Higher-order tensors in diffusion imaging

Mathematics and Visualization

Volumn 0, Issue , 2014, Pages 129-161

  Schultz, Thomas ^a,b   Fuster, Andrea ^c   Ghosh, Aurobrata ^d   Deriche, Rachid ^d   Florack, Luc ^c   Lim, Lek Heng ^e  

^a UNIVERSITY OF BONN   (Germany)

^b MAX PLANCK INSTITUTE FOR INTELLIGENT SYSTEMS   (Germany)

^c EINDHOVEN UNIVERSITY OF TECHNOLOGY   (Netherlands)

^d INRIA   (France)

^e UNIVERSITY OF CHICAGO   (United States)

Author keywords

[No Author keywords available]

Indexed keywords

DIFFUSION; DISTRIBUTION FUNCTIONS; HIGHER ORDER STATISTICS; MAGNETIC RESONANCE IMAGING;

DIFFUSION IMAGING; HIGH-ORDER; HIGHER-ORDER; HIGHER-ORDER TENSOR; LARGE DATA; MATHEMATICAL LANGUAGES; MODELLING AND ANALYSIS; MUSCLE TISSUES; NERVE TISSUE; NONINVASIVE TOOLS;

TENSORS;

EID: 84919912384     PISSN: 16123786     EISSN: 2197666X     Source Type: Book Series    
DOI: 10.1007/978-3-642-54301-2_6     Document Type: Conference Paper

References (116)

