Abstract

α-divergence-based nonnegative tensor factorization (NTF) is applied to blind multispectral image (MSI) decomposition. The matrix of spectral profiles and the matrix of spatial distributions of the materials resident in the image are identified from the factors in Tucker3 and PARAFAC models. NTF preserves local structure in the MSI that is lost as a result of vectorization of the image when nonnegative matrix factorization (NMF)- or independent component analysis (ICA)-based decompositions are used. Moreover, NTF based on the PARAFAC model is unique up to permutation and scale under mild conditions. To achieve this, NMF- and ICA-based factorizations, respectively, require enforcement of sparseness (orthogonality) and statistical independence constraints on the spatial distributions of the materials resident in the MSI, and these conditions do not hold. We demonstrate efficiency of the NTF-based factorization in relation to NMF- and ICA-based factorizations on blind decomposition of the experimental MSI with the known ground truth.

© 2009 Optical Society of America

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  1. C.-I. Chang, S.-S. Chiang, J. A. Smith, and I. W. Ginsberg, IEEE Trans. Geosci. Remote Sens. 40, 375 (2002).
    [CrossRef]
  2. T. Tu, Opt. Eng. (Bellingham) 39, 897 (2000).
    [CrossRef]
  3. H. A. L. Kiers, J. Chemom. 14, 105 (2000).
    [CrossRef]
  4. A. Cichocki, R. Zdunek, and S. Amari, IEEE Signal Process. Mag. 25, 142 (2008).
    [CrossRef]
  5. A. Cichocki and S. Amari, Adaptive Blind Signal and Image Processing (Wiley, 2002).
    [CrossRef]
  6. J. F. Cardoso and A. Soulomniac, Proc. IEE F 140, 362 (1993).
  7. J. M. P. Nascimento and J. M. Dias Bioucas, IEEE Trans. Geosci. Remote Sens. 43, 175 (2005).
    [CrossRef]
  8. N. Renard, S. Bourennane, and J. Blanc-Talon, IEEE Geosci. Remote Sens. Lett. 5, 138 (2008).
    [CrossRef]
  9. N. Renard and S. Bourennane, IEEE Trans. Geosci. Remote Sens. 46, 2407 (2008).
    [CrossRef]
  10. Q. Zhang, H. Wang, R. J. Plemons, and V. P. Pauca, J. Opt. Soc. Am. A 25, 3001 (2008).
    [CrossRef]
  11. L. R. Tucker, Psychometrika 31, 279 (1966).
    [CrossRef] [PubMed]
  12. J. D. Carrol and J. J. Chang, Psychometrika 35, 283 (1970).
    [CrossRef]
  13. R. A. Harshman, UCLA Working Papers in Phonetics 16, 1 (1970).
  14. E. Acar and B. Yener, IEEE Trans. Knowl. Data Eng. 21, 6 (2009).
    [CrossRef]
  15. J. B. Kruskal, Linear Algebr. Appl. 18, 95 (1977).
    [CrossRef]
  16. N. D. Sidiropoulos and R. Bro, J. Chemom. 14, 229 (2000).
    [CrossRef]
  17. Y. D. Kim, A. Cichocki, and S. Choi, presented at the 2008 IEEE ICASSP International Conference on Acoustics, Speech and Signal Processing, Las Vegas, Nevada, March 30-April 4, 2008, pp. 1829-1832.
  18. A. H. Phan and A. Cichocki, Lect. Notes Comput. Sci. 5264, 772 (2008).
    [CrossRef]
  19. C. A. Anderson and R. Bro, Chemom. Intell. Lab. Syst. 52, 1 (2000).
    [CrossRef]
  20. R. Zdunek and A. Cichocki, Signal Process. 87, 1904 (2007).
    [CrossRef]
  21. A. Hyvärinen, in Proceedings of the International Conference on Artificial Neural Networks (ICANN'98) (IEEE, 1998), pp. 541-546.
  22. B. W. Bader and T. G. Kolda, MATLAB Tensor Toolbox version 2.2., http://csmr.ca.sandia.gov/~tkolda/TensorToolbox.
  23. Ch. Li, Ch. Xu, Ch. Gui, and M. D. Fox, in IEEE International Conference on Computer Vision and Pattern Recognition (IEEE, 2005), pp. 430-436.

