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Volumn 87, Issue 2, 2007, Pages 220-236

Spectral decomposition of a 4th-order covariance tensor: Applications to diffusion tensor MRI

Author keywords

Anisotropy; Covariance; DT MRI; DTI; HOS; Karhunen Lo ve; Multi linear algebra; PCA; Tensor

Indexed keywords

ANISOTROPY; EIGENVALUES AND EIGENFUNCTIONS; LINEAR ALGEBRA; MAGNETIC RESONANCE IMAGING; MAXIMUM LIKELIHOOD ESTIMATION; PRINCIPAL COMPONENT ANALYSIS; TENSORS; VECTOR QUANTIZATION;

EID: 33750978683     PISSN: 01651684     EISSN: None     Source Type: Journal    
DOI: 10.1016/j.sigpro.2006.02.050     Document Type: Article
Times cited : (84)

References (57)
  • 1
    • 0000325341 scopus 로고
    • On lines and planes of closest fit to systems of points in space
    • Pearson K. On lines and planes of closest fit to systems of points in space. Philos. Mag. 6 2 (1901) 559-572
    • (1901) Philos. Mag. , vol.6 , Issue.2 , pp. 559-572
    • Pearson, K.1
  • 2
    • 58149421595 scopus 로고
    • Analysis of a complex of statistical variables into principal components
    • and 498-520
    • Hotelling H. Analysis of a complex of statistical variables into principal components. J. Educ. Psychol. 24 (1933) 417-444 and 498-520
    • (1933) J. Educ. Psychol. , vol.24 , pp. 417-444
    • Hotelling, H.1
  • 3
    • 0002965815 scopus 로고
    • The proof and measurement of association between two things
    • and 202
    • Spearman C. The proof and measurement of association between two things. Am. J. Psychol. 15 (1933) 72 and 202
    • (1933) Am. J. Psychol. , vol.15 , pp. 72
    • Spearman, C.1
  • 4
    • 0030931604 scopus 로고    scopus 로고
    • Blind separation of auditory event-related brain responses into independent components
    • Makeig S., Jung T.P., Bell A.J., Ghahremani D., and Sejnowski T.J. Blind separation of auditory event-related brain responses into independent components. Proc. Natl. Acad. Sci. USA 94 20 (1997) 10979-10984
    • (1997) Proc. Natl. Acad. Sci. USA , vol.94 , Issue.20 , pp. 10979-10984
    • Makeig, S.1    Jung, T.P.2    Bell, A.J.3    Ghahremani, D.4    Sejnowski, T.J.5
  • 5
    • 0030832881 scopus 로고    scopus 로고
    • The "independent components" of natural scenes are edge filters
    • Bell A.J., and Sejnowski T.J. The "independent components" of natural scenes are edge filters. Vision Res. 37 23 (1997) 3327-3338
    • (1997) Vision Res. , vol.37 , Issue.23 , pp. 3327-3338
    • Bell, A.J.1    Sejnowski, T.J.2
  • 6
    • 0026191274 scopus 로고
    • Blind separation of sources, part 1: an adaptive algorithm based on neuromimetic architecture
    • Herault J., and Jutten C. Blind separation of sources, part 1: an adaptive algorithm based on neuromimetic architecture. Signal Process. 24 (1991) 1-10
    • (1991) Signal Process. , vol.24 , pp. 1-10
    • Herault, J.1    Jutten, C.2
  • 7
    • 0029411030 scopus 로고
    • An information-maximization approach to blind separation and blind deconvolution
    • Bell A., and Sejnowski T. An information-maximization approach to blind separation and blind deconvolution. Neural Comput. 7 6 (1995) 1129-1159
    • (1995) Neural Comput. , vol.7 , Issue.6 , pp. 1129-1159
    • Bell, A.1    Sejnowski, T.2
  • 9
    • 0028154030 scopus 로고
    • MR diffusion tensor spectroscopy and imaging
    • Basser P.J., Mattiello J., and Le Bihan D. MR diffusion tensor spectroscopy and imaging. Biophys. J. 66 1 (1994) 259-267
    • (1994) Biophys. J. , vol.66 , Issue.1 , pp. 259-267
    • Basser, P.J.1    Mattiello, J.2    Le Bihan, D.3
  • 10
    • 0028398052 scopus 로고
    • Estimation of the effective self-diffusion tensor from the NMR spin echo
    • Basser P.J., Mattiello J., and Le Bihan D. Estimation of the effective self-diffusion tensor from the NMR spin echo. J. Magn. Reson. B 103 3 (1994) 247-254
    • (1994) J. Magn. Reson. B , vol.103 , Issue.3 , pp. 247-254
