Abstract

Although several hyperspectral anomaly detection algorithms have proven useful when illumination conditions provide for enough light, many of these same detection algorithms fail to perform well when shadows are also present. To date, no general approach to the problem has been demonstrated. In this paper, a novel hyperspectral anomaly detection algorithm that adapts the dimensionality of the spectral detection subspace to multiple illumination levels is described. The novel detection algorithm is applied to reflectance domain hyperspectral data that represents a variety of illumination conditions: well illuminated and poorly illuminated (i.e., shadowed). Detection results obtained for objects located in deep shadows and light–shadow transition areas suggest superiority of the novel algorithm over standard subspace RX detection.

© 2010 Optical Society of America

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References

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  1. M. T. Eismann, J. Meola, A. D. Stocker, S. G. Beaven, and A. P. Schaum, “Airborne hyperspectral detection of small changes,” Appl. Opt. 47, F27-F45 (2008).
    [CrossRef] [PubMed]
  2. R. Mayer, J. Antoniades, M. Baumback, D. Chester, J. Edwards, A. Goldstein, D. Haas, and S. Henderson, “Shadowed target detection for hyperspectral imagery,” Proc. SPIE 6678, 66780L (2007).
    [CrossRef]
  3. S. M. Adler-Golden, R. Y. Levine, M. W. Matthew, S. C. Richtsmeier, L. S. Bernstein, J. Gruninger, G. Felde, M. Hoke, G. P. Anderson, and A. Ratkowski, “Shadow-insensitive material detection/classification with atmospherically corrected hyperspectral imagery,” Proc. SPIE 4381, 460-469 (2001).
    [CrossRef]
  4. E. A. Ashton, B. D. Wemett, R. A. Leathers, and T. V. Downes, “A novel method for illumination suppression in hyperspectral images,” Proc. SPIE 6966, 69660C (2008).
    [CrossRef]
  5. M. J. Carlotto, “A cluster based approach for detecting man-made objects and changes in imagery,” IEEE Trans. Geosci. Remote Sens. 43, 374-387 (2005).
    [CrossRef]
  6. D. W. J. Stein, S. G. Beaven, L. E. Hoff, Edwin M. Winter, A. P. Schaum, and A. D. Stocker, “Anomaly detection from hyperspectral imagery,” IEEE Signal Process. Mag. 19, 58-69(2002).
    [CrossRef]
  7. P. C. Hytle, R. C. Hardie, M. T. Eismann, and J. Meola, “Anomaly detection in hyperspectral imagery: comparison of methods using diurnal and seasonal data,” J. Appl. Remote Sens. 3, 033546 (2009).
    [CrossRef]
  8. A. Schaum and A. D. Stocker, “Hyperspectral change detection and supervised matched filtering based on covariance equalization,” Proc. SPIE 5425, 77-90 (2004).
    [CrossRef]
  9. A. Schaum, “Joint subspace detection of hyperspectral targets,” in IEEE Aerospace Conference Proceedings (IEEE, 2004), Vol. 3, pp. 1818-1824.
  10. A. Schaum, “Autonomous hyperspectral target detection with quasi-stationarity violation at background boundaries,” in 35th Applied Imagery and Pattern Recognition Workshop (IEEE, 2006).
    [CrossRef]
  11. T. N. Pappas, “An adaptive clustering algorithm for image segmentation,” IEEE Trans. Signal Process. 40, 901-914 (1992).
    [CrossRef]
  12. M. Sezgin and B. Sankur, “Survey over image thresholding techniques and quantitative performance evaluation,” J. Electron. Imaging 13,146-165 (2004) and references therein.
    [CrossRef]
  13. E. Ensafi and A. D. Stocker, “An adaptive CFAR algorithm for real-time hyperspectral target detection,” Proc. SPIE 6966, 696605 (2008).
    [CrossRef]
  14. A. V. Kanaev, E. Allman, and J. Murray-Krezan, “Reduction of false alarms caused by background boundaries in real time subspace RX anomaly detection,” Proc. SPIE 7334, 733405(2009).
    [CrossRef]

2009 (2)