2009

E. Acar and B. Yener, IEEE Trans. Knowl. Data Eng. 21, 6 (2009).
[CrossRef]

2008

A. H. Phan and A. Cichocki, Lect. Notes Comput. Sci. 5264, 772 (2008).
[CrossRef]

Q. Zhang, H. Wang, R. J. Plemons, and V. P. Pauca, J. Opt. Soc. Am. A 25, 3001 (2008).
[CrossRef]

A. Cichocki, R. Zdunek, and S. Amari, IEEE Signal Process. Mag. 25, 142 (2008).
[CrossRef]

N. Renard, S. Bourennane, and J. Blanc-Talon, IEEE Geosci. Remote Sens. Lett. 5, 138 (2008).
[CrossRef]

N. Renard and S. Bourennane, IEEE Trans. Geosci. Remote Sens. 46, 2407 (2008).
[CrossRef]

2007

R. Zdunek and A. Cichocki, Signal Process. 87, 1904 (2007).
[CrossRef]

2005

J. M. P. Nascimento and J. M. Dias Bioucas, IEEE Trans. Geosci. Remote Sens. 43, 175 (2005).
[CrossRef]

2002

C.-I. Chang, S.-S. Chiang, J. A. Smith, and I. W. Ginsberg, IEEE Trans. Geosci. Remote Sens. 40, 375 (2002).
[CrossRef]

2000

T. Tu, Opt. Eng. (Bellingham) 39, 897 (2000).
[CrossRef]

H. A. L. Kiers, J. Chemom. 14, 105 (2000).
[CrossRef]

N. D. Sidiropoulos and R. Bro, J. Chemom. 14, 229 (2000).
[CrossRef]

C. A. Anderson and R. Bro, Chemom. Intell. Lab. Syst. 52, 1 (2000).
[CrossRef]

1993

J. F. Cardoso and A. Soulomniac, Proc. IEE F 140, 362 (1993).

1977

J. B. Kruskal, Linear Algebr. Appl. 18, 95 (1977).
[CrossRef]

1970

J. D. Carrol and J. J. Chang, Psychometrika 35, 283 (1970).
[CrossRef]

R. A. Harshman, UCLA Working Papers in Phonetics 16, 1 (1970).

1966

L. R. Tucker, Psychometrika 31, 279 (1966).
[CrossRef] [PubMed]

Acar, E.

E. Acar and B. Yener, IEEE Trans. Knowl. Data Eng. 21, 6 (2009).
[CrossRef]

Amari, S.

A. Cichocki, R. Zdunek, and S. Amari, IEEE Signal Process. Mag. 25, 142 (2008).
[CrossRef]

A. Cichocki and S. Amari, Adaptive Blind Signal and Image Processing (Wiley, 2002).
[CrossRef]

Anderson, C. A.

C. A. Anderson and R. Bro, Chemom. Intell. Lab. Syst. 52, 1 (2000).
[CrossRef]

Bader, B. W.

B. W. Bader and T. G. Kolda, MATLAB Tensor Toolbox version 2.2., http://csmr.ca.sandia.gov/~tkolda/TensorToolbox.

Blanc-Talon, J.

N. Renard, S. Bourennane, and J. Blanc-Talon, IEEE Geosci. Remote Sens. Lett. 5, 138 (2008).
[CrossRef]

Bourennane, S.

N. Renard, S. Bourennane, and J. Blanc-Talon, IEEE Geosci. Remote Sens. Lett. 5, 138 (2008).
[CrossRef]

N. Renard and S. Bourennane, IEEE Trans. Geosci. Remote Sens. 46, 2407 (2008).
[CrossRef]

Bro, R.