1. Aganj, I., Lenglet, C., Sapiro, G., Yacoub, E., Ugurbil, K., Harel, N.: Reconstruction of the orientation distribution function in single-and multiple-shell q-ball imaging within constant solid angle. Magn. Reson. Med. 64(2), 554-566 (2010)
2. Anderson, A.W.: Measurement of fiber orientation distributions using high angular resolution diffusion imaging. Magn. Reson. Med. 54(5), 1194-1206 (2005)
3. Arsigny, V., Fillard, P., Pennec, X., Ayache, N.: Log-Euclidean metrics for fast and simple calculus on diffusion tensors. Magn. Reson. Med. 56(2), 411-421 (2006)
4. Astola, L., Florack, L.: Finsler geometry on higher order tensor fields and applications to high angular resolution diffusion imaging. In: Proceedings of the International Conference on Scale Space and Variational Methods in Computer Vision (SSVM), Voss, pp. 224-234. Springer (2009)
5. Astola, L., Florack, L.: Finsler geometry on higher order tensor fields and applications to high angular resolution diffusion imaging. Int. J. Comput Vis. 92, 325-336 (2011)
6. Astola, L., Fuster, A., Florack, L.: A Riemannian scalar measure for diffusion tensor images. Pattern Recognit 44, 1885-1891 (2011)
7. Astola, L., Jalba, A., Balmashnova, E., Florack, L.: Finsler streamline tracking with single tensor orientation distribution function for high angular resolution diffusion imaging. J. Math. Imaging Vis. 41, 170-181 (2011)
8. Balmashnova, E., Fuster, A., Florack, L.: Decomposition of higher-order homogeneous tensors and applications to HARDI. In: Panagiotaki, E., O’Donnell, L., Schultz, T., Zhang, G.H. (eds.) Proceedings of the Computational Diffusion MRI (CDMRI), Nice, pp. 79-89 (2012)
9. Barmpoutis, A., Hwang, M.S., Howland, D., Forder, J.R., Vemuri, B.C.: Regularized positivedefinite fourth order tensor field estimation from DW-MRI. NeuroImage 45(1, Suppl. 1), S153-S162 (2009)
10. Barmpoutis, A., Jian, B., Vemuri, B.C., Shepherd, T.M.: Symmetric positive 4th order tensors & their estimation from diffusion weighted MRI. In: Karssemeijer, N., Lelieveldt B. (eds.) IPMI, Kerkrade. LNCS, vol. 4584, pp. 308-319 (2007)
11. Barmpoutis, A., Vemuri, B.C.: Exponential tensors: a framework for efficient higher-order DT-MRI computations. In: Proceedings of the IEEE International Symposium on Biomedical Imaging, Washington, DC, pp. 792-795 (2007)
12. Barmpoutis, A., Vemuri, B.C.: Groupwise registration and atlas construction of 4th-order tensor fields using the RC Riemannian metric. In: Yang, G.Z., Hawkes, D., Rueckert, D., Noble, A., Taylor, C. (eds.) Proceedings of the Medical Image Computing and Computer-Assisted Intervention (MICCAI), Part I, London. LNCS, vol. 5761, pp. 640-647 (2009)
13. Barmpoutis, A., Vemuri, B.C.: A unified framework for estimating diffusion tensors of any order with symmetric positive-definite constraints. In: Proceedings of the IEEE International Symposium on Biomedical Imaging, Rotterdam, pp. 1385-1388 (2010)
14. Barmpoutis, A., Vemuri, B.C., Forder, J.R.: Registration of high angular resolution diffusion MRI images using 4th order tensors. In: Ayache, N., Ourselin, S., Maeder, A. (eds.) Proceedings of the Medical Image Computing and Computer-Assisted Intervention, Part I, Brisbane. LNCS, vol. 4791, pp. 908-915 (2007)
15. Barmpoutis, A., Vemuri, B.C., Forder, J.R.: Fast displacement probability profile approximation from HARDI using 4th-order tensors. In: Proceedings of the IEEE International Symposium on Biomedical Imaging, Paris, pp. 911-914 (2008)
16. Barmpoutis, A., Zhuo, J.: Diffusion kurtosis imaging: robust estimation from DW-MRI using homogeneous polynomials. In: Proceedings of the IEEE International Symposium on Biomedical Imaging, Chicago, pp. 262-265 (2011)
17. Barnett, A.: Theory of Q-ball imaging redux: implications for fiber tracking. Magn. Reson. Med. 62(4), 910-923 (2009)
