Abstract

An important and well-studied problem in hyperspectral image data applications is to identify materials present in the object or scene being imaged and to quantify their abundance in the mixture. Due to the increasing quantity of data usually encountered in hyperspectral datasets, effective data compression is also an important consideration. In this paper, we develop novel methods based on tensor analysis that focus on all three of these goals: material identification, material abundance estimation, and data compression. Test results are reported in all three perspectives.

© 2008 Optical Society of America

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2008 (3)

A. Cichocki, R. Zdunek, and S. Amari, “Nonnegative matrix and tensor factorization,” IEEE Signal Process. Mag. 21, 142-145 (2008).
[CrossRef]

M. Chu and M. M. Lin, “Low dimensional polytope approximation and its application to nonnegative matrix factorization,” SIAM J. Comput. 30, 1131-1155 (2008).
[CrossRef]

Y. Huang, M. Ng, and Y. Wen, “A fast total variation minimization method for image restoration,” Multiscale Model. Simul. 7, 774-795 (2008).
[CrossRef]

2007 (2)

C. Lin, “Projected gradient methods for non-negative matrix factorization,” Neural Comput. 19, 2756-2779 (2007).
[CrossRef] [PubMed]

M. Berry, M. Browne, A. Langville, P. Pauca, and R. Plemmons, “A survey of algorithms and applications for approximate nonnegative matrix factorization,” Computat. Statistics Data Anal. 52, 155-173 (2007).
[CrossRef]

2006 (4)

P. Pauca, J. Piper, and R. Plemmons, “Nonnegative matrix factorization for spectral data analysis,” Linear Algebr. Appl. 416, 29-47 (2006).
[CrossRef]

T. Blake, S. Cain, M. Goda, and K. Jerkatis, “Enhancing the resolution of spectral images,” Proc. SPIE 6233, 623309 (2006).
[CrossRef]

J. Wang and C.-I. Chang, “Applications of independent component analysis (ICA) in endmember extraction and abundance quantification for hyperspectral imagery,” IEEE Trans. Geosci. Remote Sens. 44, 2601-2616 (2006).
[CrossRef]

J. Scholl, K. Hege, M. Lloyd-Hart, D. O'Conneel, W. Johnson, and E. Dereniak, “Evaluations of classification and spectral unmixing algorithms using ground based satellite imaging,” Proc. SPIE 6233, 1-12 (2006).

2005 (1)

N. Goodwin, N. C. Coops, and C. Stone, “Assessing plantation canopy condition from airborne imagery using spectral mixture analysis and fractional abundances,” Int. J. Appl. Earth Obs. Geoinf. , 7, 11-28 (2005).
[CrossRef]

2004 (2)

K. J. Abercromby, J. Africano, K. Hamada, E. Stansbery, P. Sydney and P. Kervin, “Physical properties of orbital debris from spectroscopic observations,” Adv. Space Res. 34, 1021-1025 (2004).
[CrossRef]

Y. Du, C-I. Chang, H. Ren, C-C. Chang, J. O. Jensen and F. M. D'Amico, “New hyperspectral discrimination measure for spectral characterization,” Opt. Eng. (Bellingham) 43, 1777-1786 (2004).
[CrossRef]

2003 (3)

S. Kaewpijit, J. Le Moigne, and T. El-Ghazawi, “Automatic reduction of hyperspectral imagery using wavelet spectral analysis,” IEEE Trans. Geosci. Remote Sens. 41, 863-871 (2003).
[CrossRef]

K. Hege, D. O'Connell, W. Johnson, S. Basty, and E. Dereniak, “Hyperspectral imaging for astronomy and space surveillance,” Proc. SPIE 5159, 380-391 (2003).
[CrossRef]

N. K. M. Faber, R. Bro, and P. K. Hopke, “Recent developments in CANDECOMP/PARAFAC algorithms: a critical review,” Chemom. Intell. Lab. Syst. 65, 119-137 (2003).
[CrossRef]

2002 (1)

A. Plaza, P. Martinez, R. Perez and J. Plaza, “Spatial/spectral endmember extraction by multidimensional morphological operations,” IEEE Trans. Geosci. Remote Sens. 40, 2025-2041 (2002).
[CrossRef]

2001 (1)

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

2000 (1)

L. Grippo and M. Sciandrone, “On the convergence of the block nonlinear Gauss-Seidel method under convex constraints,” Oper. Res. Lett. 26, 127-136 (2000).
[CrossRef]

1999 (2)

D. Lee and H. Seung, “Learning the parts of objects by non-negative matrix factorization,” Nature 401, 788-791 (1999).
[CrossRef] [PubMed]

M. Winter, “N-finder: an algorithm for fast autonomous spectral endmember determination in hyperspectral data,” Proc. SPIE 3753, 266-277 (1999).
[CrossRef]

1997 (1)

R. Bro and S. D. Jong, “A fast non-negativity-constrained least squares algorithm,” J. Chemom. 11, 393-401 (1997).
[CrossRef]

1996 (1)

S. Qian, B. Hollinger, D. Williams, and D. Manak, “Fast three-dimensional data compression of hyperspectral imagery using vector quantization with spectral-feature-based binary coding,” Opt. Eng. (Bellingham) 35, 3242-3249 (1996).
[CrossRef]

1994 (1)

P. Paatero and U. Tapper, “Positive matrix factorization a nonnegative factor model with optimal utilization of error-estimates of data value,” Environmetrics 5, 111-126 (1994).
[CrossRef]

1977 (1)

J. Kruskal, “Three-way arrays: rank and uniqueness of trilinear decompositions, with applications to arithmetic complexity and statistics,” Linear Algebr. Appl. 18, 95-138 (1977).
[CrossRef]

1976 (1)

D. Bertsekas, “On the Goldstein-Levitin-Polyak gradient projection method,” IEEE Trans. Autom. Control 21, 174-184 (1976).
[CrossRef]

Abercromby, K. J.

K. J. Abercromby, J. Africano, K. Hamada, E. Stansbery, P. Sydney and P. Kervin, “Physical properties of orbital debris from spectroscopic observations,” Adv. Space Res. 34, 1021-1025 (2004).
[CrossRef]

Africano, J.

K. J. Abercromby, J. Africano, K. Hamada, E. Stansbery, P. Sydney and P. Kervin, “Physical properties of orbital debris from spectroscopic observations,” Adv. Space Res. 34, 1021-1025 (2004).
[CrossRef]

Althouse, M. L. G.