P. C. Hytle, R. C. Hardie, M. T. Eismann, and J. Meola, “Anomaly detection in hyperspectral imagery: comparison of methods using diurnal and seasonal data,” J. Appl. Remote Sens. 3, 033546 (2009).
[CrossRef]

A. V. Kanaev, E. Allman, and J. Murray-Krezan, “Reduction of false alarms caused by background boundaries in real time subspace RX anomaly detection,” Proc. SPIE 7334, 733405(2009).
[CrossRef]

2008 (3)

M. T. Eismann, J. Meola, A. D. Stocker, S. G. Beaven, and A. P. Schaum, “Airborne hyperspectral detection of small changes,” Appl. Opt. 47, F27-F45 (2008).
[CrossRef] [PubMed]

E. Ensafi and A. D. Stocker, “An adaptive CFAR algorithm for real-time hyperspectral target detection,” Proc. SPIE 6966, 696605 (2008).
[CrossRef]

E. A. Ashton, B. D. Wemett, R. A. Leathers, and T. V. Downes, “A novel method for illumination suppression in hyperspectral images,” Proc. SPIE 6966, 69660C (2008).
[CrossRef]

2007 (1)

R. Mayer, J. Antoniades, M. Baumback, D. Chester, J. Edwards, A. Goldstein, D. Haas, and S. Henderson, “Shadowed target detection for hyperspectral imagery,” Proc. SPIE 6678, 66780L (2007).
[CrossRef]

2005 (1)

M. J. Carlotto, “A cluster based approach for detecting man-made objects and changes in imagery,” IEEE Trans. Geosci. Remote Sens. 43, 374-387 (2005).
[CrossRef]

2004 (2)

M. Sezgin and B. Sankur, “Survey over image thresholding techniques and quantitative performance evaluation,” J. Electron. Imaging 13,146-165 (2004) and references therein.
[CrossRef]

A. Schaum and A. D. Stocker, “Hyperspectral change detection and supervised matched filtering based on covariance equalization,” Proc. SPIE 5425, 77-90 (2004).
[CrossRef]

2002 (1)

D. W. J. Stein, S. G. Beaven, L. E. Hoff, Edwin M. Winter, A. P. Schaum, and A. D. Stocker, “Anomaly detection from hyperspectral imagery,” IEEE Signal Process. Mag. 19, 58-69(2002).
[CrossRef]

2001 (1)

S. M. Adler-Golden, R. Y. Levine, M. W. Matthew, S. C. Richtsmeier, L. S. Bernstein, J. Gruninger, G. Felde, M. Hoke, G. P. Anderson, and A. Ratkowski, “Shadow-insensitive material detection/classification with atmospherically corrected hyperspectral imagery,” Proc. SPIE 4381, 460-469 (2001).
[CrossRef]

1992 (1)

T. N. Pappas, “An adaptive clustering algorithm for image segmentation,” IEEE Trans. Signal Process. 40, 901-914 (1992).
[CrossRef]

Adler-Golden, S. M.

S. M. Adler-Golden, R. Y. Levine, M. W. Matthew, S. C. Richtsmeier, L. S. Bernstein, J. Gruninger, G. Felde, M. Hoke, G. P. Anderson, and A. Ratkowski, “Shadow-insensitive material detection/classification with atmospherically corrected hyperspectral imagery,” Proc. SPIE 4381, 460-469 (2001).
[CrossRef]

Allman, E.

A. V. Kanaev, E. Allman, and J. Murray-Krezan, “Reduction of false alarms caused by background boundaries in real time subspace RX anomaly detection,” Proc. SPIE 7334, 733405(2009).
[CrossRef]

Anderson, G. P.

S. M. Adler-Golden, R. Y. Levine, M. W. Matthew, S. C. Richtsmeier, L. S. Bernstein, J. Gruninger, G. Felde, M. Hoke, G. P. Anderson, and A. Ratkowski, “Shadow-insensitive material detection/classification with atmospherically corrected hyperspectral imagery,” Proc. SPIE 4381, 460-469 (2001).
[CrossRef]

Antoniades, J.

