FAST DISPLACEMENT PROBABILITY PROFILE APPROXIMATION FROM HARDI USING 4TH-ORDER TENSORS

Abstract

Cartesian tensor basis have been widely used to approximate spherical functions. In Medical Imaging, tensors of various orders have been used to model the diffusivity function in Diffusion-weighted MRI data sets. However, it is known that the peaks of the diffusivity do not correspond to orientations of the underlying fibers and hence the displacement probability profiles should be employed instead. In this paper, we present a novel representation of the probability profile by a 4(th) order tensor, which is a smooth spherical function that can approximate single-fibers as well as multiple-fiber structures. We also present a method for efficiently estimating the unknown tensor coefficients of the probability profile directly from a given high-angular resolution diffusion-weighted (HARDI) data set. The accuracy of our model is validated by experiments on synthetic and real HARDI datasets from a fixed rat spinal cord.

Fig. 1

3D plots of three of the basis functions from Table 1. The functions were evaluated for varying ∥ q ∥ over a unit by circle of directions q. The circle was defined fixing the elevation spherical coordinate to π/3 and varying azimuth.

Fig. 2

Probability profiles estimated by applying our method to simulated data of: left) 2-fiber crossing bundle and right) corrupted crossings for different amounts of Riccian noise.

Fig. 3

Fiber orientation errors for different SNR in the data using our method (P4) and three other existing methods: 1) DOT, 2) ODF and 3) 4^th-order DT. In the experiment we used simulated MR signal of a 2-fiber crossing, whose probability profile is shown in Fig.2(right).

Fig. 4

Estimated probability profiles from real data of a rat’s fixed spinal cord. The zoomed ROI shows single fiber distributions in white matter and other more complex tissue structures.