Q. Du, C.-I Chang, D. C. Heinz, M. L. G. Althouse, I. W. Ginsberg, “A linear mixture analysis-based compression for hyperspectral image analysis,” in Proceeding of the IEEE International Geoscience and Remote Sensing Symposium (IEEE, 2000), Vol. 2, pp. 585-587.

Amari, S.

A. Cichocki, R. Zdunek, and S. Amari, “Nonnegative matrix and tensor factorization,” IEEE Signal Process. Mag. 21, 142-145 (2008).
[CrossRef]

A. Cichocki, R. Zdunek, and S. Amari, “Hierarchical ALS algorithms for nonnegative matrix and 3D tensor factorization,” in Independent Component Analysis, Vol. 4666 of Lecture Notes in Computer Science (Springer, 2007), pp. 169-176.
[CrossRef]

Bader, B.

B. Bader, M. Berry, and M. Browne, “Discussion tracking in Enron email using PARAFAC,” in Survey of Text Mining II Clustering, Classification, and Retrieval, M.Berry and M.Castellanos, eds. (Springer, 2008), pp. 147-163.

Basty, S.

K. Hege, D. O'Connell, W. Johnson, S. Basty, and E. Dereniak, “Hyperspectral imaging for astronomy and space surveillance,” Proc. SPIE 5159, 380-391 (2003).
[CrossRef]

Berry, M.

M. Berry, M. Browne, A. Langville, P. Pauca, and R. Plemmons, “A survey of algorithms and applications for approximate nonnegative matrix factorization,” Computat. Statistics Data Anal. 52, 155-173 (2007).
[CrossRef]

B. Bader, M. Berry, and M. Browne, “Discussion tracking in Enron email using PARAFAC,” in Survey of Text Mining II Clustering, Classification, and Retrieval, M.Berry and M.Castellanos, eds. (Springer, 2008), pp. 147-163.

Bertsekas, D.

D. Bertsekas, “On the Goldstein-Levitin-Polyak gradient projection method,” IEEE Trans. Autom. Control 21, 174-184 (1976).
[CrossRef]

Blake, T.

T. Blake, S. Cain, M. Goda, and K. Jerkatis, “Enhancing the resolution of spectral images,” Proc. SPIE 6233, 623309 (2006).
[CrossRef]

T. Blake, S. Cain, M. Goda, and K. Jerkatis, “Model of the AEOS Spectral Imaging Sensor (ASIS) for spectral image deconvolution,” in Proceedings of AMOS Technical Conference (Curran Associates, 2005).

T. Blake, S. Cain, M. Goda, and K. Jerkatis, “Reconstruction of spectral images from the AEOS Spectral Imaging Sensor,” in Proceedings of AMOS Technical Conference (Curran Associates, 2006).

Boardman, J. W.

J. W. Boardman, F. A. Kruse, and R. O. Green, “Mapping target signatures via partial unmixing of AVIRIS data,” in Summaries of the Fifth JPL Airborne Earth Science Workshop, JPL Publication 1 (1995), pp. 23-26.

Bro, R.

N. K. M. Faber, R. Bro, and P. K. Hopke, “Recent developments in CANDECOMP/PARAFAC algorithms: a critical review,” Chemom. Intell. Lab. Syst. 65, 119-137 (2003).
[CrossRef]

R. Bro and S. D. Jong, “A fast non-negativity-constrained least squares algorithm,” J. Chemom. 11, 393-401 (1997).
[CrossRef]

Browne, M.

M. Berry, M. Browne, A. Langville, P. Pauca, and R. Plemmons, “A survey of algorithms and applications for approximate nonnegative matrix factorization,” Computat. Statistics Data Anal. 52, 155-173 (2007).
[CrossRef]

B. Bader, M. Berry, and M. Browne, “Discussion tracking in Enron email using PARAFAC,” in Survey of Text Mining II Clustering, Classification, and Retrieval, M.Berry and M.Castellanos, eds. (Springer, 2008), pp. 147-163.

Cain, S.

T. Blake, S. Cain, M. Goda, and K. Jerkatis, “Enhancing the resolution of spectral images,” Proc. SPIE 6233, 623309 (2006).
[CrossRef]

T. Blake, S. Cain, M. Goda, and K. Jerkatis, “Model of the AEOS Spectral Imaging Sensor (ASIS) for spectral image deconvolution,” in Proceedings of AMOS Technical Conference (Curran Associates, 2005).

T. Blake, S. Cain, M. Goda, and K. Jerkatis, “Reconstruction of spectral images from the AEOS Spectral Imaging Sensor,” in Proceedings of AMOS Technical Conference (Curran Associates, 2006).

Chang, C.-I

Q. Du, C.-I Chang, D. C. Heinz, M. L. G. Althouse, I. W. Ginsberg, “A linear mixture analysis-based compression for hyperspectral image analysis,” in Proceeding of the IEEE International Geoscience and Remote Sensing Symposium (IEEE, 2000), Vol. 2, pp. 585-587.

Chang, C.-I.

J. Wang and C.-I. Chang, “Applications of independent component analysis (ICA) in endmember extraction and abundance quantification for hyperspectral imagery,” IEEE Trans. Geosci. Remote Sens. 44, 2601-2616 (2006).
[CrossRef]

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

Chang, C-C.

Y. Du, C-I. Chang, H. Ren, C-C. Chang, J. O. Jensen and F. M. D'Amico, “New hyperspectral discrimination measure for spectral characterization,” Opt. Eng. (Bellingham) 43, 1777-1786 (2004).
[CrossRef]

Chang, C-I.

Y. Du, C-I. Chang, H. Ren, C-C. Chang, J. O. Jensen and F. M. D'Amico, “New hyperspectral discrimination measure for spectral characterization,” Opt. Eng. (Bellingham) 43, 1777-1786 (2004).
[CrossRef]

Chen, D.

D. Chen and R. Plemmons, “Nonnegativity constraints in numerical analysis.” presented at the Symposium on the Birth of Numerical Analysis, Leuven, Belgium, October 2007.

Chu, M.

M. Chu and M. M. Lin, “Low dimensional polytope approximation and its application to nonnegative matrix factorization,” SIAM J. Comput. 30, 1131-1155 (2008).
[CrossRef]

Cichocki, A.