N. D. Sidiropoulos and R. Bro, J. Chemom. 14, 229 (2000).
[CrossRef]

C. A. Anderson and R. Bro, Chemom. Intell. Lab. Syst. 52, 1 (2000).
[CrossRef]

Cardoso, J. F.

J. F. Cardoso and A. Soulomniac, Proc. IEE F 140, 362 (1993).

Carrol, J. D.

J. D. Carrol and J. J. Chang, Psychometrika 35, 283 (1970).
[CrossRef]

Chang, C.-I.

C.-I. Chang, S.-S. Chiang, J. A. Smith, and I. W. Ginsberg, IEEE Trans. Geosci. Remote Sens. 40, 375 (2002).
[CrossRef]

Chang, J. J.

J. D. Carrol and J. J. Chang, Psychometrika 35, 283 (1970).
[CrossRef]

Chiang, S.-S.

C.-I. Chang, S.-S. Chiang, J. A. Smith, and I. W. Ginsberg, IEEE Trans. Geosci. Remote Sens. 40, 375 (2002).
[CrossRef]

Choi, S.

Y. D. Kim, A. Cichocki, and S. Choi, presented at the 2008 IEEE ICASSP International Conference on Acoustics, Speech and Signal Processing, Las Vegas, Nevada, March 30-April 4, 2008, pp. 1829-1832.

Cichocki, A.

A. Cichocki, R. Zdunek, and S. Amari, IEEE Signal Process. Mag. 25, 142 (2008).
[CrossRef]

A. H. Phan and A. Cichocki, Lect. Notes Comput. Sci. 5264, 772 (2008).
[CrossRef]

R. Zdunek and A. Cichocki, Signal Process. 87, 1904 (2007).
[CrossRef]

A. Cichocki and S. Amari, Adaptive Blind Signal and Image Processing (Wiley, 2002).
[CrossRef]

Y. D. Kim, A. Cichocki, and S. Choi, presented at the 2008 IEEE ICASSP International Conference on Acoustics, Speech and Signal Processing, Las Vegas, Nevada, March 30-April 4, 2008, pp. 1829-1832.

Dias Bioucas, J. M.

J. M. P. Nascimento and J. M. Dias Bioucas, IEEE Trans. Geosci. Remote Sens. 43, 175 (2005).
[CrossRef]

Fox, M. D.

Ch. Li, Ch. Xu, Ch. Gui, and M. D. Fox, in IEEE International Conference on Computer Vision and Pattern Recognition (IEEE, 2005), pp. 430-436.

Ginsberg, I. W.

C.-I. Chang, S.-S. Chiang, J. A. Smith, and I. W. Ginsberg, IEEE Trans. Geosci. Remote Sens. 40, 375 (2002).
[CrossRef]

Gui, Ch.

Ch. Li, Ch. Xu, Ch. Gui, and M. D. Fox, in IEEE International Conference on Computer Vision and Pattern Recognition (IEEE, 2005), pp. 430-436.

Harshman, R. A.

R. A. Harshman, UCLA Working Papers in Phonetics 16, 1 (1970).

Hyvärinen, A.

A. Hyvärinen, in Proceedings of the International Conference on Artificial Neural Networks (ICANN'98) (IEEE, 1998), pp. 541-546.

Kiers, H. A. L.

H. A. L. Kiers, J. Chemom. 14, 105 (2000).
[CrossRef]

Kim, Y. D.

Y. D. Kim, A. Cichocki, and S. Choi, presented at the 2008 IEEE ICASSP International Conference on Acoustics, Speech and Signal Processing, Las Vegas, Nevada, March 30-April 4, 2008, pp. 1829-1832.

Kolda, T. G.

B. W. Bader and T. G. Kolda, MATLAB Tensor Toolbox version 2.2., http://csmr.ca.sandia.gov/~tkolda/TensorToolbox.

Kruskal, J. B.

J. B. Kruskal, Linear Algebr. Appl. 18, 95 (1977).
[CrossRef]

Li, Ch.

Ch. Li, Ch. Xu, Ch. Gui, and M. D. Fox, in IEEE International Conference on Computer Vision and Pattern Recognition (IEEE, 2005), pp. 430-436.