    • Basser, P.J.1    Mattiello, J.2    Le Bihan, D.3
  • 11
    • 0042687862 scopus 로고
    • Introduction to statistical pattern recognition
    • Booker H.G., and DeClaris N. (Eds), Academic Press Inc., New York
    • Fukunaga K. Introduction to statistical pattern recognition. In: Booker H.G., and DeClaris N. (Eds). Electrical Sciences (1972), Academic Press Inc., New York
    • (1972) Electrical Sciences
    • Fukunaga, K.1
  • 12
    • 84984030647 scopus 로고
    • Irreduzible Invarianten eines Tensors vierter Stufe
    • Betten J., and Helisch W. Irreduzible Invarianten eines Tensors vierter Stufe. (ZAMM) Z. Angew. Math. Mech. 72 1 (1992) 45-57
    • (1992) (ZAMM) Z. Angew. Math. Mech. , vol.72 , Issue.1 , pp. 45-57
    • Betten, J.1    Helisch, W.2
  • 13
    • 0028371062 scopus 로고
    • On the polynomial invariants of the elasticity tensor
    • Onat E.T., Boehler J.P., and Kirillov J.A.A. On the polynomial invariants of the elasticity tensor. J. Elasticity (1994) 97-110
    • (1994) J. Elasticity , pp. 97-110
    • Onat, E.T.1    Boehler, J.P.2    Kirillov, J.A.A.3
  • 14
    • 0029255185 scopus 로고
    • 6-Dimensional orthogonal tensor representation of the rotation about an axis in 3 dimensions
    • Mehrabadi M.M., Cowin S.C., and Jaric J. 6-Dimensional orthogonal tensor representation of the rotation about an axis in 3 dimensions. Int. J. Solids Struct. 32 3-4 (1995) 439-449
    • (1995) Int. J. Solids Struct. , vol.32 , Issue.3-4 , pp. 439-449
    • Mehrabadi, M.M.1    Cowin, S.C.2    Jaric, J.3
  • 15
    • 0041702764 scopus 로고    scopus 로고
    • Parametric and non-parametric statistical analysis of DT-MRI data
    • Pajevic S., and Basser P.J. Parametric and non-parametric statistical analysis of DT-MRI data. J. Magn. Reson. 161 1 (2003) 1-14
    • (2003) J. Magn. Reson. , vol.161 , Issue.1 , pp. 1-14
    • Pajevic, S.1    Basser, P.J.2
  • 16
    • 1942477354 scopus 로고    scopus 로고
    • Dealing with uncertainty in DT-MRI data
    • Basser P.J., and Pajevic S. Dealing with uncertainty in DT-MRI data. Israel J. Chem. 43 1-2 (2003) 129-144
    • (2003) Israel J. Chem. , vol.43 , Issue.1-2 , pp. 129-144
    • Basser, P.J.1    Pajevic, S.2
  • 17
    • 0043028196 scopus 로고    scopus 로고
    • A normal distribution for tensor-valued random variables: applications to diffusion tensor MRI
    • Basser P.J., and Pajevic S. A normal distribution for tensor-valued random variables: applications to diffusion tensor MRI. IEEE Trans. Med. Imaging 22 7 (2003) 785-794
    • (2003) IEEE Trans. Med. Imaging , vol.22 , Issue.7 , pp. 785-794
    • Basser, P.J.1    Pajevic, S.2
  • 21
    • 0004324990 scopus 로고
    • Cambridge University Press, Cambridge p. 93
    • Jeffreys H. Cartesian Tensors (1931), Cambridge University Press, Cambridge p. 93
    • (1931) Cartesian Tensors
    • Jeffreys, H.1
  • 24
    • 0001635791 scopus 로고
    • Elements of a mathematical theory of elasticity
    • (Lord Kelvin) Thomson W. Elements of a mathematical theory of elasticity. Philos. Trans. R. Soc. 166 (1856) 481
    • (1856) Philos. Trans. R. Soc. , vol.166 , pp. 481
    • (Lord Kelvin) Thomson, W.1
  • 25
    • 0027001124 scopus 로고
    • Spectral decomposition of the elasticity tensor
    • Sutcliffe S. Spectral decomposition of the elasticity tensor. Trans. ASME 59 (1992) 762-773
    • (1992) Trans. ASME , vol.59 , pp. 762-773
    • Sutcliffe, S.1
  • 26
    • 84984068311 scopus 로고
    • A note on representation for isotropic functions of 4th-order tensors in 2-dimensional space
    • Zheng Q.S. A note on representation for isotropic functions of 4th-order tensors in 2-dimensional space. Z. Angew. Math. Mech. 74 8 (1994) 357-359
    • (1994) Z. Angew. Math. Mech. , vol.74 , Issue.8 , pp. 357-359
    • Zheng, Q.S.1
  • 27
    • 0001473923 scopus 로고
    • On the identification of material symmetry for anisotropic elastic-materials
