt. Eng. 31, 1505–1516 (1992).
    [Crossref]
  15. C. Simi, J. Parish, E. M. Winter, R. Dixon, C. LaSota, M. M. Williams, “Night vision imaging spectrometer (NVIS) performance parameters and their impact on various detection algorithms,” in Algorithms for Multispectral, Hyperspectral, and Ultraspectral Imagery VI, S. S. Shen, M. R. Descour, eds., Proc. SPIE4049, 218–229 (2000).
    [Crossref]
  16. J. Pearlman, S. Carman, C. Segal, P. Jarecke, P. Clancy, W. Browne, “Overview of the Hyperion imaging spectrometer for the NASA EO-1 mission,” in Proceedings of the 2002 International Geoscience and Remote Sensing Symposium (IEEE, New York, 2002), pp. 3036–3038.
  17. J. A. Hackwell, D. W. Warren, R. P. Bongiovi, S. J. Hansel, T. L. Hayhurst, D. J. Mabry, M. G. Sivjee, J. W. Skinner, “LWIR/MWIR imaging hyperspectral sensor for airborne and ground-based remote sensing,” in Imaging Spectrometry II, M. R. Descour, J. M. Mooney, eds., Proc. SPIE2819, 102–107 (1996).
    [Crossref]
  18. P. G. Lucey, T. Williams, M. Mignard, J. Julian, D. Kokobun, G. Allen, D. Hampton, W. Schaff, M. Schlangen, E. M. Winter, A. Stocker, K. Horton, A. P. Bowman, “AHI: an airborne long wave infrared hyperspectral imager,” in Airborne Reconnaissance XXII, W. G. Fishell, A. A. Andraitis, M. S. Fagan, J. D. Greer, M. C. Norton, eds., Proc. SPIE3431, 36–43 (1998).
    [Crossref]
  19. M. T. Eismann, R. C. Hardie, “Application of the stochastic mixing model to hyperspectral resolution enhancement,” IEEE Trans. Geosci. Remote Sens. 42, 1924–1933 (2004).
    [Crossref]
  20. M. T. Eismann, R. C. Hardie, “Hyperspectral resolution enhancement using high resolution multispectral imagery with arbitrary response functions,” IEEE Trans. Geosci. Remote Sens. (to be published).

2004 (2)

R. C. Hardie, M. T. Eismann, “MAP estimation for hyperspectral image resolution enhancement using an auxiliary sensor,” IEEE Trans. Image Process. 13, 1174–1184 (2004).
[Crossref] [PubMed]

M. T. Eismann, R. C. Hardie, “Application of the stochastic mixing model to hyperspectral resolution enhancement,” IEEE Trans. Geosci. Remote Sens. 42, 1924–1933 (2004).
[Crossref]

2002 (1)

N. Keshava, J. F. Mustard, “Spectral unmixing,” IEEE Signal Process. Mag. 19, 44–57 (2002).
[Crossref]

1993 (1)

P. Masson, W. Pieczynski, “SEM algorithm and unsupervised statistical segmentation of satellite images,” IEEE Trans. Geosci. Remote Sens. 31, 618–633 (1993).
[Crossref]

1992 (1)

J. R. Schott, R. Raqueno, C. Salvaggio, “Incorporation of time-dependent thermodynamic model and radiation propagation model into infrared three-dimensional synthetic image generation,” Opt. Eng. 31, 1505–1516 (1992).
[Crossref]

1984 (1)

R. A. Redner, H. F. Walker, “Mixture densities, maximum likelihood, and the EM algorithm,” SIAM Rev. 26, 195–239 (1984).
[Crossref]

Allen, G.

P. G. Lucey, T. Williams, M. Mignard, J. Julian, D. Kokobun, G. Allen, D. Hampton, W. Schaff, M. Schlangen, E. M. Winter, A. Stocker, K. Horton, A. P. Bowman, “AHI: an airborne long wave infrared hyperspectral imager,” in Airborne Reconnaissance XXII, W. G. Fishell, A. A. Andraitis, M. S. Fagan, J. D. Greer, M. C. Norton, eds., Proc. SPIE3431, 36–43 (1998).
[Crossref]

Bongiovi, R. P.

