Abstract
Second-order tensors may be described in terms of shape and orientation. Shape is quantified by tensor invariants, which are fixed with respect to coordinate system changes. This chapter describes an anatomically-motivated method of detecting edges in diffusion tensor fields based on the gradients of invariants. Three particular invariants (the mean, variance, and skewness of the tensor eigenvalues) are described in two ways: first, as the geometric parameters of an intuitive graphical device for representing tensor shape (the eigenvalue wheel), and second, in terms of their physical and anatomical significance in diffusion tensor MRI. Tensor-valued gradients of these invariants lead to an orthonormal basis for describing changes in tensor shape. The spatial gradient of the diffusion tensor field may be projected onto this basis, producing three different measures of edge strength, selective for different kinds of anatomical boundaries. The gradient measures are grounded in standard tensor analysis, and are demonstrated on synthetic data.
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References
PJ Basser, J Mattiello, and D Le Bihan. Estimation of the effective self-diffusion tensor from the NMR spin-echo. Journal of Magnetic Resonance, B, 103(3):247–254, 1994.
GA Holzapfel. Nonlinear Solid Mechanics, Chap. 1. John Wiley and Sons, Ltd, England, 2000.
AM Uluğ and PCM van Zijl. Orientation-independent diffusion imaging without tensor diagonalization: Anisotropy definitions based on physical attributes of the diffusion ellipsoid. Journal of Magnetic Resonance Imaging, 9:804–813, 1999.
KM Hasan, PJ Basser, DL Parker, and AL Alexander. Analytical computation of the eigenvalues and eigenvectors in DT-MRI. Journal of Magnetic Resonance, 152:41–47, 2001.
EW Weisstein. CRC Concise Encyclopedia of Mathematics, pp. 362–365, 1652. CRC Press, Florida, 1999.
WH Press, BP Flannery, SA Teukolsky, and WT Vetterling. Numerical Recipes: The Art of Scientific Computing. Cambridge University Press, Cambridge (UK) and New York, 2nd edition, 1992.
RWD Nickalls. A new approach to solving the cubic: Cardan’s solution revealed. The Mathematical Gazette, 77:354–359, November 1993.
MM Bahn. Invariant and orthonormal scalar measures derived from magnetic resonance diffusion tensor imaging. Journal of Magnetic Resonance, 141:68–77, 1999.
JC Criscione, JD Humphrey, AS Douglas, and WC Hunter. An invariant basis for natural strain which yields orthogonal stress response terms in isotropic hyperelasticity. Journal of Mechanics and Physics of Solids, 48:2445–2465, 2000.
G Kindlmann. Superquadric tensor glyphs. In Proceedings IEEE TVCG/EG Symposium on Visualization 2004, pp. 147–154, May 2004.
PJ Basser and DK Jones. Diffusion-tensor MRI: theory, experimental design and data analysis — a technical review. Nuclear Magnetic Resonance in Biomedicine, 15:456–467, 2002.
C Pierpaoli, P Jezzard, PJ Basser, A Barnett, and G DiChiro. Diffusion tensor MR imaging of the human brain. Radiology, 201(3):637–648, 1996.
AM Uluğ, N Beauchamp, RN Bryan, and PCM van Zijl. Absolute quantitation of diffusion constants in human stroke. Stroke, 28(3):483–490, 1997.
C Beaulieu. The basis of anisotropic water diffusion in the nervous system — a technical review. Nuclear Magnetic Resonance in Biomedicine, 15:435–455, 2002.
C Pierpaoli and PJ Basser. Toward a quantitative assessment of diffusion anisotropy. Magnetic Resonance in Medecine, 33:893–906, 1996.
DC Alexander, GJ Barker, and SR Arridge. Detection and modeling of non-gaussian apparent diffusion coefficients profiles in human brain data. Magnetic Resonance in Medecine, 48:331–340, 2002.
AL Alexander, KM Hasan, M Lazar, JS Tsuruda, and DL Parker. Analysis of partial volume effects in diffusion-tensor MRI. Magnetic Resonance in Medicine, 45:770–780, 2001.
MR Wiegell, HBW Larsson, and VJ Wedeen. Fiber crossing in human brain depicted with diffusion tensor MR imaging. Radiology, 217(3):897–903, Dec 2000.
DS Tuch, RM Weisskoff, JW Belliveau, and VJ Wedeen. High angular resolution diffusion imaging of the human brain. In Proceedings 7th Annual Meeting of ISMRM, page 321, 1999.
S Pajevic, A Aldroubi, and PJ Basser. A continuous tensor field approximation of discrete DT-MRI data for extracting microstructural and architectural features of tissue. Journal of Magnetic Resonance, 154:85–100, 2002.
K Hoffman and R Kunze. Linear Algebra. Prentice-Hall, Inc., Englewood Cliffs, NJ, 1971.
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Kindlmann, G. (2006). Tensor Invariants and their Gradients. In: Weickert, J., Hagen, H. (eds) Visualization and Processing of Tensor Fields. Mathematics and Visualization. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-31272-2_12
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DOI: https://doi.org/10.1007/3-540-31272-2_12
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-25032-6
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