A. Cichocki, R. Zdunek, and S. Amari, “Nonnegative matrix and tensor factorization,” IEEE Signal Process. Mag. 21, 142-145 (2008).
[CrossRef]

A. Cichocki, R. Zdunek, and S. Amari, “Hierarchical ALS algorithms for nonnegative matrix and 3D tensor factorization,” in Independent Component Analysis, Vol. 4666 of Lecture Notes in Computer Science (Springer, 2007), pp. 169-176.
[CrossRef]

R. Zdunek and A. Cichocki, “Non-negative matrix factorization with quasi-Newton optimization,” in Proceedings of the Eighth International Conference on Artificial Intelligence and Soft Computing (Springer-Verlag, 2006), pp. 870-879.

Coops, N. C.

N. Goodwin, N. C. Coops, and C. Stone, “Assessing plantation canopy condition from airborne imagery using spectral mixture analysis and fractional abundances,” Int. J. Appl. Earth Obs. Geoinf. , 7, 11-28 (2005).
[CrossRef]

D'Amico, F. M.

Y. Du, C-I. Chang, H. Ren, C-C. Chang, J. O. Jensen and F. M. D'Amico, “New hyperspectral discrimination measure for spectral characterization,” Opt. Eng. (Bellingham) 43, 1777-1786 (2004).
[CrossRef]

Dereniak, E.

J. Scholl, K. Hege, M. Lloyd-Hart, D. O'Conneel, W. Johnson, and E. Dereniak, “Evaluations of classification and spectral unmixing algorithms using ground based satellite imaging,” Proc. SPIE 6233, 1-12 (2006).

K. Hege, D. O'Connell, W. Johnson, S. Basty, and E. Dereniak, “Hyperspectral imaging for astronomy and space surveillance,” Proc. SPIE 5159, 380-391 (2003).
[CrossRef]

Du, Q.

Q. Du, C.-I Chang, D. C. Heinz, M. L. G. Althouse, I. W. Ginsberg, “A linear mixture analysis-based compression for hyperspectral image analysis,” in Proceeding of the IEEE International Geoscience and Remote Sensing Symposium (IEEE, 2000), Vol. 2, pp. 585-587.

Du, Y.

Y. Du, C-I. Chang, H. Ren, C-C. Chang, J. O. Jensen and F. M. D'Amico, “New hyperspectral discrimination measure for spectral characterization,” Opt. Eng. (Bellingham) 43, 1777-1786 (2004).
[CrossRef]

El-Ghazawi, T.

S. Kaewpijit, J. Le Moigne, and T. El-Ghazawi, “Automatic reduction of hyperspectral imagery using wavelet spectral analysis,” IEEE Trans. Geosci. Remote Sens. 41, 863-871 (2003).
[CrossRef]

Faber, N. K. M.

N. K. M. Faber, R. Bro, and P. K. Hopke, “Recent developments in CANDECOMP/PARAFAC algorithms: a critical review,” Chemom. Intell. Lab. Syst. 65, 119-137 (2003).
[CrossRef]

Friedlander, M. P.

M. P. Friedlander and K. Hatz, “Computing nonnegative tensor factorizations,” Tech. Rep. (University of British Columbia, 2006).

Giffin, M.

P. Pauca, J. Piper, R. Plemmons, and M. Giffin, “Object characterization from spectral data using nonnegative factorization and information theory,” in Proceedings of AMOS Technical Conference (Curran Associates, 2004), pp. 591-600.

Ginsberg, I. W.

Q. Du, C.-I Chang, D. C. Heinz, M. L. G. Althouse, I. W. Ginsberg, “A linear mixture analysis-based compression for hyperspectral image analysis,” in Proceeding of the IEEE International Geoscience and Remote Sensing Symposium (IEEE, 2000), Vol. 2, pp. 585-587.

Goda, M.

T. Blake, S. Cain, M. Goda, and K. Jerkatis, “Enhancing the resolution of spectral images,” Proc. SPIE 6233, 623309 (2006).
[CrossRef]

T. Blake, S. Cain, M. Goda, and K. Jerkatis, “Model of the AEOS Spectral Imaging Sensor (ASIS) for spectral image deconvolution,” in Proceedings of AMOS Technical Conference (Curran Associates, 2005).

T. Blake, S. Cain, M. Goda, and K. Jerkatis, “Reconstruction of spectral images from the AEOS Spectral Imaging Sensor,” in Proceedings of AMOS Technical Conference (Curran Associates, 2006).

Goodwin, N.

N. Goodwin, N. C. Coops, and C. Stone, “Assessing plantation canopy condition from airborne imagery using spectral mixture analysis and fractional abundances,” Int. J. Appl. Earth Obs. Geoinf. , 7, 11-28 (2005).
[CrossRef]

Green, R. O.

J. W. Boardman, F. A. Kruse, and R. O. Green, “Mapping target signatures via partial unmixing of AVIRIS data,” in Summaries of the Fifth JPL Airborne Earth Science Workshop, JPL Publication 1 (1995), pp. 23-26.

Grippo, L.

L. Grippo and M. Sciandrone, “On the convergence of the block nonlinear Gauss-Seidel method under convex constraints,” Oper. Res. Lett. 26, 127-136 (2000).
[CrossRef]

Hamada, K.

K. J. Abercromby, J. Africano, K. Hamada, E. Stansbery, P. Sydney and P. Kervin, “Physical properties of orbital debris from spectroscopic observations,” Adv. Space Res. 34, 1021-1025 (2004).
[CrossRef]

Hanson, R. J.

C. L. Lawson and R. J. Hanson, Solving Least Squares Problems (Prentice Hall, 1974).

Harshman, R. A.

R. A. Harshman, “Foundations of the PARAFAC procedure: models and conditions for an explanatory multi-modal factor analysis,” UCLA Working Papers in Phonetics (1970), Vol. 16, pp. 1-84.

Hatz, K.

M. P. Friedlander and K. Hatz, “Computing nonnegative tensor factorizations,” Tech. Rep. (University of British Columbia, 2006).

Hauff, P.

R. Neville, K. Staenz, T. Szeredi, J. Lefebvre and P. Hauff, “Automatic endmember extraction from hyperspectral data for mineral exploration,” in Proceedings of the 4th International Airborne Remote Sensing Conference and Exhibition/21st Canadian Symposium on Remote Sensing (ERIM International 1999), pp. 21-24.

Hazan, T.

A. Shashua and T. Hazan, “Non-negative tensor factorization with applications to statistics and computer vision,” in Proceedings of the 22nd International Conference on Machine Learning (ACM, 2005), pp. 792-799.
[CrossRef]

Hege, K.