J. A. Hackwell, D. W. Warren, R. P. Bongiovi, S. J. Hansel, T. L. Hayhurst, D. J. Mabry, M. G. Sivjee, J. W. Skinner, “LWIR/MWIR imaging hyperspectral sensor for airborne and ground-based remote sensing,” in Imaging Spectrometry II, M. R. Descour, J. M. Mooney, eds., Proc. SPIE2819, 102–107 (1996).
[Crossref]

Bowman, A. P.

P. G. Lucey, T. Williams, M. Mignard, J. Julian, D. Kokobun, G. Allen, D. Hampton, W. Schaff, M. Schlangen, E. M. Winter, A. Stocker, K. Horton, A. P. Bowman, “AHI: an airborne long wave infrared hyperspectral imager,” in Airborne Reconnaissance XXII, W. G. Fishell, A. A. Andraitis, M. S. Fagan, J. D. Greer, M. C. Norton, eds., Proc. SPIE3431, 36–43 (1998).
[Crossref]

Browne, W.

J. Pearlman, S. Carman, C. Segal, P. Jarecke, P. Clancy, W. Browne, “Overview of the Hyperion imaging spectrometer for the NASA EO-1 mission,” in Proceedings of the 2002 International Geoscience and Remote Sensing Symposium (IEEE, New York, 2002), pp. 3036–3038.

Carman, S.

J. Pearlman, S. Carman, C. Segal, P. Jarecke, P. Clancy, W. Browne, “Overview of the Hyperion imaging spectrometer for the NASA EO-1 mission,” in Proceedings of the 2002 International Geoscience and Remote Sensing Symposium (IEEE, New York, 2002), pp. 3036–3038.

Clancy, P.

J. Pearlman, S. Carman, C. Segal, P. Jarecke, P. Clancy, W. Browne, “Overview of the Hyperion imaging spectrometer for the NASA EO-1 mission,” in Proceedings of the 2002 International Geoscience and Remote Sensing Symposium (IEEE, New York, 2002), pp. 3036–3038.

Dixon, R.

C. Simi, J. Parish, E. M. Winter, R. Dixon, C. LaSota, M. M. Williams, “Night vision imaging spectrometer (NVIS) performance parameters and their impact on various detection algorithms,” in Algorithms for Multispectral, Hyperspectral, and Ultraspectral Imagery VI, S. S. Shen, M. R. Descour, eds., Proc. SPIE4049, 218–229 (2000).
[Crossref]

Eismann, M. T.

R. C. Hardie, M. T. Eismann, “MAP estimation for hyperspectral image resolution enhancement using an auxiliary sensor,” IEEE Trans. Image Process. 13, 1174–1184 (2004).
[Crossref] [PubMed]

M. T. Eismann, R. C. Hardie, “Application of the stochastic mixing model to hyperspectral resolution enhancement,” IEEE Trans. Geosci. Remote Sens. 42, 1924–1933 (2004).
[Crossref]

M. T. Eismann, R. C. Hardie, “Hyperspectral resolution enhancement using high resolution multispectral imagery with arbitrary response functions,” IEEE Trans. Geosci. Remote Sens. (to be published).

Fukunaga, K.

K. Fukunaga, Introduction to Statistical Pattern Recognition, 2nd ed. (Academic, San Diego, Calif., 1990), Chap. 11.

Gonzalez, R. C.

J. T. Tou, R. C. Gonzalez, Pattern Recognition Principles (Addison-Wesley, Reading, Mass., 1974), p. 87.

Gruninger, J. H.

R. L. Sundberg, J. H. Gruninger, R. Haren, “Extraction of hyperspectral scene statistics and scene realization,” in Algorithms and Technologies for Multispectral, Hyperspectral, and Ultraspectral Imagery VIII, S. S. Shen, P. E. Lewis, eds., Proc. SPIE4725, 184–194 (2002).
[Crossref]

Hackwell, J. A.

J. A. Hackwell, D. W. Warren, R. P. Bongiovi, S. J. Hansel, T. L. Hayhurst, D. J. Mabry, M. G. Sivjee, J. W. Skinner, “LWIR/MWIR imaging hyperspectral sensor for airborne and ground-based remote sensing,” in Imaging Spectrometry II, M. R. Descour, J. M. Mooney, eds., Proc. SPIE2819, 102–107 (1996).
[Crossref]

Hampton, D.

