Abstract

We investigate the relationship among several popular end-member extraction algorithms, including N-FINDR, the simplex growing algorithm (SGA), vertex component analysis (VCA), automatic target generation process (ATGP), and fully constrained least squares linear unmixing (FCLSLU). We analyze the fundamental equivalence in the searching criteria of the simplex volume maximization and pixel spectral signature similarity employed by these algorithms. We point out that their performance discrepancy comes mainly from the use of a dimensionality reduction process, a parallel or sequential implementation mode, or the imposition of certain constraints. Instructive recommendations in algorithm selection for practical applications are provided.

© 2008 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. M. E. Winter, “N-FINDR: an algorithm for fast autonomous spectral end-member determination in hyperspectral data,” Proc. SPIE 3753, 266-275 (1999).
    [CrossRef]
  2. H. Ren and C.-I. Chang, “Automatic spectral target recognition in hyperspectral imagery,” IEEE Trans. Aerosp. Electron. Syst. 39, 1232-1249 (2003).
    [CrossRef]
  3. J. C. Harsanyi and C.-I. Chang, “Hyperspectral image classification and dimensionality reduction: an orthogonal subspace projection,” IEEE Trans. Geosci. Remote Sens. 32, 779-785(1994).
    [CrossRef]
  4. J. M. P. Nascimento and J. M. Bioucas Dias, “Vertex component analysis: a fast algorithm to unmix hyperspectral data,” IEEE Trans. Geosci. Remote Sens. 43, 898-910 (2005).
    [CrossRef]
  5. D. Heinz and C.-I. Chang, “Fully constrained least squares linear mixture analysis for material quantification in hyperspectral imagery,” IEEE Trans. Geosci. Remote Sens. 39, 529-545(2001).
    [CrossRef]
  6. C.-I. Chang, C.-C. Wu, W.-M. Liu, and Y.-C. Ouyang, “A new growing method for simplex-based endmember extraction algorithm,” IEEE Trans. Geosci. Remote Sens. 44, 2804-2819(2006).
    [CrossRef]
  7. R. E. Roger, “A faster way to compute the noise-adjusted principal components transform matrix,” IEEE Trans. Geosci. Remote Sens. 32, 1194-1196 (1994).
    [CrossRef]
  8. X. Tao, B. Wang, L. Zhang, and J. Zhang, “A new endmember extraction algorithm based on orthogonal bases of subspace formed by endmembers,” IEEE International Geoscience and Remote Sensing Symposium, 2007. IGARSS 2007 (IEEE, 2007), pp. 2006-2009.
  9. D. M. Rogge, B. Rivard, J. Zhang, and J. Feng, “Iterative spectral unmixing for optimizing per-pixel endmembers sets,” IEEE Trans. Geosci. Remote Sens. 44, 3725-3736 (2006).
    [CrossRef]

2006

C.-I. Chang, C.-C. Wu, W.-M. Liu, and Y.-C. Ouyang, “A new growing method for simplex-based endmember extraction algorithm,” IEEE Trans. Geosci. Remote Sens. 44, 2804-2819(2006).
[CrossRef]

D. M. Rogge, B. Rivard, J. Zhang, and J. Feng, “Iterative spectral unmixing for optimizing per-pixel endmembers sets,” IEEE Trans. Geosci. Remote Sens. 44, 3725-3736 (2006).
[CrossRef]

2005

J. M. P. Nascimento and J. M. Bioucas Dias, “Vertex component analysis: a fast algorithm to unmix hyperspectral data,” IEEE Trans. Geosci. Remote Sens. 43, 898-910 (2005).
[CrossRef]

2003

H. Ren and C.-I. Chang, “Automatic spectral target recognition in hyperspectral imagery,” IEEE Trans. Aerosp. Electron. Syst. 39, 1232-1249 (2003).
[CrossRef]

2001

D. Heinz and C.-I. Chang, “Fully constrained least squares linear mixture analysis for material quantification in hyperspectral imagery,” IEEE Trans. Geosci. Remote Sens. 39, 529-545(2001).
[CrossRef]

1999

M. E. Winter, “N-FINDR: an algorithm for fast autonomous spectral end-member determination in hyperspectral data,” Proc. SPIE 3753, 266-275 (1999).
[CrossRef]

1994

R. E. Roger, “A faster way to compute the noise-adjusted principal components transform matrix,” IEEE Trans. Geosci. Remote Sens. 32, 1194-1196 (1994).
[CrossRef]

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

Bioucas Dias, J. M.

