Quasi-interpolation by quadratic piecewise polynomials in three variables

A quasi-interpolation method for quadratic piecewise polynomials in three variables is described which can be used for the efficient reconstruction and visualization of gridded volume data. The Bernstein-Bézier coefficients of the splines are immediately available from the given data values by applying a local averaging, where no prescribed derivatives are required. Since the approach does not make use of a particular basis or a subset spanning the spline spaces, we analyze the smoothness properties of the trivariate splines. We prove that the splines yield nearly optimal approximation order while simultaneously its piecewise derivatives provide optimal approximation of the derivatives for smooth functions. The constants of the corresponding error bounds are given explicitly. Numerical tests confirm the results and the efficiency of the method.