J. Scholl, K. Hege, M. Lloyd-Hart, D. O'Conneel, W. Johnson, and E. Dereniak, “Evaluations of classification and spectral unmixing algorithms using ground based satellite imaging,” Proc. SPIE 6233, 1-12 (2006).

K. Hege, D. O'Connell, W. Johnson, S. Basty, and E. Dereniak, “Hyperspectral imaging for astronomy and space surveillance,” Proc. SPIE 5159, 380-391 (2003).
[CrossRef]

Heinz, D.

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

Heinz, D. C.

Q. Du, C.-I Chang, D. C. Heinz, M. L. G. Althouse, I. W. Ginsberg, “A linear mixture analysis-based compression for hyperspectral image analysis,” in Proceeding of the IEEE International Geoscience and Remote Sensing Symposium (IEEE, 2000), Vol. 2, pp. 585-587.

Hollinger, B.

S. Qian, B. Hollinger, D. Williams, and D. Manak, “Fast three-dimensional data compression of hyperspectral imagery using vector quantization with spectral-feature-based binary coding,” Opt. Eng. (Bellingham) 35, 3242-3249 (1996).
[CrossRef]

Hopke, P. K.

N. K. M. Faber, R. Bro, and P. K. Hopke, “Recent developments in CANDECOMP/PARAFAC algorithms: a critical review,” Chemom. Intell. Lab. Syst. 65, 119-137 (2003).
[CrossRef]

Huang, Y.

Y. Huang, M. Ng, and Y. Wen, “A fast total variation minimization method for image restoration,” Multiscale Model. Simul. 7, 774-795 (2008).
[CrossRef]

Jain, A. K.

A. K. Jain, Fundamentals of Digital Image Processing (Prentice Hall, 1989).

Jensen, J. O.

Y. Du, C-I. Chang, H. Ren, C-C. Chang, J. O. Jensen and F. M. D'Amico, “New hyperspectral discrimination measure for spectral characterization,” Opt. Eng. (Bellingham) 43, 1777-1786 (2004).
[CrossRef]

Jerkatis, K.

T. Blake, S. Cain, M. Goda, and K. Jerkatis, “Enhancing the resolution of spectral images,” Proc. SPIE 6233, 623309 (2006).
[CrossRef]

T. Blake, S. Cain, M. Goda, and K. Jerkatis, “Model of the AEOS Spectral Imaging Sensor (ASIS) for spectral image deconvolution,” in Proceedings of AMOS Technical Conference (Curran Associates, 2005).

T. Blake, S. Cain, M. Goda, and K. Jerkatis, “Reconstruction of spectral images from the AEOS Spectral Imaging Sensor,” in Proceedings of AMOS Technical Conference (Curran Associates, 2006).

Johnson, W.

J. Scholl, K. Hege, M. Lloyd-Hart, D. O'Conneel, W. Johnson, and E. Dereniak, “Evaluations of classification and spectral unmixing algorithms using ground based satellite imaging,” Proc. SPIE 6233, 1-12 (2006).

K. Hege, D. O'Connell, W. Johnson, S. Basty, and E. Dereniak, “Hyperspectral imaging for astronomy and space surveillance,” Proc. SPIE 5159, 380-391 (2003).
[CrossRef]

Jong, S. D.

R. Bro and S. D. Jong, “A fast non-negativity-constrained least squares algorithm,” J. Chemom. 11, 393-401 (1997).
[CrossRef]

Jorgersen Abercromby, K.

K. Jorgersen Abercromby, NASA Johnson Space Center (personal communication, 2006).

Kaewpijit, S.

S. Kaewpijit, J. Le Moigne, and T. El-Ghazawi, “Automatic reduction of hyperspectral imagery using wavelet spectral analysis,” IEEE Trans. Geosci. Remote Sens. 41, 863-871 (2003).
[CrossRef]

Kervin, P.

K. J. Abercromby, J. Africano, K. Hamada, E. Stansbery, P. Sydney and P. Kervin, “Physical properties of orbital debris from spectroscopic observations,” Adv. Space Res. 34, 1021-1025 (2004).
[CrossRef]

Keshava, N.

N. Keshava, “A taxonomy of spectral unmixing algorithms and performance comparisons,” Rep. HTAP-9 (Lincoln Laboratory, MIT, 2002).

Kruse, F. A.

J. W. Boardman, F. A. Kruse, and R. O. Green, “Mapping target signatures via partial unmixing of AVIRIS data,” in Summaries of the Fifth JPL Airborne Earth Science Workshop, JPL Publication 1 (1995), pp. 23-26.

Kruskal, J.

J. Kruskal, “Three-way arrays: rank and uniqueness of trilinear decompositions, with applications to arithmetic complexity and statistics,” Linear Algebr. Appl. 18, 95-138 (1977).
[CrossRef]

Kwon, H.

H. Kwon and N. M. Nasrabadi, “Hyperspectral target detection using kernel spectral matched filter,” in Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition (IEEE Computer Society, 2004), Vol. 27, pp. 127-127.

Langville, A.

M. Berry, M. Browne, A. Langville, P. Pauca, and R. Plemmons, “A survey of algorithms and applications for approximate nonnegative matrix factorization,” Computat. Statistics Data Anal. 52, 155-173 (2007).
[CrossRef]

Lawson, C. L.

C. L. Lawson and R. J. Hanson, Solving Least Squares Problems (Prentice Hall, 1974).

Le Moigne, J.

S. Kaewpijit, J. Le Moigne, and T. El-Ghazawi, “Automatic reduction of hyperspectral imagery using wavelet spectral analysis,” IEEE Trans. Geosci. Remote Sens. 41, 863-871 (2003).
[CrossRef]

Lee, D.

D. Lee and H. Seung, “Learning the parts of objects by non-negative matrix factorization,” Nature 401, 788-791 (1999).
[CrossRef] [PubMed]

Lefebvre, J.

R. Neville, K. Staenz, T. Szeredi, J. Lefebvre and P. Hauff, “Automatic endmember extraction from hyperspectral data for mineral exploration,” in Proceedings of the 4th International Airborne Remote Sensing Conference and Exhibition/21st Canadian Symposium on Remote Sensing (ERIM International 1999), pp. 21-24.

Levin, A.

A. Shashua and A. Levin, “Linear image coding for regression and classification using the tensor-rank principle,” in Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition (IEEE Computer Society, 2001), pp. 42-49.