    • Cowin S.C., and Mehrabadi M.M. On the identification of material symmetry for anisotropic elastic-materials. Q. J. Mech. Appl. Math. 40 (1987) 451-476
    • (1987) Q. J. Mech. Appl. Math. , vol.40 , pp. 451-476
    • Cowin, S.C.1    Mehrabadi, M.M.2
  • 29
    • 0025387418 scopus 로고
    • Eigentensors of linear anisotropic elastic-materials
    • Mehrabadi M.M., and Cowin S.C. Eigentensors of linear anisotropic elastic-materials. Q. J. Mech. Appl. Math. 43 (1990) 15-41
    • (1990) Q. J. Mech. Appl. Math. , vol.43 , pp. 15-41
    • Mehrabadi, M.M.1    Cowin, S.C.2
  • 31
    • 33751015982 scopus 로고    scopus 로고
    • W. (Lord Kelvin) Thomson, Elasticity, Encyclopedia Britannica, vol. 7, ninth ed., Adam and Charles Black, London, Edinburgh, 1878, pp. 796-825.
  • 33
    • 85012577031 scopus 로고
    • Principles and applications of self-diffusion measurements by nuclear magnetic resonance
    • Waugh J. (Ed), Academic Press, New York
    • Kärger J., Pfeifer H., and Heink W. Principles and applications of self-diffusion measurements by nuclear magnetic resonance. In: Waugh J. (Ed). Advances in Magnetic Resonance vol. 12 (1988), Academic Press, New York 1-89
    • (1988) Advances in Magnetic Resonance , vol.12 , pp. 1-89
    • Kärger, J.1    Pfeifer, H.2    Heink, W.3
  • 34
    • 0036787434 scopus 로고    scopus 로고
    • High angular resolution diffusion imaging reveals intravoxel white matter fiber heterogeneity
    • Tuch D.S., Reese T.G., Wiegell M.R., Makris N., Belliveau J.W., and Wedeen V.J. High angular resolution diffusion imaging reveals intravoxel white matter fiber heterogeneity. Magn. Reson. Med. 48 4 (2002) 577-582
    • (2002) Magn. Reson. Med. , vol.48 , Issue.4 , pp. 577-582
    • Tuch, D.S.1    Reese, T.G.2    Wiegell, M.R.3    Makris, N.4    Belliveau, J.W.5    Wedeen, V.J.6
  • 35
    • 0036263865 scopus 로고    scopus 로고
    • Characterization of anisotropy in high angular resolution diffusion-weighted MRI
    • Frank L.R. Characterization of anisotropy in high angular resolution diffusion-weighted MRI. Magn. Reson. Med. 47 6 (2002) 1083-1099
    • (2002) Magn. Reson. Med. , vol.47 , Issue.6 , pp. 1083-1099
    • Frank, L.R.1
  • 36
    • 33750994574 scopus 로고    scopus 로고
    • C. Pierpaoli, P.J. Basser, New invariant "lattice" index achieves significant noise reduction in measuring diffusion anisotropy, in: Proceedings of the ISMRM, New York, 1996, p. 1326.
  • 37
    • 0021948756 scopus 로고
    • Measurement of signal intensities in the presence of noise in MR images
    • Henkelman R.M. Measurement of signal intensities in the presence of noise in MR images. Med. Phys. 12 2 (1985) 232-233
    • (1985) Med. Phys. , vol.12 , Issue.2 , pp. 232-233
    • Henkelman, R.M.1
  • 38
    • 0032835247 scopus 로고    scopus 로고
    • Optimal strategies for measuring diffusion in anisotropic systems by magnetic resonance imaging
    • Jones D.K., Horsfield M.A., and Simmons A. Optimal strategies for measuring diffusion in anisotropic systems by magnetic resonance imaging. Magn. Reson. Med. 42 3 (1999) 515-525
    • (1999) Magn. Reson. Med. , vol.42 , Issue.3 , pp. 515-525
    • Jones, D.K.1    Horsfield, M.A.2    Simmons, A.3
  • 39
    • 1842533406 scopus 로고    scopus 로고
    • The effect of gradient sampling schemes on measures derived from diffusion tensor MRI: a Monte Carlo study
    • Jones D.K. The effect of gradient sampling schemes on measures derived from diffusion tensor MRI: a Monte Carlo study. Magn. Reson. Med. 51 4 (2004) 807-815
    • (2004) Magn. Reson. Med. , vol.51 , Issue.4 , pp. 807-815
    • Jones, D.K.1
  • 40
    • 0034567198 scopus 로고    scopus 로고
    • Condition number as a measure of noise performance of diffusion tensor data acquisition schemes with MRI
    • Skare S., Hedehus M., Moseley M.E., and Li T.Q. Condition number as a measure of noise performance of diffusion tensor data acquisition schemes with MRI. J. Magn. Reson. 147 2 (2000) 340-352
