Quasi-interpolation by quadratic C1-splines on truncated octahedral partitions

We describe an approximating scheme for the smooth reconstruction of discrete data on volumetric grids. A local quasi-interpolation method for quadratic C1-splines on uniform tetrahedral partitions is used to achieve a globally smooth function. The Bernstein–Bézier coefficients of the piecewise polynomials are thereby directly determined by appropriate combinations of the data values. We explicitly give a construction scheme for a family of quasi-interpolation operators and prove that the splines and their derivatives can provide an approximation order two for smooth functions. The optimal approximation of the derivatives and the simple averaging rules for the coefficients recommend this method for high quality visualization of volume data. Numerical tests confirm the approximation properties and show the efficient computation of the splines.