Lin, C.

C. Lin, “Projected gradient methods for non-negative matrix factorization,” Neural Comput. 19, 2756-2779 (2007).
[CrossRef] [PubMed]

Lin, M. M.

M. Chu and M. M. Lin, “Low dimensional polytope approximation and its application to nonnegative matrix factorization,” SIAM J. Comput. 30, 1131-1155 (2008).
[CrossRef]

Lloyd-Hart, M.

J. Scholl, K. Hege, M. Lloyd-Hart, D. O'Conneel, W. Johnson, and E. Dereniak, “Evaluations of classification and spectral unmixing algorithms using ground based satellite imaging,” Proc. SPIE 6233, 1-12 (2006).

Manak, D.

S. Qian, B. Hollinger, D. Williams, and D. Manak, “Fast three-dimensional data compression of hyperspectral imagery using vector quantization with spectral-feature-based binary coding,” Opt. Eng. (Bellingham) 35, 3242-3249 (1996).
[CrossRef]

Martinez, P.

A. Plaza, P. Martinez, R. Perez and J. Plaza, “Spatial/spectral endmember extraction by multidimensional morphological operations,” IEEE Trans. Geosci. Remote Sens. 40, 2025-2041 (2002).
[CrossRef]

Nasrabadi, N. M.

H. Kwon and N. M. Nasrabadi, “Hyperspectral target detection using kernel spectral matched filter,” in Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition (IEEE Computer Society, 2004), Vol. 27, pp. 127-127.

Neville, R.

R. Neville, K. Staenz, T. Szeredi, J. Lefebvre and P. Hauff, “Automatic endmember extraction from hyperspectral data for mineral exploration,” in Proceedings of the 4th International Airborne Remote Sensing Conference and Exhibition/21st Canadian Symposium on Remote Sensing (ERIM International 1999), pp. 21-24.

Ng, M.

Y. Huang, M. Ng, and Y. Wen, “A fast total variation minimization method for image restoration,” Multiscale Model. Simul. 7, 774-795 (2008).
[CrossRef]

O'Conneel, D.

J. Scholl, K. Hege, M. Lloyd-Hart, D. O'Conneel, W. Johnson, and E. Dereniak, “Evaluations of classification and spectral unmixing algorithms using ground based satellite imaging,” Proc. SPIE 6233, 1-12 (2006).

O'Connell, D.

K. Hege, D. O'Connell, W. Johnson, S. Basty, and E. Dereniak, “Hyperspectral imaging for astronomy and space surveillance,” Proc. SPIE 5159, 380-391 (2003).
[CrossRef]

Paatero, P.

P. Paatero and U. Tapper, “Positive matrix factorization a nonnegative factor model with optimal utilization of error-estimates of data value,” Environmetrics 5, 111-126 (1994).
[CrossRef]

Pauca, P.

M. Berry, M. Browne, A. Langville, P. Pauca, and R. Plemmons, “A survey of algorithms and applications for approximate nonnegative matrix factorization,” Computat. Statistics Data Anal. 52, 155-173 (2007).
[CrossRef]

P. Pauca, J. Piper, and R. Plemmons, “Nonnegative matrix factorization for spectral data analysis,” Linear Algebr. Appl. 416, 29-47 (2006).
[CrossRef]

P. Pauca, J. Piper, R. Plemmons, and M. Giffin, “Object characterization from spectral data using nonnegative factorization and information theory,” in Proceedings of AMOS Technical Conference (Curran Associates, 2004), pp. 591-600.

Perez, R.

A. Plaza, P. Martinez, R. Perez and J. Plaza, “Spatial/spectral endmember extraction by multidimensional morphological operations,” IEEE Trans. Geosci. Remote Sens. 40, 2025-2041 (2002).
[CrossRef]

Piper, J.

P. Pauca, J. Piper, and R. Plemmons, “Nonnegative matrix factorization for spectral data analysis,” Linear Algebr. Appl. 416, 29-47 (2006).
[CrossRef]

P. Pauca, J. Piper, R. Plemmons, and M. Giffin, “Object characterization from spectral data using nonnegative factorization and information theory,” in Proceedings of AMOS Technical Conference (Curran Associates, 2004), pp. 591-600.

Plaza, A.

A. Plaza, P. Martinez, R. Perez and J. Plaza, “Spatial/spectral endmember extraction by multidimensional morphological operations,” IEEE Trans. Geosci. Remote Sens. 40, 2025-2041 (2002).
[CrossRef]

Plaza, J.

A. Plaza, P. Martinez, R. Perez and J. Plaza, “Spatial/spectral endmember extraction by multidimensional morphological operations,” IEEE Trans. Geosci. Remote Sens. 40, 2025-2041 (2002).
[CrossRef]

Plemmons, R.

M. Berry, M. Browne, A. Langville, P. Pauca, and R. Plemmons, “A survey of algorithms and applications for approximate nonnegative matrix factorization,” Computat. Statistics Data Anal. 52, 155-173 (2007).
[CrossRef]

P. Pauca, J. Piper, and R. Plemmons, “Nonnegative matrix factorization for spectral data analysis,” Linear Algebr. Appl. 416, 29-47 (2006).
[CrossRef]

P. Pauca, J. Piper, R. Plemmons, and M. Giffin, “Object characterization from spectral data using nonnegative factorization and information theory,” in Proceedings of AMOS Technical Conference (Curran Associates, 2004), pp. 591-600.

D. Chen and R. Plemmons, “Nonnegativity constraints in numerical analysis.” presented at the Symposium on the Birth of Numerical Analysis, Leuven, Belgium, October 2007.

Qian, S.

S. Qian, B. Hollinger, D. Williams, and D. Manak, “Fast three-dimensional data compression of hyperspectral imagery using vector quantization with spectral-feature-based binary coding,” Opt. Eng. (Bellingham) 35, 3242-3249 (1996).
[CrossRef]

Ren, H.

Y. Du, C-I. Chang, H. Ren, C-C. Chang, J. O. Jensen and F. M. D'Amico, “New hyperspectral discrimination measure for spectral characterization,” Opt. Eng. (Bellingham) 43, 1777-1786 (2004).
[CrossRef]

Roggemann, M. C.

M. C. Roggemann and B. Welsh, Imaging Through Turbulence (CRC Press, 1996).

Scholl, J.

