A novel Riemannian metric for analyzing HARDI data

We propose a novel Riemannian framework for analyzing orientation distribution functions (ODFs) in HARDI data sets, for use in comparing, interpolating, averaging, and denoising ODFs. A recently used Fisher-Rao metric does not provide physically feasible solutions, and we suggest a modification that removes orientations from ODFs and treats them as separate variables. This way a comparison of any two ODFs is based on separate comparisons of their shapes and orientations. Furthermore, this provides an explicit orientation at each voxel for use in tractography. We demonstrate these ideas by computing geodesics between ODFs and Karcher means of ODFs, for both the original Fisher-Rao and the proposed framework.

[1]  D. Tuch High Angular Resolution Diffusion Imaging of the Human Brain , 1999 .

[2]  N. Makris,et al.  High angular resolution diffusion imaging reveals intravoxel white matter fiber heterogeneity , 2002, Magnetic resonance in medicine.

[3]  D. Tuch Q‐ball imaging , 2004, Magnetic resonance in medicine.

[4]  Rachid Deriche,et al.  A Riemannian Framework for Orientation Distribution Function Computing , 2009, MICCAI.

[5]  H. Karcher Riemannian center of mass and mollifier smoothing , 1977 .

[6]  L. Frank Characterization of anisotropy in high angular resolution diffusion‐weighted MRI , 2002, Magnetic resonance in medicine.

[7]  R. Vidal,et al.  A nonparametric Riemannian framework for processing high angular resolution diffusion images (HARDI) , 2009, 2009 IEEE Conference on Computer Vision and Pattern Recognition.

[8]  Anuj Srivastava,et al.  Riemannian Analysis of Probability Density Functions with Applications in Vision , 2007, 2007 IEEE Conference on Computer Vision and Pattern Recognition.

[9]  P. Thomas Fletcher,et al.  Riemannian geometry for the statistical analysis of diffusion tensor data , 2007, Signal Process..

[10]  I. Dryden,et al.  Non-Euclidean statistics for covariance matrices, with applications to diffusion tensor imaging , 2009, 0910.1656.

[11]  Maher Moakher,et al.  A Differential Geometric Approach to the Geometric Mean of Symmetric Positive-Definite Matrices , 2005, SIAM J. Matrix Anal. Appl..

[12]  C. R. Rao,et al.  Information and the Accuracy Attainable in the Estimation of Statistical Parameters , 1992 .

[13]  Xavier Pennec,et al.  A Riemannian Framework for Tensor Computing , 2005, International Journal of Computer Vision.

[14]  S. Arridge,et al.  Detection and modeling of non‐Gaussian apparent diffusion coefficient profiles in human brain data , 2002, Magnetic resonance in medicine.

[15]  P. Hagmann,et al.  Mapping complex tissue architecture with diffusion spectrum magnetic resonance imaging , 2005, Magnetic resonance in medicine.

[16]  R. Deriche,et al.  Regularized, fast, and robust analytical Q‐ball imaging , 2007, Magnetic resonance in medicine.

[17]  Paul M. Thompson,et al.  Mapping genetic influences on brain fiber architecture with high angular resolution diffusion imaging (HARDI) , 2008, 2008 5th IEEE International Symposium on Biomedical Imaging: From Nano to Macro.