P. G. Lucey, T. Williams, M. Mignard, J. Julian, D. Kokobun, G. Allen, D. Hampton, W. Schaff, M. Schlangen, E. M. Winter, A. Stocker, K. Horton, A. P. Bowman, “AHI: an airborne long wave infrared hyperspectral imager,” in Airborne Reconnaissance XXII, W. G. Fishell, A. A. Andraitis, M. S. Fagan, J. D. Greer, M. C. Norton, eds., Proc. SPIE3431, 36–43 (1998).
[Crossref]

Hansel, S. J.

J. A. Hackwell, D. W. Warren, R. P. Bongiovi, S. J. Hansel, T. L. Hayhurst, D. J. Mabry, M. G. Sivjee, J. W. Skinner, “LWIR/MWIR imaging hyperspectral sensor for airborne and ground-based remote sensing,” in Imaging Spectrometry II, M. R. Descour, J. M. Mooney, eds., Proc. SPIE2819, 102–107 (1996).
[Crossref]

Hardie, R. C.

M. T. Eismann, R. C. Hardie, “Application of the stochastic mixing model to hyperspectral resolution enhancement,” IEEE Trans. Geosci. Remote Sens. 42, 1924–1933 (2004).
[Crossref]

R. C. Hardie, M. T. Eismann, “MAP estimation for hyperspectral image resolution enhancement using an auxiliary sensor,” IEEE Trans. Image Process. 13, 1174–1184 (2004).
[Crossref] [PubMed]

M. T. Eismann, R. C. Hardie, “Hyperspectral resolution enhancement using high resolution multispectral imagery with arbitrary response functions,” IEEE Trans. Geosci. Remote Sens. (to be published).

Haren, R.

R. L. Sundberg, J. H. Gruninger, R. Haren, “Extraction of hyperspectral scene statistics and scene realization,” in Algorithms and Technologies for Multispectral, Hyperspectral, and Ultraspectral Imagery VIII, S. S. Shen, P. E. Lewis, eds., Proc. SPIE4725, 184–194 (2002).
[Crossref]

Hayhurst, T. L.

J. A. Hackwell, D. W. Warren, R. P. Bongiovi, S. J. Hansel, T. L. Hayhurst, D. J. Mabry, M. G. Sivjee, J. W. Skinner, “LWIR/MWIR imaging hyperspectral sensor for airborne and ground-based remote sensing,” in Imaging Spectrometry II, M. R. Descour, J. M. Mooney, eds., Proc. SPIE2819, 102–107 (1996).
[Crossref]

Horton, K.

P. G. Lucey, T. Williams, M. Mignard, J. Julian, D. Kokobun, G. Allen, D. Hampton, W. Schaff, M. Schlangen, E. M. Winter, A. Stocker, K. Horton, A. P. Bowman, “AHI: an airborne long wave infrared hyperspectral imager,” in Airborne Reconnaissance XXII, W. G. Fishell, A. A. Andraitis, M. S. Fagan, J. D. Greer, M. C. Norton, eds., Proc. SPIE3431, 36–43 (1998).
[Crossref]

Jarecke, P.

J. Pearlman, S. Carman, C. Segal, P. Jarecke, P. Clancy, W. Browne, “Overview of the Hyperion imaging spectrometer for the NASA EO-1 mission,” in Proceedings of the 2002 International Geoscience and Remote Sensing Symposium (IEEE, New York, 2002), pp. 3036–3038.

Julian, J.

P. G. Lucey, T. Williams, M. Mignard, J. Julian, D. Kokobun, G. Allen, D. Hampton, W. Schaff, M. Schlangen, E. M. Winter, A. Stocker, K. Horton, A. P. Bowman, “AHI: an airborne long wave infrared hyperspectral imager,” in Airborne Reconnaissance XXII, W. G. Fishell, A. A. Andraitis, M. S. Fagan, J. D. Greer, M. C. Norton, eds., Proc. SPIE3431, 36–43 (1998).
[Crossref]

Keshava, N.

N. Keshava, J. F. Mustard, “Spectral unmixing,” IEEE Signal Process. Mag. 19, 44–57 (2002).
[Crossref]

Kokobun, D.

P. G. Lucey, T. Williams, M. Mignard, J. Julian, D. Kokobun, G. Allen, D. Hampton, W. Schaff, M. Schlangen, E. M. Winter, A. Stocker, K. Horton, A. P. Bowman, “AHI: an airborne long wave infrared hyperspectral imager,” in Airborne Reconnaissance XXII, W. G. Fishell, A. A. Andraitis, M. S. Fagan, J. D. Greer, M. C. Norton, eds., Proc. SPIE3431, 36–43 (1998).
[Crossref]

LaSota, C.