J. M. P. Nascimento and J. M. Bioucas Dias, “Vertex component analysis: a fast algorithm to unmix hyperspectral data,” IEEE Trans. Geosci. Remote Sens. 43, 898-910 (2005).
[CrossRef]

Chang, C.-I.

C.-I. Chang, C.-C. Wu, W.-M. Liu, and Y.-C. Ouyang, “A new growing method for simplex-based endmember extraction algorithm,” IEEE Trans. Geosci. Remote Sens. 44, 2804-2819(2006).
[CrossRef]

H. Ren and C.-I. Chang, “Automatic spectral target recognition in hyperspectral imagery,” IEEE Trans. Aerosp. Electron. Syst. 39, 1232-1249 (2003).
[CrossRef]

D. Heinz and C.-I. Chang, “Fully constrained least squares linear mixture analysis for material quantification in hyperspectral imagery,” IEEE Trans. Geosci. Remote Sens. 39, 529-545(2001).
[CrossRef]

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

Feng, J.

D. M. Rogge, B. Rivard, J. Zhang, and J. Feng, “Iterative spectral unmixing for optimizing per-pixel endmembers sets,” IEEE Trans. Geosci. Remote Sens. 44, 3725-3736 (2006).
[CrossRef]

Harsanyi, J. C.

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

Heinz, D.

D. Heinz and C.-I. Chang, “Fully constrained least squares linear mixture analysis for material quantification in hyperspectral imagery,” IEEE Trans. Geosci. Remote Sens. 39, 529-545(2001).
[CrossRef]

Liu, W.-M.

C.-I. Chang, C.-C. Wu, W.-M. Liu, and Y.-C. Ouyang, “A new growing method for simplex-based endmember extraction algorithm,” IEEE Trans. Geosci. Remote Sens. 44, 2804-2819(2006).
[CrossRef]

Nascimento, J. M. P.

J. M. P. Nascimento and J. M. Bioucas Dias, “Vertex component analysis: a fast algorithm to unmix hyperspectral data,” IEEE Trans. Geosci. Remote Sens. 43, 898-910 (2005).
[CrossRef]

Ouyang, Y.-C.

C.-I. Chang, C.-C. Wu, W.-M. Liu, and Y.-C. Ouyang, “A new growing method for simplex-based endmember extraction algorithm,” IEEE Trans. Geosci. Remote Sens. 44, 2804-2819(2006).
[CrossRef]

Ren, H.

H. Ren and C.-I. Chang, “Automatic spectral target recognition in hyperspectral imagery,” IEEE Trans. Aerosp. Electron. Syst. 39, 1232-1249 (2003).
[CrossRef]

Rivard, B.

D. M. Rogge, B. Rivard, J. Zhang, and J. Feng, “Iterative spectral unmixing for optimizing per-pixel endmembers sets,” IEEE Trans. Geosci. Remote Sens. 44, 3725-3736 (2006).
[CrossRef]

Roger, R. E.

R. E. Roger, “A faster way to compute the noise-adjusted principal components transform matrix,” IEEE Trans. Geosci. Remote Sens. 32, 1194-1196 (1994).
[CrossRef]

Rogge, D. M.

D. M. Rogge, B. Rivard, J. Zhang, and J. Feng, “Iterative spectral unmixing for optimizing per-pixel endmembers sets,” IEEE Trans. Geosci. Remote Sens. 44, 3725-3736 (2006).
[CrossRef]

Tao, X.

X. Tao, B. Wang, L. Zhang, and J. Zhang, “A new endmember extraction algorithm based on orthogonal bases of subspace formed by endmembers,” IEEE International Geoscience and Remote Sensing Symposium, 2007. IGARSS 2007 (IEEE, 2007), pp. 2006-2009.

Wang, B.

