Quasi-interpolation for Volumetric Data Reconstruction in S_4^2(Delta_3)

In this paper we propose a method based on basis in S₄²(Δ₃) for reconstructing volumetric data sampled on the BCC lattice. In particular we implement numerical representation of a trivariate box spline reconstruction kernel in S₄²(Δ₃). It is proved that the box spline have an uniform property and reconstruction that can be considered as a three dimensional extension of the well-known Zwart-Powell element in 2D. At the same time, we obtain the quasi-interpolation operators in S₄²(Δ₃) and some supporting numerical results are presented.