C. Simi, J. Parish, E. M. Winter, R. Dixon, C. LaSota, M. M. Williams, “Night vision imaging spectrometer (NVIS) performance parameters and their impact on various detection algorithms,” in Algorithms for Multispectral, Hyperspectral, and Ultraspectral Imagery VI, S. S. Shen, M. R. Descour, eds., Proc. SPIE4049, 218–229 (2000).
[Crossref]

Lucey, P. G.

P. G. Lucey, T. Williams, M. Mignard, J. Julian, D. Kokobun, G. Allen, D. Hampton, W. Schaff, M. Schlangen, E. M. Winter, A. Stocker, K. Horton, A. P. Bowman, “AHI: an airborne long wave infrared hyperspectral imager,” in Airborne Reconnaissance XXII, W. G. Fishell, A. A. Andraitis, M. S. Fagan, J. D. Greer, M. C. Norton, eds., Proc. SPIE3431, 36–43 (1998).
[Crossref]

Mabry, D. J.

J. A. Hackwell, D. W. Warren, R. P. Bongiovi, S. J. Hansel, T. L. Hayhurst, D. J. Mabry, M. G. Sivjee, J. W. Skinner, “LWIR/MWIR imaging hyperspectral sensor for airborne and ground-based remote sensing,” in Imaging Spectrometry II, M. R. Descour, J. M. Mooney, eds., Proc. SPIE2819, 102–107 (1996).
[Crossref]

Masson, P.

P. Masson, W. Pieczynski, “SEM algorithm and unsupervised statistical segmentation of satellite images,” IEEE Trans. Geosci. Remote Sens. 31, 618–633 (1993).
[Crossref]

Mignard, M.

P. G. Lucey, T. Williams, M. Mignard, J. Julian, D. Kokobun, G. Allen, D. Hampton, W. Schaff, M. Schlangen, E. M. Winter, A. Stocker, K. Horton, A. P. Bowman, “AHI: an airborne long wave infrared hyperspectral imager,” in Airborne Reconnaissance XXII, W. G. Fishell, A. A. Andraitis, M. S. Fagan, J. D. Greer, M. C. Norton, eds., Proc. SPIE3431, 36–43 (1998).
[Crossref]

Mustard, J. F.

N. Keshava, J. F. Mustard, “Spectral unmixing,” IEEE Signal Process. Mag. 19, 44–57 (2002).
[Crossref]

Parish, J.

C. Simi, J. Parish, E. M. Winter, R. Dixon, C. LaSota, M. M. Williams, “Night vision imaging spectrometer (NVIS) performance parameters and their impact on various detection algorithms,” in Algorithms for Multispectral, Hyperspectral, and Ultraspectral Imagery VI, S. S. Shen, M. R. Descour, eds., Proc. SPIE4049, 218–229 (2000).
[Crossref]

Pearlman, J.

J. Pearlman, S. Carman, C. Segal, P. Jarecke, P. Clancy, W. Browne, “Overview of the Hyperion imaging spectrometer for the NASA EO-1 mission,” in Proceedings of the 2002 International Geoscience and Remote Sensing Symposium (IEEE, New York, 2002), pp. 3036–3038.

Pieczynski, W.

P. Masson, W. Pieczynski, “SEM algorithm and unsupervised statistical segmentation of satellite images,” IEEE Trans. Geosci. Remote Sens. 31, 618–633 (1993).
[Crossref]

Raqueno, R.

J. R. Schott, R. Raqueno, C. Salvaggio, “Incorporation of time-dependent thermodynamic model and radiation propagation model into infrared three-dimensional synthetic image generation,” Opt. Eng. 31, 1505–1516 (1992).
[Crossref]

Redner, R. A.

R. A. Redner, H. F. Walker, “Mixture densities, maximum likelihood, and the EM algorithm,” SIAM Rev. 26, 195–239 (1984).
[Crossref]

Salvaggio, C.

J. R. Schott, R. Raqueno, C. Salvaggio, “Incorporation of time-dependent thermodynamic model and radiation propagation model into infrared three-dimensional synthetic image generation,” Opt. Eng. 31, 1505–1516 (1992).
[Crossref]

Schaff, W.

