Nonnegative diffusion orientation distribution function

Liqun Qi, Gaohang Yu, Yi Xu

Research output: Journal article publicationJournal articleAcademic researchpeer-review

34 Citations (Scopus)


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 nonnega-tivity 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.
Original languageEnglish
Pages (from-to)103-113
Number of pages11
JournalJournal of Mathematical Imaging and Vision
Issue number2
Publication statusPublished - 1 Jan 2013


  • High angular resolution diffusion imaging (HARDI)
  • Higher order diffusion tensor
  • Nonnegativity
  • Orientation distribution function (ODF)
  • Principal direction

ASJC Scopus subject areas

  • Statistics and Probability
  • Modelling and Simulation
  • Condensed Matter Physics
  • Computer Vision and Pattern Recognition
  • Geometry and Topology
  • Applied Mathematics


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