R. Mayer, J. Antoniades, M. Baumback, D. Chester, J. Edwards, A. Goldstein, D. Haas, and S. Henderson, “Shadowed target detection for hyperspectral imagery,” Proc. SPIE 6678, 66780L (2007).
[CrossRef]

Ashton, E. A.

E. A. Ashton, B. D. Wemett, R. A. Leathers, and T. V. Downes, “A novel method for illumination suppression in hyperspectral images,” Proc. SPIE 6966, 69660C (2008).
[CrossRef]

Baumback, M.

R. Mayer, J. Antoniades, M. Baumback, D. Chester, J. Edwards, A. Goldstein, D. Haas, and S. Henderson, “Shadowed target detection for hyperspectral imagery,” Proc. SPIE 6678, 66780L (2007).
[CrossRef]

Beaven, S. G.

M. T. Eismann, J. Meola, A. D. Stocker, S. G. Beaven, and A. P. Schaum, “Airborne hyperspectral detection of small changes,” Appl. Opt. 47, F27-F45 (2008).
[CrossRef] [PubMed]

D. W. J. Stein, S. G. Beaven, L. E. Hoff, Edwin M. Winter, A. P. Schaum, and A. D. Stocker, “Anomaly detection from hyperspectral imagery,” IEEE Signal Process. Mag. 19, 58-69(2002).
[CrossRef]

Bernstein, L. S.

S. M. Adler-Golden, R. Y. Levine, M. W. Matthew, S. C. Richtsmeier, L. S. Bernstein, J. Gruninger, G. Felde, M. Hoke, G. P. Anderson, and A. Ratkowski, “Shadow-insensitive material detection/classification with atmospherically corrected hyperspectral imagery,” Proc. SPIE 4381, 460-469 (2001).
[CrossRef]

Carlotto, M. J.

M. J. Carlotto, “A cluster based approach for detecting man-made objects and changes in imagery,” IEEE Trans. Geosci. Remote Sens. 43, 374-387 (2005).
[CrossRef]

Chester, D.

R. Mayer, J. Antoniades, M. Baumback, D. Chester, J. Edwards, A. Goldstein, D. Haas, and S. Henderson, “Shadowed target detection for hyperspectral imagery,” Proc. SPIE 6678, 66780L (2007).
[CrossRef]

Downes, T. V.

E. A. Ashton, B. D. Wemett, R. A. Leathers, and T. V. Downes, “A novel method for illumination suppression in hyperspectral images,” Proc. SPIE 6966, 69660C (2008).
[CrossRef]

Edwards, J.

R. Mayer, J. Antoniades, M. Baumback, D. Chester, J. Edwards, A. Goldstein, D. Haas, and S. Henderson, “Shadowed target detection for hyperspectral imagery,” Proc. SPIE 6678, 66780L (2007).
[CrossRef]

Eismann, M. T.

P. C. Hytle, R. C. Hardie, M. T. Eismann, and J. Meola, “Anomaly detection in hyperspectral imagery: comparison of methods using diurnal and seasonal data,” J. Appl. Remote Sens. 3, 033546 (2009).
[CrossRef]

M. T. Eismann, J. Meola, A. D. Stocker, S. G. Beaven, and A. P. Schaum, “Airborne hyperspectral detection of small changes,” Appl. Opt. 47, F27-F45 (2008).
[CrossRef] [PubMed]

Ensafi, E.

E. Ensafi and A. D. Stocker, “An adaptive CFAR algorithm for real-time hyperspectral target detection,” Proc. SPIE 6966, 696605 (2008).
[CrossRef]

Felde, G.

S. M. Adler-Golden, R. Y. Levine, M. W. Matthew, S. C. Richtsmeier, L. S. Bernstein, J. Gruninger, G. Felde, M. Hoke, G. P. Anderson, and A. Ratkowski, “Shadow-insensitive material detection/classification with atmospherically corrected hyperspectral imagery,” Proc. SPIE 4381, 460-469 (2001).
[CrossRef]

Goldstein, A.

R. Mayer, J. Antoniades, M. Baumback, D. Chester, J. Edwards, A. Goldstein, D. Haas, and S. Henderson, “Shadowed target detection for hyperspectral imagery,” Proc. SPIE 6678, 66780L (2007).
[CrossRef]

Gruninger, J.