P. G. Lucey, T. Williams, M. Mignard, J. Julian, D. Kokobun, G. Allen, D. Hampton, W. Schaff, M. Schlangen, E. M. Winter, A. Stocker, K. Horton, A. P. Bowman, “AHI: an airborne long wave infrared hyperspectral imager,” in Airborne Reconnaissance XXII, W. G. Fishell, A. A. Andraitis, M. S. Fagan, J. D. Greer, M. C. Norton, eds., Proc. SPIE3431, 36–43 (1998).
[Crossref]

Schaum, A. P.

A. D. Stocker, A. P. Schaum, “Application of stochastic mixing models to hyperspectral detection problems,” in Algorithms and Technologies for Multispectral, Hyperspectral, and Ultraspectral Imagery III, A. E. Iverson, S. S. Shen, eds., Proc. SPIE3071, 47–60 (1997).

Schlangen, M.

P. G. Lucey, T. Williams, M. Mignard, J. Julian, D. Kokobun, G. Allen, D. Hampton, W. Schaff, M. Schlangen, E. M. Winter, A. Stocker, K. Horton, A. P. Bowman, “AHI: an airborne long wave infrared hyperspectral imager,” in Airborne Reconnaissance XXII, W. G. Fishell, A. A. Andraitis, M. S. Fagan, J. D. Greer, M. C. Norton, eds., Proc. SPIE3431, 36–43 (1998).
[Crossref]

Schott, J. R.

J. R. Schott, R. Raqueno, C. Salvaggio, “Incorporation of time-dependent thermodynamic model and radiation propagation model into infrared three-dimensional synthetic image generation,” Opt. Eng. 31, 1505–1516 (1992).
[Crossref]

Schowengerdt, R. A.

R. A. Schowengerdt, Remote Sensing: Models and Methods for Image Processing, 2nd ed. (Academic, San Diego, Calif., 1997).

Segal, C.

J. Pearlman, S. Carman, C. Segal, P. Jarecke, P. Clancy, W. Browne, “Overview of the Hyperion imaging spectrometer for the NASA EO-1 mission,” in Proceedings of the 2002 International Geoscience and Remote Sensing Symposium (IEEE, New York, 2002), pp. 3036–3038.

Simi, C.

C. Simi, J. Parish, E. M. Winter, R. Dixon, C. LaSota, M. M. Williams, “Night vision imaging spectrometer (NVIS) performance parameters and their impact on various detection algorithms,” in Algorithms for Multispectral, Hyperspectral, and Ultraspectral Imagery VI, S. S. Shen, M. R. Descour, eds., Proc. SPIE4049, 218–229 (2000).
[Crossref]

Sivjee, M. G.

J. A. Hackwell, D. W. Warren, R. P. Bongiovi, S. J. Hansel, T. L. Hayhurst, D. J. Mabry, M. G. Sivjee, J. W. Skinner, “LWIR/MWIR imaging hyperspectral sensor for airborne and ground-based remote sensing,” in Imaging Spectrometry II, M. R. Descour, J. M. Mooney, eds., Proc. SPIE2819, 102–107 (1996).
[Crossref]

Skinner, J. W.

J. A. Hackwell, D. W. Warren, R. P. Bongiovi, S. J. Hansel, T. L. Hayhurst, D. J. Mabry, M. G. Sivjee, J. W. Skinner, “LWIR/MWIR imaging hyperspectral sensor for airborne and ground-based remote sensing,” in Imaging Spectrometry II, M. R. Descour, J. M. Mooney, eds., Proc. SPIE2819, 102–107 (1996).
[Crossref]

Stark, H.

H. Stark, J. W. Woods, Probability and Random Processes with Applications to Signal Processing (Prentice-Hall, Upper Saddle River, N.J., 2002), pp. 28–30.

Stein, D. W.

D. W. Stein, “Stochastic compositional models applied to subpixel analysis of hyperspectral imagery,” in Imaging Spectrometry VII, M. R. Descour, S. S. Shen, eds., Proc. SPIE4480, 49–56 (2002).
[Crossref]

Stocker, A.