X. Tao, B. Wang, L. Zhang, and J. Zhang, “A new endmember extraction algorithm based on orthogonal bases of subspace formed by endmembers,” IEEE International Geoscience and Remote Sensing Symposium, 2007. IGARSS 2007 (IEEE, 2007), pp. 2006-2009.

Winter, M. E.

M. E. Winter, “N-FINDR: an algorithm for fast autonomous spectral end-member determination in hyperspectral data,” Proc. SPIE 3753, 266-275 (1999).
[CrossRef]

Wu, C.-C.

C.-I. Chang, C.-C. Wu, W.-M. Liu, and Y.-C. Ouyang, “A new growing method for simplex-based endmember extraction algorithm,” IEEE Trans. Geosci. Remote Sens. 44, 2804-2819(2006).
[CrossRef]

Zhang, J.

D. M. Rogge, B. Rivard, J. Zhang, and J. Feng, “Iterative spectral unmixing for optimizing per-pixel endmembers sets,” IEEE Trans. Geosci. Remote Sens. 44, 3725-3736 (2006).
[CrossRef]

X. Tao, B. Wang, L. Zhang, and J. Zhang, “A new endmember extraction algorithm based on orthogonal bases of subspace formed by endmembers,” IEEE International Geoscience and Remote Sensing Symposium, 2007. IGARSS 2007 (IEEE, 2007), pp. 2006-2009.

Zhang, L.

X. Tao, B. Wang, L. Zhang, and J. Zhang, “A new endmember extraction algorithm based on orthogonal bases of subspace formed by endmembers,” IEEE International Geoscience and Remote Sensing Symposium, 2007. IGARSS 2007 (IEEE, 2007), pp. 2006-2009.

IEEE Trans. Aerosp. Electron. Syst.

H. Ren and C.-I. Chang, “Automatic spectral target recognition in hyperspectral imagery,” IEEE Trans. Aerosp. Electron. Syst. 39, 1232-1249 (2003).
[CrossRef]

IEEE Trans. Geosci. Remote Sens.

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

J. M. P. Nascimento and J. M. Bioucas Dias, “Vertex component analysis: a fast algorithm to unmix hyperspectral data,” IEEE Trans. Geosci. Remote Sens. 43, 898-910 (2005).
[CrossRef]

D. Heinz and C.-I. Chang, “Fully constrained least squares linear mixture analysis for material quantification in hyperspectral imagery,” IEEE Trans. Geosci. Remote Sens. 39, 529-545(2001).
[CrossRef]

C.-I. Chang, C.-C. Wu, W.-M. Liu, and Y.-C. Ouyang, “A new growing method for simplex-based endmember extraction algorithm,” IEEE Trans. Geosci. Remote Sens. 44, 2804-2819(2006).
[CrossRef]

R. E. Roger, “A faster way to compute the noise-adjusted principal components transform matrix,” IEEE Trans. Geosci. Remote Sens. 32, 1194-1196 (1994).
[CrossRef]

IEEE Trans. Geosci. Remote Sens.

D. M. Rogge, B. Rivard, J. Zhang, and J. Feng, “Iterative spectral unmixing for optimizing per-pixel endmembers sets,” IEEE Trans. Geosci. Remote Sens. 44, 3725-3736 (2006).
[CrossRef]

Proc. SPIE

M. E. Winter, “N-FINDR: an algorithm for fast autonomous spectral end-member determination in hyperspectral data,” Proc. SPIE 3753, 266-275 (1999).
[CrossRef]

Other

X. Tao, B. Wang, L. Zhang, and J. Zhang, “A new endmember extraction algorithm based on orthogonal bases of subspace formed by endmembers,” IEEE International Geoscience and Remote Sensing Symposium, 2007. IGARSS 2007 (IEEE, 2007), pp. 2006-2009.

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (4)

Fig. 1
Fig. 1

(a) HYDICE panel scene that contains 15 panels. (b) Ground truth map of spatial locations of the 15 panels.

Fig. 2
Fig. 2

End members extracted by the N-FINDR algorithm in different versions (highlighted in red circles): (a) parallel mode with 20 PCs; (b) sequential mode with 20 PCs; (c) parallel mode with all bands; (d) sequential mode with all bands.

