Nonnegative Diffusion Orientation Distribution Function

Authors:
Liqun Qi;Gaohang Yu;Yi Xu
Affiliations:
Department of Applied Mathematics, The Hong Kong Polytechnic University, Kowloon, Hong Kong;Department of Applied Mathematics, The Hong Kong Polytechnic University, Kowloon, Hong Kong and Jiangxi Key Laboratory of Numerical Simulation Technology, School of Mathematics and Computer Scienc ...;Department of Applied Mathematics, The Hong Kong Polytechnic University, Kowloon, Hong Kong
Venue:
Journal of Mathematical Imaging and Vision
Year:
2013

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Abstract

Because of the well-known limitations of diffusion tensor imaging (DTI) in regions of low anisotropy and multiple fiber crossing, high angular resolution diffusion imaging (HARDI) and Q-Ball Imaging (QBI) are used to estimate the probability density function (PDF) of the average spin displacement of water molecules. In particular, QBI is used to obtain the diffusion orientation distribution function (ODF) of these multiple fiber crossing. As a probability distribution function, the orientation distribution function should be nonnegative which is not guaranteed in the existing methods. This paper proposes a novel technique to guarantee the nonnegative property of ODF by solving a convex optimization problem, which has a convex quadratic objective function and a constraint involving the nonnegativity requirement on the smallest Z-eigenvalue of the diffusivity tensor. Using convex analysis and optimization techniques, we first derive the optimality conditions of this convex optimization problem. Then, we propose a gradient descent algorithm to solve this problem. We also present formulas for determining the principal directions (maxima) of the ODF. Numerical examples on synthetic data as well as MRI data are displayed to demonstrate the significance of our approach.