Nascimento, J. M. P.

J. M. P. Nascimento and J. M. Dias Bioucas, IEEE Trans. Geosci. Remote Sens. 43, 175 (2005).
[CrossRef]

Pauca, V. P.

Phan, A. H.

A. H. Phan and A. Cichocki, Lect. Notes Comput. Sci. 5264, 772 (2008).
[CrossRef]

Plemons, R. J.

Renard, N.

N. Renard, S. Bourennane, and J. Blanc-Talon, IEEE Geosci. Remote Sens. Lett. 5, 138 (2008).
[CrossRef]

N. Renard and S. Bourennane, IEEE Trans. Geosci. Remote Sens. 46, 2407 (2008).
[CrossRef]

Sidiropoulos, N. D.

N. D. Sidiropoulos and R. Bro, J. Chemom. 14, 229 (2000).
[CrossRef]

Smith, J. A.

C.-I. Chang, S.-S. Chiang, J. A. Smith, and I. W. Ginsberg, IEEE Trans. Geosci. Remote Sens. 40, 375 (2002).
[CrossRef]

Soulomniac, A.

J. F. Cardoso and A. Soulomniac, Proc. IEE F 140, 362 (1993).

Tu, T.

T. Tu, Opt. Eng. (Bellingham) 39, 897 (2000).
[CrossRef]

Tucker, L. R.

L. R. Tucker, Psychometrika 31, 279 (1966).
[CrossRef] [PubMed]

Wang, H.

Xu, Ch.

Ch. Li, Ch. Xu, Ch. Gui, and M. D. Fox, in IEEE International Conference on Computer Vision and Pattern Recognition (IEEE, 2005), pp. 430-436.

Yener, B.

E. Acar and B. Yener, IEEE Trans. Knowl. Data Eng. 21, 6 (2009).
[CrossRef]

Zdunek, R.

A. Cichocki, R. Zdunek, and S. Amari, IEEE Signal Process. Mag. 25, 142 (2008).
[CrossRef]

R. Zdunek and A. Cichocki, Signal Process. 87, 1904 (2007).
[CrossRef]

Zhang, Q.

Chemom. Intell. Lab. Syst.

C. A. Anderson and R. Bro, Chemom. Intell. Lab. Syst. 52, 1 (2000).
[CrossRef]

IEEE Geosci. Remote Sens. Lett.

N. Renard, S. Bourennane, and J. Blanc-Talon, IEEE Geosci. Remote Sens. Lett. 5, 138 (2008).
[CrossRef]

IEEE Signal Process. Mag.

A. Cichocki, R. Zdunek, and S. Amari, IEEE Signal Process. Mag. 25, 142 (2008).
[CrossRef]

IEEE Trans. Geosci. Remote Sens.

C.-I. Chang, S.-S. Chiang, J. A. Smith, and I. W. Ginsberg, IEEE Trans. Geosci. Remote Sens. 40, 375 (2002).
[CrossRef]

N. Renard and S. Bourennane, IEEE Trans. Geosci. Remote Sens. 46, 2407 (2008).
[CrossRef]

J. M. P. Nascimento and J. M. Dias Bioucas, IEEE Trans. Geosci. Remote Sens. 43, 175 (2005).
[CrossRef]

IEEE Trans. Knowl. Data Eng.

E. Acar and B. Yener, IEEE Trans. Knowl. Data Eng. 21, 6 (2009).
[CrossRef]

J. Chemom.

H. A. L. Kiers, J. Chemom. 14, 105 (2000).
[CrossRef]

N. D. Sidiropoulos and R. Bro, J. Chemom. 14, 229 (2000).
[CrossRef]

J. Opt. Soc. Am. A

Lect. Notes Comput. Sci.

A. H. Phan and A. Cichocki, Lect. Notes Comput. Sci. 5264, 772 (2008).
[CrossRef]

Linear Algebr. Appl.