J. Scholl, K. Hege, M. Lloyd-Hart, D. O'Conneel, W. Johnson, and E. Dereniak, “Evaluations of classification and spectral unmixing algorithms using ground based satellite imaging,” Proc. SPIE 6233, 1-12 (2006).

Sciandrone, M.

L. Grippo and M. Sciandrone, “On the convergence of the block nonlinear Gauss-Seidel method under convex constraints,” Oper. Res. Lett. 26, 127-136 (2000).
[CrossRef]

Seung, H.

D. Lee and H. Seung, “Learning the parts of objects by non-negative matrix factorization,” Nature 401, 788-791 (1999).
[CrossRef] [PubMed]

Shashua, A.

A. Shashua and A. Levin, “Linear image coding for regression and classification using the tensor-rank principle,” in Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition (IEEE Computer Society, 2001), pp. 42-49.

A. Shashua and T. Hazan, “Non-negative tensor factorization with applications to statistics and computer vision,” in Proceedings of the 22nd International Conference on Machine Learning (ACM, 2005), pp. 792-799.
[CrossRef]

Staenz, K.

R. Neville, K. Staenz, T. Szeredi, J. Lefebvre and P. Hauff, “Automatic endmember extraction from hyperspectral data for mineral exploration,” in Proceedings of the 4th International Airborne Remote Sensing Conference and Exhibition/21st Canadian Symposium on Remote Sensing (ERIM International 1999), pp. 21-24.

Stansbery, E.

K. J. Abercromby, J. Africano, K. Hamada, E. Stansbery, P. Sydney and P. Kervin, “Physical properties of orbital debris from spectroscopic observations,” Adv. Space Res. 34, 1021-1025 (2004).
[CrossRef]

Stone, C.

N. Goodwin, N. C. Coops, and C. Stone, “Assessing plantation canopy condition from airborne imagery using spectral mixture analysis and fractional abundances,” Int. J. Appl. Earth Obs. Geoinf. , 7, 11-28 (2005).
[CrossRef]

Sydney, P.

K. J. Abercromby, J. Africano, K. Hamada, E. Stansbery, P. Sydney and P. Kervin, “Physical properties of orbital debris from spectroscopic observations,” Adv. Space Res. 34, 1021-1025 (2004).
[CrossRef]

Szeredi, T.

R. Neville, K. Staenz, T. Szeredi, J. Lefebvre and P. Hauff, “Automatic endmember extraction from hyperspectral data for mineral exploration,” in Proceedings of the 4th International Airborne Remote Sensing Conference and Exhibition/21st Canadian Symposium on Remote Sensing (ERIM International 1999), pp. 21-24.

Tapper, U.

P. Paatero and U. Tapper, “Positive matrix factorization a nonnegative factor model with optimal utilization of error-estimates of data value,” Environmetrics 5, 111-126 (1994).
[CrossRef]

Wang, H.

H. Wang, “Nonnegative tensor factorization for hyperspectral data analysis,” Master's Thesis (Wake Forest University, 2007).

Wang, J.

J. Wang and C.-I. Chang, “Applications of independent component analysis (ICA) in endmember extraction and abundance quantification for hyperspectral imagery,” IEEE Trans. Geosci. Remote Sens. 44, 2601-2616 (2006).
[CrossRef]

Welsh, B.

M. C. Roggemann and B. Welsh, Imaging Through Turbulence (CRC Press, 1996).

Wen, Y.

Y. Huang, M. Ng, and Y. Wen, “A fast total variation minimization method for image restoration,” Multiscale Model. Simul. 7, 774-795 (2008).
[CrossRef]

Williams, D.

S. Qian, B. Hollinger, D. Williams, and D. Manak, “Fast three-dimensional data compression of hyperspectral imagery using vector quantization with spectral-feature-based binary coding,” Opt. Eng. (Bellingham) 35, 3242-3249 (1996).
[CrossRef]

Winter, M.

M. Winter, “N-finder: an algorithm for fast autonomous spectral endmember determination in hyperspectral data,” Proc. SPIE 3753, 266-277 (1999).
[CrossRef]

Zdunek, R.

A. Cichocki, R. Zdunek, and S. Amari, “Nonnegative matrix and tensor factorization,” IEEE Signal Process. Mag. 21, 142-145 (2008).
[CrossRef]

A. Cichocki, R. Zdunek, and S. Amari, “Hierarchical ALS algorithms for nonnegative matrix and 3D tensor factorization,” in Independent Component Analysis, Vol. 4666 of Lecture Notes in Computer Science (Springer, 2007), pp. 169-176.
[CrossRef]

R. Zdunek and A. Cichocki, “Non-negative matrix factorization with quasi-Newton optimization,” in Proceedings of the Eighth International Conference on Artificial Intelligence and Soft Computing (Springer-Verlag, 2006), pp. 870-879.

Adv. Space Res. (1)

K. J. Abercromby, J. Africano, K. Hamada, E. Stansbery, P. Sydney and P. Kervin, “Physical properties of orbital debris from spectroscopic observations,” Adv. Space Res. 34, 1021-1025 (2004).
[CrossRef]

Chemom. Intell. Lab. Syst. (1)

N. K. M. Faber, R. Bro, and P. K. Hopke, “Recent developments in CANDECOMP/PARAFAC algorithms: a critical review,” Chemom. Intell. Lab. Syst. 65, 119-137 (2003).
[CrossRef]

Computat. Statistics Data Anal. (1)

M. Berry, M. Browne, A. Langville, P. Pauca, and R. Plemmons, “A survey of algorithms and applications for approximate nonnegative matrix factorization,” Computat. Statistics Data Anal. 52, 155-173 (2007).
[CrossRef]

Environmetrics (1)

P. Paatero and U. Tapper, “Positive matrix factorization a nonnegative factor model with optimal utilization of error-estimates of data value,” Environmetrics 5, 111-126 (1994).
[CrossRef]

IEEE Signal Process. Mag. (1)

A. Cichocki, R. Zdunek, and S. Amari, “Nonnegative matrix and tensor factorization,” IEEE Signal Process. Mag. 21, 142-145 (2008).
[CrossRef]

IEEE Trans. Autom. Control (1)

D. Bertsekas, “On the Goldstein-Levitin-Polyak gradient projection method,” IEEE Trans. Autom. Control 21, 174-184 (1976).
[CrossRef]

IEEE Trans. Geosci. Remote Sens. (4)

A. Plaza, P. Martinez, R. Perez and J. Plaza, “Spatial/spectral endmember extraction by multidimensional morphological operations,” IEEE Trans. Geosci. Remote Sens. 40, 2025-2041 (2002).
[CrossRef]