P. G. Lucey, T. Williams, M. Mignard, J. Julian, D. Kokobun, G. Allen, D. Hampton, W. Schaff, M. Schlangen, E. M. Winter, A. Stocker, K. Horton, A. P. Bowman, “AHI: an airborne long wave infrared hyperspectral imager,” in Airborne Reconnaissance XXII, W. G. Fishell, A. A. Andraitis, M. S. Fagan, J. D. Greer, M. C. Norton, eds., Proc. SPIE3431, 36–43 (1998).
[Crossref]

Stocker, A. D.

A. D. Stocker, A. P. Schaum, “Application of stochastic mixing models to hyperspectral detection problems,” in Algorithms and Technologies for Multispectral, Hyperspectral, and Ultraspectral Imagery III, A. E. Iverson, S. S. Shen, eds., Proc. SPIE3071, 47–60 (1997).

Sundberg, R. L.

R. L. Sundberg, J. H. Gruninger, R. Haren, “Extraction of hyperspectral scene statistics and scene realization,” in Algorithms and Technologies for Multispectral, Hyperspectral, and Ultraspectral Imagery VIII, S. S. Shen, P. E. Lewis, eds., Proc. SPIE4725, 184–194 (2002).
[Crossref]

Tou, J. T.

J. T. Tou, R. C. Gonzalez, Pattern Recognition Principles (Addison-Wesley, Reading, Mass., 1974), p. 87.

Walker, H. F.

R. A. Redner, H. F. Walker, “Mixture densities, maximum likelihood, and the EM algorithm,” SIAM Rev. 26, 195–239 (1984).
[Crossref]

Warren, D. W.

J. A. Hackwell, D. W. Warren, R. P. Bongiovi, S. J. Hansel, T. L. Hayhurst, D. J. Mabry, M. G. Sivjee, J. W. Skinner, “LWIR/MWIR imaging hyperspectral sensor for airborne and ground-based remote sensing,” in Imaging Spectrometry II, M. R. Descour, J. M. Mooney, eds., Proc. SPIE2819, 102–107 (1996).
[Crossref]

Williams, M. M.

C. Simi, J. Parish, E. M. Winter, R. Dixon, C. LaSota, M. M. Williams, “Night vision imaging spectrometer (NVIS) performance parameters and their impact on various detection algorithms,” in Algorithms for Multispectral, Hyperspectral, and Ultraspectral Imagery VI, S. S. Shen, M. R. Descour, eds., Proc. SPIE4049, 218–229 (2000).
[Crossref]

Williams, T.

P. G. Lucey, T. Williams, M. Mignard, J. Julian, D. Kokobun, G. Allen, D. Hampton, W. Schaff, M. Schlangen, E. M. Winter, A. Stocker, K. Horton, A. P. Bowman, “AHI: an airborne long wave infrared hyperspectral imager,” in Airborne Reconnaissance XXII, W. G. Fishell, A. A. Andraitis, M. S. Fagan, J. D. Greer, M. C. Norton, eds., Proc. SPIE3431, 36–43 (1998).
[Crossref]

Winter, E. M.

P. G. Lucey, T. Williams, M. Mignard, J. Julian, D. Kokobun, G. Allen, D. Hampton, W. Schaff, M. Schlangen, E. M. Winter, A. Stocker, K. Horton, A. P. Bowman, “AHI: an airborne long wave infrared hyperspectral imager,” in Airborne Reconnaissance XXII, W. G. Fishell, A. A. Andraitis, M. S. Fagan, J. D. Greer, M. C. Norton, eds., Proc. SPIE3431, 36–43 (1998).
[Crossref]

C. Simi, J. Parish, E. M. Winter, R. Dixon, C. LaSota, M. M. Williams, “Night vision imaging spectrometer (NVIS) performance parameters and their impact on various detection algorithms,” in Algorithms for Multispectral, Hyperspectral, and Ultraspectral Imagery VI, S. S. Shen, M. R. Descour, eds., Proc. SPIE4049, 218–229 (2000).
[Crossref]

Winter, M. E.

M. E. Winter, “Fast autonomous endmember determination in hyperspectral data,” in Proceedings of the 13th International Conference on Applied Geological Remote Sensing (Environmental Research Institute of Michigan, Ann Arbor, Mich., 1999), Vol. II, pp. 337–344.

Woods, J. W.

H. Stark, J. W. Woods, Probability and Random Processes with Applications to Signal Processing (Prentice-Hall, Upper Saddle River, N.J., 2002), pp. 28–30.

