Third-order tensors as operators on matrices: A theoretical and computational framework with applications in imaging

SIAM Journal on Matrix Analysis and Applications

Volumn 34, Issue 1, 2012, Pages 148-172

  Kilmer, Misha E ^a   Braman, Karen ^b   Hao, Ning ^a   Hoover, Randy C ^b  

^a TUFTS UNIVERSITY   (United States)

^b South Dakota School of Mines and Technology   (United States)

Author keywords

Eigendecomposition; Krylov methods; Multidimensional arrays; Singular value decomposition; Tensor decomposition; Tensor SVD

Indexed keywords

COMPUTATIONAL FRAMEWORK; EIGEN DECOMPOSITION; IMAGE DEBLURRING PROBLEMS; KRYLOV METHOD; KRYLOV SUBSPACE METHOD; MULTIDIMENSIONAL ARRAYS; ORTHOGONAL PROJECTION; TENSOR DECOMPOSITION;

IMAGE ENHANCEMENT; ITERATIVE METHODS; OBJECT RECOGNITION; SINGULAR VALUE DECOMPOSITION;

TENSORS;

EID: 84877332017     PISSN: 08954798     EISSN: 10957162     Source Type: Journal    
DOI: 10.1137/110837711     Document Type: Article

References (35)

1. K. Braman, Third-order tensors as linear operators on a space of matrices, Linear Algebra Appl., 433 (2010), pp. 1241-1253.
2. K. Braman and R. Hoover, Tensor decomposition and application in signal and image processing, abstract, Invited Minisymposium Presentation, 2010 SIAM Annual Meeting; http://www.siam.org/meetings/an10/An10abstracts.pdf, p. 58, MS35.
3. K. Braman and M. Kilmer, Householder abstract booklet, abstract and talk, Householder Symposium, 2011.
4. J. Carroll and J. Chang, Analysis of individual differences in multidimensional scaling via an n-way generalization of "Eckart-Young" decomposition, Psychometrika, 35 (1970), pp. 283-319.
5. C.-Y. Chang, A. A. Maciejewski, and V. Balakrishnan, Fast eigenspace decomposition of correlated images, IEEE Trans. Image Process., 9 (2000), pp. 1937-1949.
6. P. Comon, Tensor decompositions, in Mathematics in Signal Processing V, J. G. McWhirter and I. K. Proudler, eds., Clarendon Press, Oxford, UK, 2002, pp. 1-24.
7. L. De Lathauwer and B. De Moor, From matrix to tensor: Multilinear algebra and signal processing, in Mathematics in Signal Processing IV, J. McWhirter and I. K. Proudler, eds., Clarendon Press, Oxford, UK, 1998, pp. 1-15.
8. L. De Lathauwer, B. De Moor, and J. Vandewalle, A multilinear singular value decomposition, SIAM J. Matrix Anal. Appl., 21 (2000), pp. 1253-1278.
9. D. F. Gleich, C. Greif, and J. M. Varah, The power and Arnoldi methods in an algebra of circulants, Numer. Linear Algebra Appl., doi: 10.1002/nla.1845.
10. N. Hao and M. Kilmer, Tensor-SVD with applications in image processing, InvitedMinisymposium Presentation, 2010 SIAM Annual Meeting; http://www.siam.org/meetings/an10/ An10abstracts.pdf, p. 58, MS35.
11. N. Hao, M. E. Kilmer, K. Braman, and R. C. Hoover, Facial recognition using tensor-tensor decompositions, SIAM J. Imaging Sci., 6 (2013), pp. 437-463.
12. R. Harshman, Foundations of the PARAFAC procedure: Model and conditions for an "explanatory" multi-mode factor analysis, UCLA Working Papers in Phonetics, 16 (1970), pp. 1-84.
13. W. S. Hoge and C.-F. Westin, Identification of translational displacements between Ndimensional data sets using the high order SVD and phase correlation, IEEE Trans. Image Process., 14 (2005), pp. 884-889.
