NDLI: A normal distribution for tensor-valued random variables: applications to diffusion tensor MRI

Content Provider	IEEE Xplore Digital Library
Author	Basser, P.J. Pajevic, S.
Copyright Year	1982
Abstract	Diffusion tensor magnetic resonance imaging (DT-MRI) provides a statistical estimate of a symmetric, second-order diffusion tensor of water, D, in each voxel within an imaging volume. We propose a new normal distribution, p(D) /spl prop/ exp(-1/2 D : A : D), which describes the variability of D in an ideal DT-MRI experiment. The scalar invariant, D : A : D, is the contraction of a positive definite symmetric, fourth-order precision tensor, A, and D. A correspondence is established between D : A : D and the elastic strain energy density function in continuum mechanics-specifically between D and the second-order infinitesimal strain tensor, and between A and the fourth-order tensor of elastic coefficients. We show that A can be further classified according to different classical elastic symmetries (i.e., isotropy, transverse isotropy, orthotropy, planar symmetry, and anisotropy). When A is an isotropic fourth-order tensor, we derive an explicit analytic expression for p(D), and for the distribution of the three eigenvalues of D, p(/spl gamma//sub 1/, /spl gamma//sub 2/, /spl gamma//sub 3/), which are confirmed by Monte Carlo simulations. We show how A can be estimated from either real or synthetic DT-MRI data for any given experimental design. Here we propose a new criterion for an optimal experimental design: that A be an isotropic fourth-order tensor. This condition ensures that the statistical properties of D (and quantities derived from it) are rotationally invariant. We also investigate the degree of isotropy of several DT-MRI experimental designs. Finally, we show that the univariate and multivariate distributions are special cases of the more general tensor-variate normal distribution, and suggest how to generalize p(D) to treat normal random tensor variables that are of third- (or higher) order. We expect that this new distribution, p(D), should be useful in feature extraction; in developing a hypothesis testing framework for segmenting and classifying noisy, discrete tensor data; and in designing experiments to measure tensor quantities.
Sponsorship	IEEE Engineering in Medicine and Biology Society IEEE Nuclear and Plasma Sciences Society IEEE Signal Processing Society IEEE Ultrasonics, Ferroelectrics, and Frequency Control Society
Page Count	10
File Size	557442
Starting Page	785
Ending Page	794
File Format	PDF
ISSN	02780062
Volume Number	22
Issue Number	7
Language	English
Publisher	Institute of Electrical and Electronics Engineers, Inc. (IEEE)
Publisher Date	2003-07-01
Publisher Place	U.S.A.
Access Restriction	One Nation One Subscription (ONOS)
Rights Holder	Institute of Electrical and Electronics Engineers, Inc. (IEEE)
Subject Keyword	Gaussian distribution Random variables Tensile stress Magnetic resonance imaging Diffusion tensor imaging Design for experiments Capacitive sensors Density functional theory Anisotropic magnetoresistance Eigenvalues and eigenfunctions
Content Type	Text
Resource Type	Article
Subject	Electrical and Electronic Engineering Computer Science Applications Radiological and Ultrasound Technology Software

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4	Project PI / Joint PI	Principal Investigator and Joint Principal Investigators of the project	Dr. B. Sutradhar bsutra@ndl.gov.in Prof. Saswat Chakrabarti will be added soon
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