18. Basser, P.J., Mattiello, J., Le Bihan, D.: Estimation of the effective self-diffusion tensor from the NMR spin echo. J. Magn. Reson. B 103, 247-254 (1994)
19. Basser, P.J., Pajevic, S.: A normal distribution for tensor-valued random variables: applications to diffusion tensor MRI. IEEE Trans. Med. Imaging 22, 785-795 (2003)
20. Basser, P.J., Pajevic, S.: Spectral decomposition of a 4th-order covariance tensor: applications to diffusion tensor MRI. Signal Process. 87, 220-236 (2007)
21. Basser, P.J., Pierpaoli, C.: Microstructural and physiological features of tissues elucidated by quantitative-diffusion-tensor MRI. J. Magn. Reson. Ser. B 111, 209-219 (1996)
22. Behrens, T.E.J., Johansen-Berg, H., Jbabdi, S., Rushworth, M.F.S., Woolrich, M.W.: Probabilistic diffusion tractography with multiple fibre orientations: what can we gain? NeuroImage 34, 144-155 (2007)
23. Bloy, L., Verma, R.: On computing the underlying fiber directions from the diffusion orientation distribution function. In: Metaxas, D.N., Axel, L., Fichtinger, G., Székely, G. (eds.) Medical Image Computing and Computer-Assisted Intervention (MICCAI), New York. LNCS, vol. 5241, pp. 1-8. Springer (2008)
24. Boothby, W.: An Introduction to Differentiable Manifolds and Riemannian Geometry. Pure and Applied Mathematics, vol. 120, 2nd edn. Academic, Orlando (1986)
25. Callaghan, P.T., Eccles, C.D., Xia, Y.: NMR microscopy of dynamic displacements: k-space and q-space imaging. J. Phys. E 21(8), 820-822 (1988)
26. Comon, P., Golub, G., Lim, L.H., Mourrain, B.: Symmetric tensors and symmetric tensor rank. SIAM J. Matrix Anal. Appl. 30(3), 1254-1279 (2008)
27. Correia, M.M., Newcombe, V.F., Williams, G.B.: Contrast-to-noise ratios for indices of anisotropy obtained from diffusion MRI: a study with standard clinical b-values at 3T. NeuroImage 57(3), 1103-1115 (2011)
28. Descoteaux, M., Angelino, E., Fitzgibbons, S., Deriche, R.: Apparent diffusion coefficients from high angular resolution diffusion imaging: estimation and applications. Magn. Reson. Med. 56, 395-410 (2006)
29. Descoteaux, M., Angelino, E., Fitzgibbons, S., Deriche, R.: Regularized, fast, and robust analytical Q-Ball imaging. Magn. Reson. Med. 58, 497-510 (2007)
30. Descoteaux, M., Deriche, R., Knösche, T.R., Anwander, A.: Deterministic and probabilistic tractography based on complex fibre orientation distributions. IEEE Trans. Med. Imaging 28(2), 269-286 (2009)
31. De Silva, V., Lim, L.H.: Tensor rank and the ill-posedness of the best low-rank approximation problem. SIAM J. Matrix Anal. Appl. 30(3), 1084-1127 (2008)
32. Ellingson, B.M., Cloughesy, T.F., Lai, A., Nghiemphu, P.L., Liau, L.M., Pope, W.B.: High order diffusion tensor imaging in human glioblastoma. Acad. Radiol. 18(8), 947-954 (2011)
33. Essen, D.V., Ugurbil, K., Auerbach, E., Barch, D., Behrens, T., Bucholz, R., Chang, A., Chen, L., Corbetta, M., Curtiss, S., Penna, S.D., Feinberg, D., Glasser, M., Harel, N., Heath, A., Larson-Prior, L., Marcus, D., Michalareas, G., Moeller, S., Oostenveld, R., Petersen, S., Prior, F., Schlaggar, B., Smith, S., Snyder, A., Xu, J., Yacoub, E.: The human connectome project: a data acquisition perspective. NeuroImage 62(4), 2222-2231 (2012)
34. Florack, L., Balmashnova, E., Astola, L., Brunenberg, E.: A new tensorial framework for single-shell high angular resolution diffusion imaging. J. Math. Imaging Vis. 38, 171-181 (2010)
35. Fuster, A., Astola, L., Florack, L.: A Riemannian scalar measure for diffusion tensor images. In: Jiang, X., Petkov, N. (eds.) Computer Analysis of Images and Patterns. LNCS, vol. 5702, pp. 419-426. Springer, Berlin/New York (2009)
36. Fuster, A., van de Sande, J., Astola, L., Poupon, C., Velterop, J., ter Haar Romeny, B.M.: Fourth-order tensor invariants in high angular resolution diffusion imaging. In: Zhang, G.H., Adluru, N. (eds.) Proceedings of the MICCAI Workshop on Computational Diffusion MRI, Toronto, pp. 54-63 (2011)