S. M. Adler-Golden, R. Y. Levine, M. W. Matthew, S. C. Richtsmeier, L. S. Bernstein, J. Gruninger, G. Felde, M. Hoke, G. P. Anderson, and A. Ratkowski, “Shadow-insensitive material detection/classification with atmospherically corrected hyperspectral imagery,” Proc. SPIE 4381, 460-469 (2001).
[CrossRef]

Haas, D.

R. Mayer, J. Antoniades, M. Baumback, D. Chester, J. Edwards, A. Goldstein, D. Haas, and S. Henderson, “Shadowed target detection for hyperspectral imagery,” Proc. SPIE 6678, 66780L (2007).
[CrossRef]

Hardie, R. C.

P. C. Hytle, R. C. Hardie, M. T. Eismann, and J. Meola, “Anomaly detection in hyperspectral imagery: comparison of methods using diurnal and seasonal data,” J. Appl. Remote Sens. 3, 033546 (2009).
[CrossRef]

Henderson, S.

R. Mayer, J. Antoniades, M. Baumback, D. Chester, J. Edwards, A. Goldstein, D. Haas, and S. Henderson, “Shadowed target detection for hyperspectral imagery,” Proc. SPIE 6678, 66780L (2007).
[CrossRef]

Hoff, L. E.

D. W. J. Stein, S. G. Beaven, L. E. Hoff, Edwin M. Winter, A. P. Schaum, and A. D. Stocker, “Anomaly detection from hyperspectral imagery,” IEEE Signal Process. Mag. 19, 58-69(2002).
[CrossRef]

Hoke, M.

S. M. Adler-Golden, R. Y. Levine, M. W. Matthew, S. C. Richtsmeier, L. S. Bernstein, J. Gruninger, G. Felde, M. Hoke, G. P. Anderson, and A. Ratkowski, “Shadow-insensitive material detection/classification with atmospherically corrected hyperspectral imagery,” Proc. SPIE 4381, 460-469 (2001).
[CrossRef]

Hytle, P. C.

P. C. Hytle, R. C. Hardie, M. T. Eismann, and J. Meola, “Anomaly detection in hyperspectral imagery: comparison of methods using diurnal and seasonal data,” J. Appl. Remote Sens. 3, 033546 (2009).
[CrossRef]

Kanaev, A. V.

A. V. Kanaev, E. Allman, and J. Murray-Krezan, “Reduction of false alarms caused by background boundaries in real time subspace RX anomaly detection,” Proc. SPIE 7334, 733405(2009).
[CrossRef]

Leathers, R. A.

E. A. Ashton, B. D. Wemett, R. A. Leathers, and T. V. Downes, “A novel method for illumination suppression in hyperspectral images,” Proc. SPIE 6966, 69660C (2008).
[CrossRef]

Levine, R. Y.

S. M. Adler-Golden, R. Y. Levine, M. W. Matthew, S. C. Richtsmeier, L. S. Bernstein, J. Gruninger, G. Felde, M. Hoke, G. P. Anderson, and A. Ratkowski, “Shadow-insensitive material detection/classification with atmospherically corrected hyperspectral imagery,” Proc. SPIE 4381, 460-469 (2001).
[CrossRef]

Matthew, M. W.

S. M. Adler-Golden, R. Y. Levine, M. W. Matthew, S. C. Richtsmeier, L. S. Bernstein, J. Gruninger, G. Felde, M. Hoke, G. P. Anderson, and A. Ratkowski, “Shadow-insensitive material detection/classification with atmospherically corrected hyperspectral imagery,” Proc. SPIE 4381, 460-469 (2001).
[CrossRef]

Mayer, R.

R. Mayer, J. Antoniades, M. Baumback, D. Chester, J. Edwards, A. Goldstein, D. Haas, and S. Henderson, “Shadowed target detection for hyperspectral imagery,” Proc. SPIE 6678, 66780L (2007).
[CrossRef]

Meola, J.

