Improving smoothness and accuracy of Modified Butterfly subdivision scheme

Motivated by the increasing request of surface representation techniques suitable for biomedical imaging applications, we construct a non-stationary subdivision scheme for regular 3-directional grids, which enjoys the following properties: (i) interpolation, (ii) affine invariance, (iii) C2 smoothness, (iv) approximation order 6 and (v) the capability of reproducing several trigonometric surfaces, especially ellipsoids. To study the smoothness properties of this new scheme via existing analysis tools, we also construct an auxiliary stationary subdivision scheme enjoying properties (i)-(iv). Taking into account that, when applied on regular 3-directional grids, the Modified Butterfly scheme is C1 and has approximation order 4, the subdivision schemes derived in this paper can be considered improved variants of the Modified Butterfly scheme.