37. Gelfand, I., Kapranov, M., Zelevinsky, A.: Discriminants, Resultants, and Multidimensional Determinants. Birkhäuser, Boston (1994)
38. Geroch, R.: Mathematical Physics. Chicago Lectures in Physics. University of Chicago Press, Chicago (1985)
39. Ghosh, A., Deriche, R.: From second to higher order tensors in diffusion-MRI. In: Aja-Fernández, S., de Luis García, R., Tao, D., Li, X. (eds.) Tensors in Image Processing and Computer Vision, pp. 315-334. Springer, London (2009)
40. Ghosh, A., Deriche, R.: Fast and closed-form ensemble-average-propagator approximation from the 4th-order diffusion tensor. In: Proc. IEEE Int’l Symposium on Biomedical Imaging, pp. 1105-1108 (2010)
41. Ghosh, A., Deriche, R.: Extracting geometrical features & peak fractional anisotropy from the ODF for white matter characterization. In: Proceedings of the IEEE International Symposium on Biomedical Imaging, Chicago, pp. 266-271 (2011)
42. Ghosh, A., Deriche, R.: Generalized invariants of a 4th order tensor: building blocks for new biomarkers in dMRI. In: Panagiotaki, E., O’Donnell, L., Schultz, T., Zhang, G.H. (eds.) Proceedings of the Computational Diffusion MRI (CDMRI), Nice, pp. 165-173 (2012)
43. Ghosh, A., Deriche, R., Moakher, M.: Ternary quartic approach for positive 4th order diffusion tensors revisited. In: Proceedings of the IEEE International Symposium on Biomedical Imaging, Boston, pp. 618-621 (2009)
44. Ghosh, A., Descoteaux, M., Deriche, R.: Riemannian framework for estimating symmetric positive definite 4th order diffusion tensors. In: Metaxas, D. (ed.) MICCAI, Part I, New York, LNCS, vol. 5241, pp. 858-865 (2008)
45. Ghosh, A., Özarslan, E., Deriche, R.: Challenges in reconstructing the propagator via a cumulant expansion of the one-dimensional qspace MR signal. In: Proceedings of the International Society for Magnetic Resonance in Medicine (ISMRM), Stockholm (2010)
46. Ghosh, A., Tsigaridas, E., Mourrain, B., Deriche, R.: A polynomial approach for extracting the extrema of a spherical function and its application in diffusion MRI. Med. Image Anal. (2013, in press). doi:10.1016/j.media.2013.03.004
47. Ghosh, A., Wassermann, D., Deriche, R.: A polynomial approach for maxima extraction and its application to tractography in HARDI. In: Székely, G., Hahn, H.K. (eds.) IPMI, Kloster Irsee. LNCS, vol. 6801, pp. 723-734 (2011)
48. Grigis, A., Renard, F., Noblet, V., Heinrich, C., Heitz, F., Armspach, J.P.: A new high order tensor decomposition: application to reorientation. In: Proceedings of the IEEE International Symposium on Biomedical Imaging, Chicago, pp. 258-261 (2011)
49. Gur, Y., Jiao, F., Zhu, S.X., Johnson, C.R.: White matter structure assessment from reduced HARDI data using low-rank polynomial approximations. In: Panagiotaki, E., O’Donnell, L., Schultz, T., Zhang, G.H. (eds.) Proceedings of the Computational Diffusion MRI (CDMRI), Nice, pp. 186-197 (2012)
50. Hess, C.P., Mukherjee, P., Han, E.T., Xu, D., Vigneron, D.B.: Q-ball reconstruction of multimodal fiber orientations using the spherical harmonic basis. Magn. Reson. Med. 56, 104-117 (2006)
51. Hillar, C., Lim, L.H.: Most tensor problems are NP-hard. JACM 60(6), Article No. 45 (2012). Preprint, arXiv:0911.1393v2 x
52. Hitchcock, F.L.: The expression of a tensor or a polyadic as a sum of products. J. Math. Phys. 6(1), 164-189 (1927)
53. Hitchcock, F.L.: Multiple invariants and generalized rank of a p-way matrix or tensor. J.Math. Phys. 7(1), 39-79 (1927)
54. Hlawitschka, M., Scheuermann, G.: HOT-lines: tracking lines in higher order tensor fields. In: Silva, C., Gröller, E., Rushmeier, H. (eds.) Proceedings of the IEEE Visualization, Minneapolis, pp. 27-34 (2005)