Fig. 3
Fig. 3

AVIRIS Cuprite scene with five major minerals: alunite (A), buddingtonite (B), calcite (C), kaolinite (K), and muscovite (M).

Fig. 4
Fig. 4

AVIRIS Lunar Lake subscene including six end members named Cinder (C), Playa Lake (P), Rhyolite (R), Vegetation (V), Shade (S), and an Anomaly (A).

Tables (5)

Tables Icon

Table 1 Number of Panel Signature Types Extracted by Algorithms, for Variousp

Tables Icon

Table 2 Minimum Value of p Required for Algorithms to Extract All Five Panel Signatures

Tables Icon

Table 3 Similarity (Spectral Angles) between Five Mineral Signatures and Corresponding Extracted Signatures in the AVIRIS Cuprite Experiment

Tables Icon

Table 4 Similarity (Spectral Angles) among Six Extracted End Members in the AVIRIS Lunar Lake Experiment

Tables Icon

Table 5 Similarity (Simplex Volume) among Six Extracted End Members in the AVIRIS Lunar Lake Experiment

Equations (16)

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

V ( E ( 0 ) ) = | det ( E ( 0 ) ) | / n ! ,
E ( 0 ) = [ 1 1 1 e 0 ( 0 ) e 1 ( 0 ) e n ( 0 ) ] .
P e 0 = I e 0 ( e 0 T e 0 ) 1 e 0 T ,
V ( E ^ ) = | det ( E ^ ) | / n ! ,
| det ( E ^ ) | = | a 1 T a 1 a 1 T a 2 a 1 T a n a n T a 1 a n T a 2 a n T a n | 1 / 2 ,
| det ( E ^ ) | = | | a ^ 1 | 2 0 0 0 0 | a ^ n | 2 | 1 / 2 = | a ^ 1 | | a ^ 2 | | a ^ n | .
[ A B B T D ] 1 = [ A 1 + A 1 B ( D B T A 1 B ) 1 B T A 1 A 1 B ( D B T A 1 B ) 1 ( D B T A 1 B ) 1 B T A 1 ( D B T A 1 B ) 1 ] ,
M T M = [ UU T U T d d T U d T d ] .
( M T M ) 1 = [ ( U T U ) 1 + ( U T U ) 1 U T dd T U ( U T U ) 1 P U d 2 ( U T U ) 1 U T d P U d 2 d T U ( U T U ) 1 P U d 2 1 P U d 2 ] ,
P M = M ( M T M ) 1 M T = [ U d ] [ ( U T U ) 1 + ( U T U ) 1 U T dd T U ( U T U ) 1 P U d 2 ( U T U ) 1 U T d P U d 2 d T U ( U T U ) 1 P U d 2 1 P U d 2 ] [ U T d T ] = P U + P U dd T P U P U d 2 dd T P U P U d 2 P U dd T P U d 2 + dd T P U d 2 = P U + ( I P U ) dd T ( I P U ) P U d 2 = P U + P U dd T P U P U d 2 .
P M = P U + d ^ d ^ T d ^ 2 .
a ^ 2 = a 2 a ^ 1 a ^ 1 T a ^ 1 T a ^ 1 a 2 = a 2 a 1 a 1 T a 1 T a 1 a 2 = ( I a 1 ( a 1 T a 1 ) 1 a 1 T ) a 2 = P a 1 a 2 = P Z 1 a 2 .
a ^ 3 = a 3 a ^ 1 a ^ 1 T a ^ 1 T a ^ 1 a 3 a ^ 2 a ^ 2 T a ^ 2 T a ^ 2 a 3 = ( I a 1 a 1 T a 1 T a 1 P a 1 a 2 a 2 T P a 1 P a 1 a 2 2 ) a 3 = ( I P Z 1 P a 1 a 2 a 2 T P a 1 P a 1 a 2 2 ) a 3 .
a ^ 3 = ( I P Z 2 ) a 3 = P Z 2 a 3 ,
P Z 2 = P Z 1 + P a 1 a 2 a 2 T P a 1 P a 1 a 2 2 = P Z 1 + a ^ 2 a ^ 2 T a ^ 2 2 .
α = ( M T M ) 1 M T r = M # r ,

Metrics