P. C. Hytle, R. C. Hardie, M. T. Eismann, and J. Meola, “Anomaly detection in hyperspectral imagery: comparison of methods using diurnal and seasonal data,” J. Appl. Remote Sens. 3, 033546 (2009).
[CrossRef]

M. T. Eismann, J. Meola, A. D. Stocker, S. G. Beaven, and A. P. Schaum, “Airborne hyperspectral detection of small changes,” Appl. Opt. 47, F27-F45 (2008).
[CrossRef] [PubMed]

Murray-Krezan, J.

A. V. Kanaev, E. Allman, and J. Murray-Krezan, “Reduction of false alarms caused by background boundaries in real time subspace RX anomaly detection,” Proc. SPIE 7334, 733405(2009).
[CrossRef]

Pappas, T. N.

T. N. Pappas, “An adaptive clustering algorithm for image segmentation,” IEEE Trans. Signal Process. 40, 901-914 (1992).
[CrossRef]

Ratkowski, A.

S. M. Adler-Golden, R. Y. Levine, M. W. Matthew, S. C. Richtsmeier, L. S. Bernstein, J. Gruninger, G. Felde, M. Hoke, G. P. Anderson, and A. Ratkowski, “Shadow-insensitive material detection/classification with atmospherically corrected hyperspectral imagery,” Proc. SPIE 4381, 460-469 (2001).
[CrossRef]

Richtsmeier, S. C.

S. M. Adler-Golden, R. Y. Levine, M. W. Matthew, S. C. Richtsmeier, L. S. Bernstein, J. Gruninger, G. Felde, M. Hoke, G. P. Anderson, and A. Ratkowski, “Shadow-insensitive material detection/classification with atmospherically corrected hyperspectral imagery,” Proc. SPIE 4381, 460-469 (2001).
[CrossRef]

Sankur, B.

M. Sezgin and B. Sankur, “Survey over image thresholding techniques and quantitative performance evaluation,” J. Electron. Imaging 13,146-165 (2004) and references therein.
[CrossRef]

Schaum, A.

A. Schaum and A. D. Stocker, “Hyperspectral change detection and supervised matched filtering based on covariance equalization,” Proc. SPIE 5425, 77-90 (2004).
[CrossRef]

A. Schaum, “Autonomous hyperspectral target detection with quasi-stationarity violation at background boundaries,” in 35th Applied Imagery and Pattern Recognition Workshop (IEEE, 2006).
[CrossRef]

A. Schaum, “Joint subspace detection of hyperspectral targets,” in IEEE Aerospace Conference Proceedings (IEEE, 2004), Vol. 3, pp. 1818-1824.

Schaum, A. P.

M. T. Eismann, J. Meola, A. D. Stocker, S. G. Beaven, and A. P. Schaum, “Airborne hyperspectral detection of small changes,” Appl. Opt. 47, F27-F45 (2008).
[CrossRef] [PubMed]

D. W. J. Stein, S. G. Beaven, L. E. Hoff, Edwin M. Winter, A. P. Schaum, and A. D. Stocker, “Anomaly detection from hyperspectral imagery,” IEEE Signal Process. Mag. 19, 58-69(2002).
[CrossRef]

Sezgin, M.

M. Sezgin and B. Sankur, “Survey over image thresholding techniques and quantitative performance evaluation,” J. Electron. Imaging 13,146-165 (2004) and references therein.
[CrossRef]

Stein, D. W. J.

D. W. J. Stein, S. G. Beaven, L. E. Hoff, Edwin M. Winter, A. P. Schaum, and A. D. Stocker, “Anomaly detection from hyperspectral imagery,” IEEE Signal Process. Mag. 19, 58-69(2002).
[CrossRef]

Stocker, A. D.

E. Ensafi and A. D. Stocker, “An adaptive CFAR algorithm for real-time hyperspectral target detection,” Proc. SPIE 6966, 696605 (2008).
[CrossRef]

M. T. Eismann, J. Meola, A. D. Stocker, S. G. Beaven, and A. P. Schaum, “Airborne hyperspectral detection of small changes,” Appl. Opt. 47, F27-F45 (2008).
[CrossRef] [PubMed]

A. Schaum and A. D. Stocker, “Hyperspectral change detection and supervised matched filtering based on covariance equalization,” Proc. SPIE 5425, 77-90 (2004).
[CrossRef]

D. W. J. Stein, S. G. Beaven, L. E. Hoff, Edwin M. Winter, A. P. Schaum, and A. D. Stocker, “Anomaly detection from hyperspectral imagery,” IEEE Signal Process. Mag. 19, 58-69(2002).
[CrossRef]

Wemett, B. D.