55. Hlawitschka, M., Scheuermann, G., Anwander, A., Knösche, T., Tittgemeyer, M., Hamann, B.: Tensor lines in tensor fields of arbitrary order. In: Bebis, G., et al. (eds.) Advances in Visual Computing. LNCS, vol. 4841, pp. 341-350. Springer, Berlin/New York (2007)
56. Hui, E.S., Cheung, M.M., Qi, L., Wu, E.X.: Towards better MR characterization of neural tissues using directional diffusion kurtosis analysis. NeuroImage 42, 122-134 (2008)
57. Hungerford, T.: Algebra. Graduate Texts in Mathematics, vol. 73. Springer, New York (1980)
58. Jansons, K.M., Alexander, D.C.: Persistent angular structure: new insights from diffusion magnetic resonance imaging data. Inverse Probl. 19, 1031-1046 (2003)
59. Jayachandra, M.R., Rehbein, N., Herweh, C., Heiland, S.: Fiber tracking of human brain using fourth-order tensor and high angular resolution diffusion imaging. Magn. Reson. Med. 60(5), 1207-1217 (2008)
60. Jensen, J.H., Helpern, J.A.: MRI quantification of non-Gaussian water diffusion by kurtosis analysis. NMR Biomed. 23(7), 698-710 (2010)
61. Jensen, J.H., Helpern, J.A., Ramani, A., Lu, H., Kaczynski, K.: Diffusional kurtosis imaging: the quantification of non-Gaussian water diffusion by means of magnetic resonance imaging. Magn. Reson. Med. 53, 1432-1440 (2005)
62. Jiao, F., Gur, Y., Johnson, C.R., Joshi, S.: Detection of crossing white matter fibers with highorder tensors and rank-k decompositions. In: Székely, G., Hahn, H.K. (eds.) IPMI, Kloster Irsee. LNCS, vol. 6801, pp. 538-549 (2011)
63. Kindlmann, G., Ennis, D., Whitaker, R., Westin, C.F.: Diffusion tensor analysis with invariant gradients and rotation tangents. IEEE Trans. Med. Imaging 26(11), 1483-1499 (2007)
64. Kroonenberg, P.: Applied Multiway Data Analysis. Wiley, Hoboken (2008)
65. Kuder, T.A., Stieltjes, B., Bachert, P., Semmler, W., Laun, F.B.: Advanced fit of the diffusion kurtosis tensor by directional weighting and regularization. Magn. Reson. Med. 67(5), 1401-1411 (2012)
66. Landsberg, J.M.: Tensors: Geometry and Applications. Graduate Studies in Mathematics, vol. 128. American Mathematical Society, Providence (2012)
67. Lang, S.: Differential and Riemannian Manifolds. Graduate Texts in Mathematics, vol. 160, 3rd edn. Springer, New York (1995)
68. Lang, S.: Algebra. Graduate Texts inMathematics, vol. 211, rev. 3rd edn. Springer, New York (2002)
69. Lazar, M., Jensen, J.H., Xuan, L., Helpern, J.A.: Estimation of the orientation distribution function from diffusional kurtosis imaging. Magn. Reson. Med. 60, 774-781 (2008)
70. Le Bihan, D., Breton, E., Lallemand, D., Grenier, P., Cabanis, E., Laval-Jeantet, M.: MR imaging of intravoxel incoherent motions: application to diffusion and perfusion in neurologic disorders. Radiology 161(2), 401-407 (1986)
71. Lenglet, C., Rousson, M., Deriche, R., Faugeras, O.: Statistics on the manifold of multivariate normal distributions: theory and application to diffusion tensor MRI processing. J. Math. Imaging Vis. 25, 423-444 (2006)
72. Lim, L.H.: Singular values and eigenvalues of tensors: a variational approach. In: Proceedings of the IEEE International Workshop on Computational Advances in Multi-Sensor Adaptive Processing (CAMSAP), Puerto Vallarta, pp. 129-132 (2005)
73. Lim, L.H.: Tensors and hypermatrices. In: Hogben, L. (ed.) Handbook of Linear Algebra, Boca Raton (2013)
74. Lim, L.H., Comon, P.: Nonnegative approximations of nonnegative tensors. J. Chemom. 23(7-8), 432-441 (2009)
75. Lim, L.H., Comon, P.: Multisensor array processing: tensor decomposition meets compressed sensing. C. R. Acad. Sci. Paris 338(6), 311-320 (2010)
76. Lim, L.H., Schultz, T.: Moment tensors and high angular resolution diffusion imaging (2013). Preprint
77. Liu, C., Bammer, R., Acar, B., Moseley, M.E.: Characterizing non-Gaussian diffusion by using generalized diffusion tensors. Magn. Reson. Med. 51(5), 924-937 (2004)
