GPU-Based Ray-Casting of Spherical Functions Applied to High Angular Resolution Diffusion Imaging

Any sufficiently smooth, positive, real-valued function \psi : S^2 \rightarrow {\hbox{\rlap{I}\kern 2.0pt{\hbox{R}}}}^+ on a sphere S^2 can be expanded by a Laplace expansion into a sum of spherical harmonics. Given the Laplace expansion coefficients, we provide a CPU and GPU-based algorithm that renders the radial graph of \psi in a fast and efficient way by ray-casting the glyph of \psi in the fragment shader of a GPU. The proposed rendering algorithm has proven highly useful in the visualization of high angular resolution diffusion imaging (HARDI) data. Our implementation of the rendering algorithm can display simultaneously thousands of glyphs depicting the local diffusivity of water. The rendering is fast enough to allow for interactive manipulation of large HARDI data sets.