E. A. Ashton, B. D. Wemett, R. A. Leathers, and T. V. Downes, “A novel method for illumination suppression in hyperspectral images,” Proc. SPIE 6966, 69660C (2008).
[CrossRef]

Winter, Edwin M.

D. W. J. Stein, S. G. Beaven, L. E. Hoff, Edwin M. Winter, A. P. Schaum, and A. D. Stocker, “Anomaly detection from hyperspectral imagery,” IEEE Signal Process. Mag. 19, 58-69(2002).
[CrossRef]

Appl. Opt. (1)

IEEE Signal Process. Mag. (1)

D. W. J. Stein, S. G. Beaven, L. E. Hoff, Edwin M. Winter, A. P. Schaum, and A. D. Stocker, “Anomaly detection from hyperspectral imagery,” IEEE Signal Process. Mag. 19, 58-69(2002).
[CrossRef]

IEEE Trans. Geosci. Remote Sens. (1)

M. J. Carlotto, “A cluster based approach for detecting man-made objects and changes in imagery,” IEEE Trans. Geosci. Remote Sens. 43, 374-387 (2005).
[CrossRef]

IEEE Trans. Signal Process. (1)

T. N. Pappas, “An adaptive clustering algorithm for image segmentation,” IEEE Trans. Signal Process. 40, 901-914 (1992).
[CrossRef]

J. Appl. Remote Sens. (1)

P. C. Hytle, R. C. Hardie, M. T. Eismann, and J. Meola, “Anomaly detection in hyperspectral imagery: comparison of methods using diurnal and seasonal data,” J. Appl. Remote Sens. 3, 033546 (2009).
[CrossRef]

J. Electron. Imaging (1)

M. Sezgin and B. Sankur, “Survey over image thresholding techniques and quantitative performance evaluation,” J. Electron. Imaging 13,146-165 (2004) and references therein.
[CrossRef]

Proc. SPIE (6)

E. Ensafi and A. D. Stocker, “An adaptive CFAR algorithm for real-time hyperspectral target detection,” Proc. SPIE 6966, 696605 (2008).
[CrossRef]

A. V. Kanaev, E. Allman, and J. Murray-Krezan, “Reduction of false alarms caused by background boundaries in real time subspace RX anomaly detection,” Proc. SPIE 7334, 733405(2009).
[CrossRef]

A. Schaum and A. D. Stocker, “Hyperspectral change detection and supervised matched filtering based on covariance equalization,” Proc. SPIE 5425, 77-90 (2004).
[CrossRef]

R. Mayer, J. Antoniades, M. Baumback, D. Chester, J. Edwards, A. Goldstein, D. Haas, and S. Henderson, “Shadowed target detection for hyperspectral imagery,” Proc. SPIE 6678, 66780L (2007).
[CrossRef]

S. M. Adler-Golden, R. Y. Levine, M. W. Matthew, S. C. Richtsmeier, L. S. Bernstein, J. Gruninger, G. Felde, M. Hoke, G. P. Anderson, and A. Ratkowski, “Shadow-insensitive material detection/classification with atmospherically corrected hyperspectral imagery,” Proc. SPIE 4381, 460-469 (2001).
[CrossRef]

E. A. Ashton, B. D. Wemett, R. A. Leathers, and T. V. Downes, “A novel method for illumination suppression in hyperspectral images,” Proc. SPIE 6966, 69660C (2008).
[CrossRef]

Other (2)

A. Schaum, “Joint subspace detection of hyperspectral targets,” in IEEE Aerospace Conference Proceedings (IEEE, 2004), Vol. 3, pp. 1818-1824.

