Local quasi-interpolation by cubic C1 splines on type-6 tetrahedral partitions

We describe an approximating scheme based on cubic C1 splines on type-6 tetrahedral partitions using data on volumetric grids. The quasi-interpolating piecewise polynomials are directly determined by setting their Bernstein–Bézier coefficients to appropriate combinations of the data values. Hence, each polynomial piece of the approximating spline is immediately available from local portions of the data, without using prescribed derivatives at any point of the domain. The locality of the method and the uniform boundedness of the associated operator provide an error bound, which shows that the approach can be used to approximate and reconstruct trivariate functions. Simultaneously, we show that the derivatives of the quasi-interpolating splines yield nearly optimal approximation order. Numerical tests with up to 17 × 106 data sites show that the method can be used for efficient approximation.