J. Wang and C.-I. Chang, “Applications of independent component analysis (ICA) in endmember extraction and abundance quantification for hyperspectral imagery,” IEEE Trans. Geosci. Remote Sens. 44, 2601-2616 (2006).
[CrossRef]

S. Kaewpijit, J. Le Moigne, and T. El-Ghazawi, “Automatic reduction of hyperspectral imagery using wavelet spectral analysis,” IEEE Trans. Geosci. Remote Sens. 41, 863-871 (2003).
[CrossRef]

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

Int. J. Appl. Earth Obs. Geoinf. (1)

N. Goodwin, N. C. Coops, and C. Stone, “Assessing plantation canopy condition from airborne imagery using spectral mixture analysis and fractional abundances,” Int. J. Appl. Earth Obs. Geoinf. , 7, 11-28 (2005).
[CrossRef]

J. Chemom. (1)

R. Bro and S. D. Jong, “A fast non-negativity-constrained least squares algorithm,” J. Chemom. 11, 393-401 (1997).
[CrossRef]

Linear Algebr. Appl. (2)

J. Kruskal, “Three-way arrays: rank and uniqueness of trilinear decompositions, with applications to arithmetic complexity and statistics,” Linear Algebr. Appl. 18, 95-138 (1977).
[CrossRef]

P. Pauca, J. Piper, and R. Plemmons, “Nonnegative matrix factorization for spectral data analysis,” Linear Algebr. Appl. 416, 29-47 (2006).
[CrossRef]

Multiscale Model. Simul. (1)

Y. Huang, M. Ng, and Y. Wen, “A fast total variation minimization method for image restoration,” Multiscale Model. Simul. 7, 774-795 (2008).
[CrossRef]

Nature (1)

D. Lee and H. Seung, “Learning the parts of objects by non-negative matrix factorization,” Nature 401, 788-791 (1999).
[CrossRef] [PubMed]

Neural Comput. (1)

C. Lin, “Projected gradient methods for non-negative matrix factorization,” Neural Comput. 19, 2756-2779 (2007).
[CrossRef] [PubMed]

Oper. Res. Lett. (1)

L. Grippo and M. Sciandrone, “On the convergence of the block nonlinear Gauss-Seidel method under convex constraints,” Oper. Res. Lett. 26, 127-136 (2000).
[CrossRef]

Opt. Eng. (Bellingham) (2)

S. Qian, B. Hollinger, D. Williams, and D. Manak, “Fast three-dimensional data compression of hyperspectral imagery using vector quantization with spectral-feature-based binary coding,” Opt. Eng. (Bellingham) 35, 3242-3249 (1996).
[CrossRef]

Y. Du, C-I. Chang, H. Ren, C-C. Chang, J. O. Jensen and F. M. D'Amico, “New hyperspectral discrimination measure for spectral characterization,” Opt. Eng. (Bellingham) 43, 1777-1786 (2004).
[CrossRef]

Proc. SPIE (4)

K. Hege, D. O'Connell, W. Johnson, S. Basty, and E. Dereniak, “Hyperspectral imaging for astronomy and space surveillance,” Proc. SPIE 5159, 380-391 (2003).
[CrossRef]

J. Scholl, K. Hege, M. Lloyd-Hart, D. O'Conneel, W. Johnson, and E. Dereniak, “Evaluations of classification and spectral unmixing algorithms using ground based satellite imaging,” Proc. SPIE 6233, 1-12 (2006).

T. Blake, S. Cain, M. Goda, and K. Jerkatis, “Enhancing the resolution of spectral images,” Proc. SPIE 6233, 623309 (2006).
[CrossRef]

M. Winter, “N-finder: an algorithm for fast autonomous spectral endmember determination in hyperspectral data,” Proc. SPIE 3753, 266-277 (1999).
[CrossRef]

SIAM J. Comput. (1)

M. Chu and M. M. Lin, “Low dimensional polytope approximation and its application to nonnegative matrix factorization,” SIAM J. Comput. 30, 1131-1155 (2008).
[CrossRef]

Other (22)

R. Zdunek and A. Cichocki, “Non-negative matrix factorization with quasi-Newton optimization,” in Proceedings of the Eighth International Conference on Artificial Intelligence and Soft Computing (Springer-Verlag, 2006), pp. 870-879.

D.R.Lide, ed., CRC Handbook of Chemistry and Physics, 83rd ed. (CRC Press, 2002).

H. Wang, “Nonnegative tensor factorization for hyperspectral data analysis,” Master's Thesis (Wake Forest University, 2007).

K. Jorgersen Abercromby, NASA Johnson Space Center (personal communication, 2006).

A. K. Jain, Fundamentals of Digital Image Processing (Prentice Hall, 1989).

M. C. Roggemann and B. Welsh, Imaging Through Turbulence (CRC Press, 1996).

R. Neville, K. Staenz, T. Szeredi, J. Lefebvre and P. Hauff, “Automatic endmember extraction from hyperspectral data for mineral exploration,” in Proceedings of the 4th International Airborne Remote Sensing Conference and Exhibition/21st Canadian Symposium on Remote Sensing (ERIM International 1999), pp. 21-24.

N. Keshava, “A taxonomy of spectral unmixing algorithms and performance comparisons,” Rep. HTAP-9 (Lincoln Laboratory, MIT, 2002).

T. Blake, S. Cain, M. Goda, and K. Jerkatis, “Reconstruction of spectral images from the AEOS Spectral Imaging Sensor,” in Proceedings of AMOS Technical Conference (Curran Associates, 2006).

H. Kwon and N. M. Nasrabadi, “Hyperspectral target detection using kernel spectral matched filter,” in Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition (IEEE Computer Society, 2004), Vol. 27, pp. 127-127.

Q. Du, C.-I Chang, D. C. Heinz, M. L. G. Althouse, I. W. Ginsberg, “A linear mixture analysis-based compression for hyperspectral image analysis,” in Proceeding of the IEEE International Geoscience and Remote Sensing Symposium (IEEE, 2000), Vol. 2, pp. 585-587.

R. A. Harshman, “Foundations of the PARAFAC procedure: models and conditions for an explanatory multi-modal factor analysis,” UCLA Working Papers in Phonetics (1970), Vol. 16, pp. 1-84.