78. Liu, C., Bammer, R., Moseley, M.E.: Generalized diffusion tensor imaging (GDTI): a method for characterizing and imaging diffusion anisotropy caused by non-Gaussian diffusion. Isr. J. Chem. 43(1-2), 145-154 (2003)
79. Liu, C., Bammer, R., Moseley, M.E.: Limitations of apparent diffusion coefficient-based models in characterizing non-Gaussian diffusion. Magn. Reson. Med. 54, 419-428 (2005)
80. Liu, C., Mang, S.C., Moseley, M.E.: In vivo generalized diffusion tensor imaging (GDTI) using higher-order tensors (HOT). Magn. Reson. Med. 63, 243-252 (2010)
81. Liu, Y., Chen, L., Yu, Y.: Diffusion kurtosis imaging based on adaptive spherical integral. IEEE Signal Process. Lett. 18(4), 243-246 (2011)
82. Lu, H., Jensen, J.H., Ramani, A., Helpern, J.A.: Three-dimensional characterization of non-Gaussian water diffusion in humans using diffusion kurtosis imaging. NMR Biomed. 19, 236-247 (2006)
83. Minati, L., Aquino, D., Rampoldi, S., Papa, S., Grisoli, M., Bruzzone, M.G., Maccagnano, E.: Biexponential and diffusional kurtosis imaging, and generalised diffusion-tensor imaging (GDTI) with rank-4 tensors: a study in a group of healthy subjects. Magn. Reson. Mater. Phys. Biol. Med. 20, 241-253 (2007)
84. Minati, L., Banasik, T., Brzezinski, J., Mandelli, M.L., Bizzi, A., Bruzzone, M.G., Konopka, M., Jasinski, A.: Elevating tensor rank increases anisotropy in brain areas associated with intra-voxel orientational heterogeneity (IVOH): a generalised DTI (GDTI) study. NMR Biomed. 21(1), 2-14 (2008)
85. Mørup, M., Hansen, L., Arnfred, S., Lim, L.H., Madsen, K.: Shift invariant multilinear decomposition of neuroimaging data. NeuroImage 42(4), 1439-1450 (2008)
86. Özarslan, E., Mareci, T.: Generalized diffusion tensor imaging and analytical relationships between diffusion tensor imaging and high angular resolution diffusion imaging. Magn. Reson. Med. 50, 955-965 (2003)
87. Özarslan, E., Shepherd, T.M., Vemuri, B.C., Blackband, S.J., Mareci, T.H.: Resolution of complex tissue microarchitecture using the diffusion orientation transform (DOT). NeuroImage 31, 1086-1103 (2006)
88. Özarslan, E., Vemuri, B.C., Mareci, T.H.: Fiber orientation mapping using generalized diffusion tensor imaging. In: Proceedings of the IEEE International Symposium on Biomedical Imaging, Arlington, pp. 1036-1039 (2004)
89. Özarslan, E., Vemuri, B.C., Mareci, T.H.: Generalized scalar measures for diffusion MRI using trace, variance, and entropy. Magn. Reson. Med. 53, 866-876 (2005)
90. Özarslan, E., Vemuri, B.C., Mareci, T.H.: Higher rank tensors in diffusion MRI. In:Weickert, J., Hagen, H. (eds.) Visualization and Processing of Tensor Fields, chap. 10, pp. 177-187. Springer, Berlin (2006)
91. Poot, D.H.J., den Dekker, A.J., Achten, E., Verhoye, M., Sijbers, J.: Optimal experimental design for diffusion kurtosis imaging. IEEE Trans. Med. Imaging 29(3), 819-829 (2010)
92. Qi, L.: Eigenvalues of a real supersymmetric tensor. J. Symb. Comput. 40, 1302-1324 (2005)
93. Qi, L., Han, D., Wu, E.X.: Principal invariants and inherent parameters of diffusion kurtosis tensors. J. Math. Anal. Appl. 349, 165-180 (2009)
94. Qi, L., Wang, Y., Wu, E.X.: D-eigenvalues of diffusion kurtosis tensors. J. Comput. Appl. Math. 221, 150-157 (2008)
95. Qi, L., Yu, G., Wu, E.X.: Higher order positive semidefinite diffusion tensor imaging. SIAM J. Imaging Sci. 3(3), 416-433 (2010)
96. Renard, F., Noblet, V., Heinrich, C., Kremer, S.: Reorientation strategies for high order tensors. In: Proceedings of the IEEE International Symposium on Biomedical Imaging, Rotterdam, pp. 1185-1188 (2010)
97. Schultz, T.: Learning a reliable estimate of the number of fiber directions in diffusionMRI. In: Ayache, N., et al. (eds.) Proceedings of theMedical Image Computing and Computer-Assisted Intervention (MICCAI) Part III, Nice. LNCS, vol. 7512, pp. 493-500 (2012)