IEEE Signal Process. Mag. (1)

N. Keshava, J. F. Mustard, “Spectral unmixing,” IEEE Signal Process. Mag. 19, 44–57 (2002).
[Crossref]

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Figures (17)

Fig. 1
Fig. 1

Two-dimensional scatterplot of data conforming to a deterministic LMM.

Fig. 2
Fig. 2

Two-dimensional scatterplot of data requiring a statistical model.

Fig. 3
Fig. 3

Number of mixture classes for various mixture fraction constraints with six endmembers.

Fig. 4
Fig. 4

Endmember reflectance spectra for synthetic test data: (a) first endmember, (b) second endmember, (c) third endmember, (d) fourth endmember.

Fig. 5
Fig. 5

True fraction maps for synthetic test data: (a) first endmember, (b) second endmember, (c) third endmember, (d) fourth endmember.

Fig. 6
Fig. 6

Leading principal component images of AVIRIS Yorktown data: (a) first component, (b) second component, (c) third component.

Fig. 7
Fig. 7

Convergence for synthetic test data for various mean initialization methods: (a) log-likelihood and (b) separability metrics.

Fig. 8
Fig. 8

Output fraction maps for synthetic test data for which global mean initialization was used: (a) first endmember, (b) second endmember, (c) third endmember, (d) fourth endmember.

Fig. 9
Fig. 9

Output fraction maps for synthetic test data for which random pixel mean initialization was used: (a) first endmember, (b) second endmember, (c) third endmember, (d) fourth endmember.

Fig. 10
Fig. 10

Output fraction maps for synthetic test data for which NFINDR mean initialization was used: (a) first endmember, (b) second endmember, (c) third endmember, (d) fourth endmember.

Fig. 11
Fig. 11

Effect of covariance matrix scaling on pure class separability for AVIRIS Yorktown data.

Fig. 12
Fig. 12

Effect of prior-probability updating on pure class separability for synthetic test data.

Fig. 13
Fig. 13

Output fraction maps for synthetic test data with fixed prior probabilities: (a) first endmember, (b) second endmember, (c) third endmember, (d) fourth endmember.

Fig. 14
Fig. 14

Effect of mixture constraints on pure class separability for AVIRIS Yorktown data.

Fig. 15
Fig. 15

Fraction maps for AVIRIS Yorktown data with three-mixture constraint: (a) first endmember, (b) second endmember, (c) third endmember, (d) fourth endmember.

Fig. 16
Fig. 16

Fraction maps for AVIRIS Yorktown data with six-endmember classes: (a) first endmember, (b) second endmember, (c) third endmember, (d) fourth endmember, (e) fifth endmember, (f) sixth endmember.

Fig. 17
Fig. 17

Fraction maps for AHI Yuma data with six-end-member classes: (a) first endmember, (b) second endmember, (c) third endmember, (d) fourth endmember, (e) fifth endmember, (f) sixth endmember.

Tables (2)

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Table 1 Outline of SMM Algorithm Steps and Investigated Modifications

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Table 2 Metrics for AVIRIS Yorktown Data with Various Numbers of Endmember Classes

Equations (16)

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

ω|q=m=1Ne amqεm,
mω|q=m=1Ne amqmεm,
Cω|q=m=1Ne am2qCεm,
pωω=q=1Nc Pqpω|qω|q,
pω|qω|q=1 2π K/21 Cω|q 1/2 exp-12 ω-mω|q TCω|q-1ω-mω|q.
V=11y1yeN.
Nc=Nlevels+Ne-1!Nlevels! Ne-1!.
Pˆnzi|q=Pˆn-1qpˆzi|qn-1zi|qq=1Nc Pˆn-1qpˆzi|qn-1zi|q.
Pˆ nq=NqnNz,
mˆεmn=1Nqmni=1iΩmnNqmnzi,
Ĉεmn=1Nqmn-1i=1iΩmnNqmnzi-mˆεmn zi-mˆεmn T,
Ln=1Ni=1Nzlnq=1Nc  Pˆ nq pˆ zi|qnzi|q,
Jn=trace Swn -1Sbn,
Swn=m=1Ne  Pˆ nqmĈεmn,
Sbn=m=1Ne  Pˆ nqm mˆ εmn- mˆ 0n mˆ εmn- mˆ 0n T,
mˆ 0n=m=1Ne  Pˆnqm mˆ εmn.

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