Uniform Sampling and Reconstruction of Trivariate Functions

The Body Centered Cubic (BCC) and Face Centered Cubic (FCC) lattices have been known to outperform the commonly-used Cartesian sampling lattice due to their improved spectral sphere packing properties. However, the Cartesian lattice has been widely used for sampling of trivariate functions with applications in areas such as biomedical imaging, scientific data visualization and computer graphics. The widespread use of Cartesian lattice is partly due to the availability of tensor-product approach that readily extend the univariate reconstruction methods to trivariate setting. In this paper we report on recent advances on non-separable reconstruction algorithms, based on box splines, for reconstruction of data sampled on the BCC and FCC lattices. It turns out that these box spline reconstructions are faster than the corresponding tensorproduct B-spline reconstructions on the Cartesian lattice. This suggests that not only the BCC and FCC lattices are more accurate sampling patterns, their respective reconstruction methods are also more computationally efficient than the tensor-product reconstructions – a fact which is contrary to the common assumption among practitioners.