C. L. Lawson and R. J. Hanson, Solving Least Squares Problems (Prentice Hall, 1974).

J. W. Boardman, F. A. Kruse, and R. O. Green, “Mapping target signatures via partial unmixing of AVIRIS data,” in Summaries of the Fifth JPL Airborne Earth Science Workshop, JPL Publication 1 (1995), pp. 23-26.

P. Pauca, J. Piper, R. Plemmons, and M. Giffin, “Object characterization from spectral data using nonnegative factorization and information theory,” in Proceedings of AMOS Technical Conference (Curran Associates, 2004), pp. 591-600.

T. Blake, S. Cain, M. Goda, and K. Jerkatis, “Model of the AEOS Spectral Imaging Sensor (ASIS) for spectral image deconvolution,” in Proceedings of AMOS Technical Conference (Curran Associates, 2005).

B. Bader, M. Berry, and M. Browne, “Discussion tracking in Enron email using PARAFAC,” in Survey of Text Mining II Clustering, Classification, and Retrieval, M.Berry and M.Castellanos, eds. (Springer, 2008), pp. 147-163.

A. Cichocki, R. Zdunek, and S. Amari, “Hierarchical ALS algorithms for nonnegative matrix and 3D tensor factorization,” in Independent Component Analysis, Vol. 4666 of Lecture Notes in Computer Science (Springer, 2007), pp. 169-176.
[CrossRef]

M. P. Friedlander and K. Hatz, “Computing nonnegative tensor factorizations,” Tech. Rep. (University of British Columbia, 2006).

A. Shashua and T. Hazan, “Non-negative tensor factorization with applications to statistics and computer vision,” in Proceedings of the 22nd International Conference on Machine Learning (ACM, 2005), pp. 792-799.
[CrossRef]

A. Shashua and A. Levin, “Linear image coding for regression and classification using the tensor-rank principle,” in Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition (IEEE Computer Society, 2001), pp. 42-49.

D. Chen and R. Plemmons, “Nonnegativity constraints in numerical analysis.” presented at the Symposium on the Birth of Numerical Analysis, Leuven, Belgium, October 2007.

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Figures (12)

Fig. 1
Fig. 1

Illustration of 3-D tensor approximate factorization.

Fig. 2
Fig. 2

Blurred and noisy simulated hyperspectral image of the HST.

Fig. 3
Fig. 3

Total variation and cumulative variation for the simulated data.

Fig. 4
Fig. 4

Spectral signatures of materials assigned to a HST model.

Fig. 5
Fig. 5

(a) Simulated image at wavelength 0.4 μ m . (b) Materials assigned to pixels in the HST.

Fig. 6
Fig. 6

Comparison of frames from the original and the reconstructed data.

Fig. 7
Fig. 7

Relative residual norm error descent through the iterations.

Fig. 8
Fig. 8

Reconstructed images from the NTF at wavelength 0.4212 μ m , with and without resampling.

Fig. 9
Fig. 9

Comparison of reconstruction error with and without resampling in preprocessing.

Fig. 10
Fig. 10

Matched endmembers using the noisy dataset.

Fig. 11
Fig. 11

Reconstructed color image of the HST for materials identification.

Fig. 12
Fig. 12

Results from a linear unmixing method with and without assuming prior knowledge.

Tables (7)

Tables Icon

Table 1 Algorithm 1: Alternating Least-Squares for NTF

Tables Icon

Table 2 Algorithm 2: An Improved Projected Gradient Method

Tables Icon

Table 1 Complexity Analysis of Armijo Rules Defined in Eqs. (6, 7)

Tables Icon

Table 2 Benchmarks for Two Improvements to the Projected Gradient Descent Method a

Tables Icon

Table 3 Materials, Colors and Fractional Abundances Used for Hubble Satellite Simulation

Tables Icon

Table 4 Numbers of Matched Z-Factors in the Four Datasets

Tables Icon

Table 5 True and Recovered Material Prevalences (in Percent)

Equations (27)

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

T = x y z R n × m × l ,
T = i = 1 k x ( i ) y ( i ) z ( i ) ,
x y = ( x 1 y , x 2 y , , x n y ) .
X Y = ( x 1 y 1 , , x n y n ) R m n × k ,
( T 1 ) ( k 1 ) D 3 + j , i = t i j k , T 1 R D 2 D 3 × D 1 ,
( T 2 ) ( k 1 ) D 3 + i , j = t i j k , T 2 R D 1 D 3 × D 2 ,
( T 3 ) ( j 1 ) D 2 + i , k = t i j k , T 3 R D 1 D 2 × D 3 ,
T F = i j k t i j k 2 .
min T ̃ T T ̃ F 2 , subject to T ̃ 0 .
X = [ x ( 1 ) , , x ( k ) ] R D 1 × k , Y = [ y ( 1 ) , , y ( k ) ] R D 2 × k ,
Z = [ z ( 1 ) , , z ( k ) ] R D 3 × k ,
T T ̃ F 2 = T 1 ( Z Y ) X T F 2 = T 2 ( Z X ) Y T F 2 = T 3 ( X Y ) Z T F 2 .
min H Φ ( H ) = A W H F 2 , subject to H 0 .
Ψ ( A , W ) = min H A W H F 2 , subject to H 0 .
min H Φ ( H ) = A W H F 2 , subject to 0 H U ,
Ψ b ( A , W ) = min H A W H F 2 .
Φ ( H ( p + 1 ) ) Φ ( H ( p ) ) σ vec ( Φ ( H ( p ) ) ) , vec ( H ( p + 1 ) H ( p ) ) ,
( 1 σ ) vec ( Φ ( H ) ) , vec ( H ̃ H )
+ 1 2 vec ( H ̃ H ) , vec ( ( W T W ) ( H ̃ H ) ) 0 .
v ( λ ) = Ω t ( x , y , λ ) λ d x d y ,
v ( λ k ) = v k i = 1 D 1 j = 1 D 2 k = 1 D 3 1 t i j , k + 1 t i j k ,
c ( λ ) = λ 1 λ v ( λ ̂ ) d λ ̂ ,
t ̃ i j k = t i j k + η i j k ( 1 ) t i j k + η i j k ( 1 ) ,
T 3 ( X Y ) Z .
t i j l = 1 k ( x i l y j l ) z l ,
min h Φ ( h ) = t i j W h 2 2 + α h T e , subject to 0 h u ,
min I ( i ) , z ( i ) T i = 1 k I ( i ) z ( i ) , subject to I ( i ) 0 , z ( i ) 0 ,

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