A. Schaum, “Autonomous hyperspectral target detection with quasi-stationarity violation at background boundaries,” in 35th Applied Imagery and Pattern Recognition Workshop (IEEE, 2006).
[CrossRef]

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Figures (12)

Fig. 1
Fig. 1

RGB representation of the first hyperspectral data set: one object (on the left) is conspicuous and two objects (on the right) are in the shadow area.

Fig. 2
Fig. 2

Segmentation of the first data set using adaptive clustering.

Fig. 3
Fig. 3

Dark-cluster pixel distribution in three-dimensional spectral space formed by the three highest PCs: background pixels are green, better illuminated object is blue, two objects in the shadow are red and magenta.

Fig. 4
Fig. 4

Anomaly detector output for (a) SARX, (b) SSRX, and (c) SSRX applied separately to dark and bright clusters.

Fig. 5
Fig. 5

Thresholded detection results corresponding to a 2.9 × 10 4 false-alarm rate for (a) SARX, (b) SSRX, and (c) SSRX applied separately to dark and bright clusters. Detected objects are marked by gray ellipses.

Fig. 6
Fig. 6

SARX performance as a function of the number of PCs used for SARX detection in the dark-cluster data. False-alarm rates (FA) are provided in units of false alarms per pixel.

Fig. 7
Fig. 7

RGB representation of the second hyperspectral data set.

Fig. 8
Fig. 8

Segmentation of the second data set using adaptive clustering.

Fig. 9
Fig. 9

Anomaly detector output for (a) SARX and (b) SSRX.

Fig. 10
Fig. 10

Thresholded detection results for (a) SARX and (b) SSRX, both at 1 × 10 4 false alarms per pixel. Objects in the shadow, detected by SARX but missed by SSRX, are outlined by the ellipse in (a).

Fig. 11
Fig. 11

SARX detection probability ( P d ) as a function of the number of PCs that comprise the dark-cluster spectral subspace.

Fig. 12
Fig. 12

SARX detection probability ( P d ) as a function of the boundary region size, where SMM detection is applied, where N is the half-width of a square neighborhood that defines the boundary region. N = 0 means SMM is not used.

Equations (15)

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

H 1 :   X = T t + n , H 0 :   X = B b .
p B ( x ) = ( 2 π ) J / 2 C 1 / 2 exp ( 1 2 ( x μ ) C 1 ( x μ ) ) .
μ = 1 M i = 1 M x i , C = 1 M i = 1 M ( x i μ ) ( x i μ ) .
max p T ( x :   { t } ) { t } p B ( x ) > < k ,
( x μ ) C 1 ( x μ ) > < k .
p ( x s | y s , x q , q N s ) exp { 1 2 σ 2 [ y s μ s x s ] 2 x s C V C ( x ) } ,
V c ( x ) = { β , if     x s = x q and s , q C , q s + β , if     x s x q and s , q C , q s .
μ = f μ 1 + ( 1 f ) μ 2 , C = f 2 C 1 + ( 1 f ) 2 C 2 ,
max p T ( x :   { t } ) { t } max p B ( x :   { b } ) { b } > < k .
max { f } { ln C + ( x μ ) C 1 ( x μ ) } ,
Λ D Λ C 1 1 / 2 C 2 C 1 1 / 2 ,
max { f } { i = 1 , 2 ( ln [ f 2 + ( 1 f ) 2 D i i ] ) + Λ ( C 1 1 / 2 ) ( x [ f μ 1 + ( 1 f ) μ 2 ] ) × [ f 2 + ( 1 f ) 2 D i i ] 1 Λ C 1 1 / 2 ( x [ f μ 1 + ( 1 f ) μ 2 ] ) } .
i = 1 , 2 { f ( 1 f ) D i i + [ u i + f w i ] w i f 2 + ( 1 f ) 2 D i i [ u i + f w i ] 2 [ f 2 ( 1 f ) 2 D i i ] [ f 2 + ( 1 f ) 2 D i i ] 2 } = 0 ,
u Λ C 1 1 / 2 ( x μ 2 ) , w Λ C 1 1 / 2 ( μ 2 μ 1 ) .
( x μ ) C 1 ( x μ ) + i ln { [ f 2 + ( 1 f ) 2 D i i ] } > k < k .

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