14. R. C. Hoover, K. S. Braman, and N. Hao, Pose estimation from a single image using tensor decomposition and an algebra of circulants, in Proceedings of the IEEE International Conference on Intelligent Robots and Systems (IROS), 2011, pp. 2928-2939.
15. M. E. Kilmer, K. Braman, and N. Hao, Third Order Tensors as Operators on Matrices: A Theoretical and Computational Framework with Applications in Imaging, Tech. Report CS-TR-01-11, Tufts University, 2011.
16. M. E. Kilmer and C. D. Martin, Factorization strategies for third-order tensors, Linear Algebra Appl., 435 (2011), pp. 641-658.
17. M. E. Kilmer, C. D. Martin, and L. Perrone, A Third-Order Generalization of the Matrix SVD as a Product of Third-Order Tensors, Tech. Report TR-2008-4, Tufts University, 2008.
18. T. G. Kolda and B. W. Bader, Tensor decompositions and applications, SIAM Rev., 51 (2009), pp. 455-500.
19. T. G. Kolda and J. R. Mayo, Shifted power method for computing tensor eigenpairs, SIAM J. Matrix Anal. Appl., 32 (2011), pp. 1095-1124.
20. P. Kroonenberg, Three-Mode Principal Component Analysis: Theory and Applications, DSWO Press, Leiden, 1983.
21. K. Legaz, 3-D Model Database for Blender, http://3dmodels.katorlegaz.com/ , 2007.
22. L.-H. Lim, Singular values and eigenvalues of tensors: A variational approach, in CAMSA P'05: Proceeding of the IEEE International Workshop on Computational Advances in Multi-sensor Adaptive Processing, 2005, pp. 129-132.
23. J. Nagy and M. Kilmer, Kronecker product approximation for preconditioning in threedimensional imaging applications, IEEE Trans. Image Process., 15 (2006), pp. 604-613.
24. M. Ng, L. Qi, and G. Zhou, Finding the largest eigenvalue of a nonnegative tensor, SIAM J. Matrix Anal. Appl., 31 (2009), pp. 1090-1099.
25. L. Qi, Eigenvalues and invariants of tensors, J. Math. Anal. Appl., 325 (2007), pp. 1363-1367.
26. M. Rezghi and L. Eldèn, Diagonalization of tensors with circulant structure, Linear Algebra Appl., 435 (2011), pp. 422-447
27. B. Savas and L. Eldén, Krylov Subspace Methods for Tensor Computations, Tech. Report LITH-MAT-R-2009-02-SE, Department of Mathematics, Linköpings Universitet, 2009.
28. B. Savas and L. Eldén, Krylov-type methods for tensor computations I, Linear Algebra Appl., 438 (2013), pp. 891-918.
29. N. Sidiropoulos, R. Bro, and G. Giannakis, Parallel factor analysis in sensor array processing, IEEE Trans. Signal Process., 48 (2000), pp. 2377-2388.
30. A. Smilde, R. Bro, and P. Geladi, Multi-way Analysis: Applications in the Chemical Sciences, John Wiley, Chichester, UK, 2004.
31. L. N. Trefethen and D. Bau, III, Numerical Linear Algebra, SIAM, Philadelphia, 1997.
32. L. Tucker, Some mathematical notes on three-mode factor analysis, Psychometrika, 31 (1966), pp. 279-311.
33. M. Vasilescu and D. Terzopoulos, Multilinear analysis of image ensembles: Tensorfaces, in Proceedings of the 7th European Conference on Computer Vision (ECCV 2002), Lecture Notes in Comput. Sci. 2350, Springer, New York, 2002, pp. 447-460.
34. M. Vasilescu and D. Terzopoulos, Multilinear image analysis for facial recognition, in Proceedings of the 16th International Conference on Pattern Recognition (ICPR 2002), Vol. 2, IEEE Computer Society Press, 2002, pp. 511-514.
35. M. Vasilescu and D. Terzopoulos, Multilinear subspace analysis of image ensembles, in Proceedings of the 2003 IEEE Computer Society Conference on Computer Vision and Pattern Recognition (CVPR 2003), IEEE Computer Society Press, 2003, pp. 93-99.