98. Schultz, T., Kindlmann, G.: A maximum enhancing higher-order tensor glyph. Comput. Graph. Forum 29(3), 1143-1152 (2010)
99. Schultz, T., Schlaffke, L., Schölkopf, B., Schmidt-Wilcke, T.: HiFiVE: a hilbert space embedding of fiber variability estimates for uncertainty modeling and visualization. Comput. Graph. Forum 32(3), 121-130 (2013)
100. Schultz, T., Seidel, H.P.: Estimating crossing fibers: a tensor decomposition approach. IEEE Trans. Vis. Comput. Graph. 14(6), 1635-1642 (2008)
101. Schultz, T., Westin, C.F., Kindlmann, G.: Multi-diffusion-tensor fitting via spherical deconvolution: a unifying framework. In: Jiang, T., et al. (eds.) Proceedings of the Medical Image Computing and Computer-Assisted Intervention (MICCAI), Beijing. LNCS, vol. 6361, pp. 673-680. Springer (2010)
102. Sidiropoulos, N., Bro, R., Giannakis, G.: Parallel factor analysis in sensor array processing. IEEE Trans. Signal Process. 48(8), 2377-2388 (2000)
103. Smilde, A., Bro, R., Geladi, P.: Multi-way Analysis: Applications in the Chemical Sciences. Wiley, West Sussex (2004)
104. Struik, D.J.: Schouten, Levi-Civita and the emergence of tensor calculus. In: Rowe, D., McCleary, J. (eds.) History of Modern Mathematics, vol. 2, pp. 99-105. Academic, Boston (1989)
105. Tabesh, A., Jensen, J.H., Ardekani, B.A., Helpern, J.A.: Estimation of tensors and tensorderived measures in diffusional kurtosis imaging. Magn. Reson. Med. 65, 823-836 (2011)
106. Tournier, J.D., Calamante, F., Connelly, A.: Robust determination of the fibre orientation distribution in diffusion MRI: non-negativity constrained super-resolved spherical deconvolution. NeuroImage 35, 1459-1472 (2007)
107. Tournier, J.D., Calamante, F., Gadian, D.G., Connelly, A.: Direct estimation of the fiber orientation density function from diffusion-weighted MRI data using spherical deconvolution. NeuroImage 23, 1176-1185 (2004)
108. Tristan-Vega, A., Westin, C.F., Aja-Fernandez, S.: A new methodology for the estimation of fiber populations in the white matter of the brain with the funk-radon transform. NeuroImage 49(2), 1301-1315 (2010)
109. Tuch, D.S.: Q-ball imaging. Magn. Reson. Med. 52, 1358-1372 (2004)
110. Vasilescu, M., Terzopoulos, D.: Multilinear image analysis for facial recognition. Proc. Int. Conf. Pattern Recognit (ICPR) 2, 511-514 (2002)
111. Veraart, J., Van Hecke, W., Sijbers, J.: Constrained maximum likelihood estimation of the diffusion kurtosis tensor using a Rician noise model. Magn. Reson. Med. 66, 678-686 (2011)
112. Warner, F.: Foundations of Differentiable Manifolds and Lie Groups. Graduate Texts in Mathematics, vol. 94. Springer, New York/Berlin (1983)
113. Weldeselassie, Y.T., Barmpoutis, A., Atkins, M.S.: Symmetric positive-definite Cartesian tensor orientation distribution functions (CT-ODF). In: Jiang, T., Navab, N., Pluim, J.P.W., Viergever, M.A. (eds.) Proceedings of the Medical Image Computing and Computer-Assisted Intervention (MICCAI), Beijing. LNCS, vol. 6361, pp. 582-589 (2010)
114. Westin, C.F., Peled, S., Gudbjartsson, H., Kikinis, R., Jolesz, F.A.: Geometrical diffusion measures for MRI from tensor basis analysis. In: Proceedings of the International Society for Magnetic Resonance in Medicine (ISMRM), Vancouver, p. 1742 (1997)
115. Ying, L., Zou, Y.M., Klemer, D.P., Wang, J.J.: Determination of fiber orientation in MRI diffusion tensor imaging based on higher-order tensor decomposition. In: Proceedings of the International Conference on IEEE Engineering in Medicine and Biology Society (EMBS), Lyon, pp. 2065-2068 (2007)
116. Yokonuma, T.: Tensor Spaces and Exterior Algebra. Translations of Mathematical Monographs, vol. 108. American Mathematical Society, Providence (1992)