HARDI models


Diffusion MRI is a MR modality that has made possible to characterize the white matter architecture in the brain in vivo. A popular model of the diffusion profile is based on the Gaussian assumption, which allows the diffusion to be modeled with a single covariance matrix, namely the diffusion tensor. With the advent of parallel imaging and better scanners, it is today feasible to acquire a larger number of diffusion weighted images in clinical time, the so-called High Angular Resolution Diffusion Imaging (HARDI). An advantage of HARDI is that it can better model more complex fiber architectures others than one single fiber bundle at each image voxel, such as crossing, bending, or kissing fibers.

A number of HARDI techniques have recently appeared, and among them Q-Ball imaging, which is based on the Funk–Radon Transform, has gained popularity mainly because it can be robustly computed with closed-form expressions and does not require any assumptions about the behavior of the diffusion signal outside of the sampled sphere. Q-Balls use the FRT to approximate the Orientation Distribution Function (ODF) as the radial projection (integral) of the 3D Probability Density Function (PDF). An alternative approach to represent the orientation distribution is to marginalize the radial part of the PDF. Although these two approaches at first seem equivalent, a major difference for the latter is that the Jacobian of the spherical coordinates is included in the radial integral defining the margin- alization, computing an angular function which is a true marginal PDF with a valid probabilistic interpretation. An estimator of such a function is the Orientation Probability Density Function (OPDF).

Like Q-Balls, the OPDF is based on the FRT, so only very weak assumptions have to be made about the behavior of the diffusion signal outside the sampled sphere. Our results indicate that this probabilistic approach better resolves fiber crossings compared to other related approaches (Q-Balls and Diffusion Orientation Trans- form), especially for low values of the diffusion weighting parameter. Results on brain HARDI data also show promising results indicating the ability to visualize regions of crossing fibers.



  • A. Tristán-Vega, C-F Westin, S Aja-Fernández, Estimation of fiber Orientation Probability Density Functions in High Angular Resolution Diffusion Imaging, NeuroImage, Vol. 47 (2), Aug. 2009, pp. 638-650. [doi]
  • A. Tristán-Vega, C.-F. Westin and S. Aja-Fernández. A new methodology for the estimation of fiber populations in the white matter of the brain with the Funk-Radon transform. NeuroImage, Vol. 49, No. 2, Jan. 2010, pp. 1301-1315. [doi]
  • A. Tristán-Vega, S. Aja-Fernández, and C.-F. Westin. On the blurring of the Funk–Radon transform in Q–ball imaging. In Medical Image Computing and Computer-Assisted Intervention, Lecture Notes in Computer Science, Vol. 5762, pp. 415-422. Springer–Verlag, Sep. 2009.
  • Tristán-Vega, A., S. Aja-Fernández, and C-F. Westin, "Deblurring of probabilistic ODFs in quantitative diffusion MRI", Biomedical Imaging (ISBI), 2012 9th IEEE International Symposium on: IEEE, pp. 932–935, 2012.