    • (2000) J. Magn. Reson. , vol.147 , Issue.2 , pp. 340-352
    • Skare, S.1    Hedehus, M.2    Moseley, M.E.3    Li, T.Q.4
  • 41
    • 0033870213 scopus 로고    scopus 로고
    • Noise considerations in the determination of diffusion tensor anisotropy
    • Skare S., Li T., Nordell B., and Ingvar M. Noise considerations in the determination of diffusion tensor anisotropy. Magn. Reson. Imaging 18 6 (2000) 659-669
    • (2000) Magn. Reson. Imaging , vol.18 , Issue.6 , pp. 659-669
    • Skare, S.1    Li, T.2    Nordell, B.3    Ingvar, M.4
  • 42
    • 0036295606 scopus 로고    scopus 로고
    • A continuous tensor field approximation of discrete DT-MRI data for extracting microstructural and architectural features of tissue
    • Pajevic S., Aldroubi A., and Basser P.J. A continuous tensor field approximation of discrete DT-MRI data for extracting microstructural and architectural features of tissue. J. Magn. Reson. 154 1 (2002) 85-100
    • (2002) J. Magn. Reson. , vol.154 , Issue.1 , pp. 85-100
    • Pajevic, S.1    Aldroubi, A.2    Basser, P.J.3
  • 44
    • 84947780359 scopus 로고    scopus 로고
    • C. Poupon, J.-F. Mangin, V. Frouin, F. Regis, M. Pachot-Clouard, D. Le Bihan, I. Bloch, Regularization of MR diffusion tensor maps for tracking brain white matter bundles, in: Proceedings of MICCAI'98, 1998, pp. 489-498.
  • 45
    • 0002648277 scopus 로고    scopus 로고
    • Reconstruction of vector and tensor fields from sampled discrete data
    • Baggett L.W., and Larson D.R. (Eds), AMS, Providence, RI
    • Aldroubi A., and Basser P.J. Reconstruction of vector and tensor fields from sampled discrete data. In: Baggett L.W., and Larson D.R. (Eds). Contemporary Mathematics vol. 247 (1999), AMS, Providence, RI 1-15
    • (1999) Contemporary Mathematics , vol.247 , pp. 1-15
    • Aldroubi, A.1    Basser, P.J.2
  • 46
    • 33750998121 scopus 로고    scopus 로고
    • S. Pajevic, A. Aldroubi, P.J. Basser, A continuous tensor field approximation of discrete DT-MRI data, in: D.J. Weickert (Universität des Saarlandes), D. H. Hagen (Universität Kaiserslautern) (Eds.), Visualization and Image Processing of Tensor Fields, Springer Series on Mathematics and Visualization, Springer, Berlin, 2005.
  • 49
    • 0037281553 scopus 로고    scopus 로고
    • Means and averaging in the group of rotations
    • Moakher M. Means and averaging in the group of rotations. SIAM J. Matrix Anal. Appl. 24 1 (2002) 1-16
    • (2002) SIAM J. Matrix Anal. Appl. , vol.24 , Issue.1 , pp. 1-16
    • Moakher, M.1
  • 50
    • 33751007771 scopus 로고    scopus 로고
    • C. Lenglet, M. Rousson, R. Deriche, O. Faugeras, Statistics on multivariate normal distributions: a geometric approach and its application to diffusion tensor MRI, INRIA, Sophia Antipolis No. 5242, June, 2004.
  • 51
    • 33751000977 scopus 로고    scopus 로고
    • X. Pennec, Probabilities and statistics on Riemannian manifolds: a geometric approach, INRIA, Sophia Antipolis No. 5093, January, 2004.
  • 52
    • 0027539974 scopus 로고
    • Strain-energy density bounds for linear anisotropic elastic-materials
    • Mehrabadi M.M., Cowin S.C., and Horgan C.O. Strain-energy density bounds for linear anisotropic elastic-materials. J. Elasticity 30 2 (1993) 191-196
    • (1993) J. Elasticity , vol.30 , Issue.2 , pp. 191-196
    • Mehrabadi, M.M.1    Cowin, S.C.2    Horgan, C.O.3
  • 53
    • 33750979230 scopus 로고    scopus 로고
    • L.D. Landau, E.M. Lifshitz, Theory of Elasticity, vol. 7, third ed., Pergamon Press, Oxford, 1986.
  • 55
    • 0000074076 scopus 로고
    • Estimation of second-order tensors with related tests and designs
    • Hext G.R. Estimation of second-order tensors with related tests and designs. Biometrika 50 3-4 (1963) 353-373
    • (1963) Biometrika , vol.50 , Issue.3-4 , pp. 353-373
    • Hext, G.R.1


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