Sharpening Fibers in Diffusion Weighted MRI via Erosion

  • Thomas C. J. Dela Haije
  • Remco Duits
  • Chantal M. W. Tax
Conference paper
Part of the Mathematics and Visualization book series (MATHVISUAL)


In this chapter erosion is generalized to the space of diffusion weighted MRI data. This is done effectively by solving a Hamilton-Jacobi-Bellman (HJB) system (erosion) on the coupled space of three dimensional positions and orientations, embedded as a quotient in the group of three dimensional rigid body motions. The solution to the HJB equations is given by a well-posed morphological convolution. We present two numerical approaches to solve the HJB equations: analytical kernels, and finite differences. Proof of concept is given by showing improved visibility of major fiber bundles in both artificial and human data. Furthermore, the method is shown to significantly improve the output of a probabilistic tractography algorithm used to extract the optic radiation.


Diffusion weighted MRI Erosion Hamilton-Jacobi-Bellman equations Lie groups Regularization Sharpening Sub-Riemannian Geometry 



The authors are grateful to the Kempenhaeghe Epilepsy Center, who supplied the MRI data. Furthermore, the authors would like to thank the anonymous reviewers for their helpful comments, and the editors for their work in making this publication happen.

Supplementary material


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Copyright information

© Springer-Verlag Berlin Heidelberg 2014

Authors and Affiliations

  • Thomas C. J. Dela Haije
    • 1
  • Remco Duits
    • 1
  • Chantal M. W. Tax
    • 2
  1. 1.Imaging Science and Technology Eindhoven (IST/e)Eindhoven University of TechnologyEindhovenThe Netherlands
  2. 2.Image Sciences Institute (ISI)University Medical Center UtrechtUtrechtThe Netherlands

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