High angular resolution diffusion imaging (HARDI) has become an important technique for imaging complex oriented structures in the brain and other anatomical tissues. This has motivated the recent development of several methods for computing the orientation probability density function (PDF) at each voxel. However, much less work has been done on developing techniques for filtering, interpolation, averaging and principal geodesic analysis of orientation PDF fields. In this paper, we present a Riemannian framework for performing such operations. The proposed framework does not require that the orientation PDFs be represented by any fixed parameterization, such as a mixture of von Mises-Fisher distributions or a spherical harmonic expansion. Instead, we use a nonparametric representation of the orientation PDF. We exploit the fact that under the square-root re-parameterization, the space of orientation PDFs forms a Riemannian manifold: the positive orthant of the unit Hilbert sphere. We show that various orientation PDF processing operations, such as filtering, interpolation, averaging and principal geodesic analysis, may be posed as optimization problems on the Hilbert sphere, and can be solved using Riemannian gradient descent. We illustrate these concepts with numerous experiments on synthetic, phantom and real datasets. We show their application to studying left/right brain asymmetries.
Bibliographical noteFunding Information:
We would like to thank Neda Jahanshad of UCLA for providing us with the registered datasets and Jennifer Campbell of the McConnell Brain Imaging Centre, McGill University for providing us with the biological phantom data. In addition, the first author would like to thank Professor Guillermo Sapiro of the University of Minnesota for inviting her for an extremely fruitful visit to his lab where the ODF based morphometry work was conceptualized. This work was funded by startup funds from JHU , by grants NSF CAREER IIS-0447739 , NIH R01 HD050735 , NIH R01 EB007813 , NIH R01 EB008432 , NIH P41 RR008079 , NIH P30 NS057091 , ONR N00014-05-10836 and ONR N00014-09-1-0084 , and the University of Minnesota Institute for Translational Neuroscience .
- Diffusion weighted MRI
- High angular resolution diffusion imaging