Constructing 3D elliptical gaussians for irregular data

Volumetric datasets obtained from scientific simulation and partial differential equation solvers are typically given in the form of non-rectilinear grids. The splatting technique is a popular direct volume rendering algorithm, which can provide high quality rendering results, but has been mainly described for rectilinear grids. In splatting, each voxel is represented by a 3D kernel weighted by the discrete voxel value. While the 3D reconstruction kernels for rectilinear grids can be easily constructed based on the distance among the aligned voxels, for irregular grids the kernel construction is significantly more complicated. In this paper, we propose a novel method based on a 3D Delaunay triangulation to create 3D elliptical Gaussian kernels, which then can be used by a splatting algorithm for the rendering of irregular grids. Our method does not require a resampling of the irregular grid. Instead, we use a weighted least squares method to fit a 3D elliptical Gaussian centered at each grid point, approximating its Voronoi cell. The resulting 3D elliptical Gaussians are represented using a convenient matrix representation, which allows them to be seamlessly incorporated into our elliptical splatting rendering system.