Using A Hyperspherical Harmonic Basis for Sparse Multi-Voxel Modeling of Diffusion MRI

Evan Schwab^{1,2}, Hasan Ertan Cetingul^{2}, Rene Vidal^{3}, and Mariappan Nadar^{2}

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Figure 1: Illustration of multi-voxel parameterization for a patch of 4 single-shell HARDI signals (top view) with center $$$p_0$$$, and q-space vector $$$q$$$. The global coordinate $$$g = p_0 + q$$$ depends on voxel size and gradient direction $$$q$$$. Any array of 3D points can be reconstructed with this parameterization.

Figure 2: Examples of HSH signal basis elements (after conversion to ODFs for visualization) for a 1×2 patch of voxels. Notice that all different types of ODF configurations are presented for a given HSH atom and that neighboring ODFs, regardless of similarity, can be jointly modeled with very few atoms.

Figure 3: (Top) $$$L_2$$$ reconstruction of a 10×10×3 synthetic HARDI signals with SNR=5. SH order L=4 requires 15×(10×10×3)=4,500 coefficients. For HSH order N=16, we used four 5×5×3 non-overlapping patches requiring 1 (0-fiber quad) + 2×121 (1-fiber quads) + 125 (2-fiber quad) = 368 coefficients. (Bottom) HSH coefficients for each quadrant.

Table 1: Results of experiments on synthetic, phantom and real HARDI datasets comparing the total number of coefficients and the average MSE of the entire datasets using single-voxel SH and multi-voxel HSH $$$L_2$$$ and $$$L_0$$$ reconstruction. We can see a drastic decrease in the number of coefficients while maintaining error.

Proc. Intl. Soc. Mag. Reson. Med. 24 (2016)

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