J. B. Kruskal, Linear Algebr. Appl. 18, 95 (1977).
[CrossRef]

Opt. Eng. (Bellingham)

T. Tu, Opt. Eng. (Bellingham) 39, 897 (2000).
[CrossRef]

Proc. IEE F

J. F. Cardoso and A. Soulomniac, Proc. IEE F 140, 362 (1993).

Psychometrika

L. R. Tucker, Psychometrika 31, 279 (1966).
[CrossRef] [PubMed]

J. D. Carrol and J. J. Chang, Psychometrika 35, 283 (1970).
[CrossRef]

Signal Process.

R. Zdunek and A. Cichocki, Signal Process. 87, 1904 (2007).
[CrossRef]

UCLA Working Papers in Phonetics

R. A. Harshman, UCLA Working Papers in Phonetics 16, 1 (1970).

Other

A. Cichocki and S. Amari, Adaptive Blind Signal and Image Processing (Wiley, 2002).
[CrossRef]

Y. D. Kim, A. Cichocki, and S. Choi, presented at the 2008 IEEE ICASSP International Conference on Acoustics, Speech and Signal Processing, Las Vegas, Nevada, March 30-April 4, 2008, pp. 1829-1832.

A. Hyvärinen, in Proceedings of the International Conference on Artificial Neural Networks (ICANN'98) (IEEE, 1998), pp. 541-546.

B. W. Bader and T. G. Kolda, MATLAB Tensor Toolbox version 2.2., http://csmr.ca.sandia.gov/~tkolda/TensorToolbox.

Ch. Li, Ch. Xu, Ch. Gui, and M. D. Fox, in IEEE International Conference on Computer Vision and Pattern Recognition (IEEE, 2005), pp. 430-436.

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Figures (2)

Fig. 1
Fig. 1

Experimental fluorescent MSI (RGB) image of skin tumor: (a) high-intensity version, (b) low-intensity version. Spatial maps of the tumor extracted from Fig. 1b by means of (c) α-NTF algorithm [18] with α = 0.1 , (d) SO NMF algorithm [5], (e) DCA algorithm [7, 21]. (f) Evolution curve calculated by level set method on Fig. 1b after 1000 iterations. Dark red color indicates that tumor is present with probability 1, while dark blue color indicates that tumor is present with probability 0.

Fig. 2
Fig. 2

ROC curves calculated for spatial maps of the tumor shown in Figs. 1c, 1d, 1e: open squares, α-NTF algorithm based on Tucker3 model with α = 0.1 ; asterisks, DCA algorithm [6, 21]; open triangles, SO NMF algorithm.

Equations (11)

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

X ̱ G ̱ × 1 A ( 1 ) × 2 A ( 2 ) × 3 A ( 3 ) ,
X ( 3 ) A S ,
A A ( 3 ) ,
S ̱ G ̱ × 1 A ( 1 ) × 2 A ( 2 ) = X ̱ × 3 ( A ( 3 ) ) 1 ,
X ( 3 ) A ( 3 ) G ( 3 ) [ A ( 2 ) A ( 1 ) ] T ,
A A ( 3 ) ,
S G ( 3 ) [ A ( 2 ) A ( 1 ) ] T = ( A ( 3 ) ) 1 X ( 3 ) .
G ̱ G ̱ { ( X ̱ X ̱ ̂ ) . α × 1 A ( 1 ) T × 2 A ( 2 ) T × 3 A ( 3 ) T E ̱ × 1 A ( 1 ) T × 2 A ( 2 ) T × 3 A ( 3 ) T } . 1 α ,
A ( n ) A ( n ) { [ ( X ̱ X ̱ ̂ ) . α ] ( n ) G A ( n ) T 11 T G A ( n ) T } . 1 α ,
[ ( X ̱ X ̱ ̂ ) . α ] ( n ) G A ( n ) T = [ ( X ̱ X ̱ ̂ ) . α × m n A ( m ) T ] n G ( n ) T ,
1 T G A ( n ) T = [ G ̱ × m n 1 T A ( m ) ] ( n ) T ,

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