International journal of wavelets, multiresolution and information processing vol:2 issue:1 pages:23-42
Recently we developped a subdivision scheme for Powell-Sabin splines. It is a triadic scheme and it is general in the sense that it is not restricted to uniform triangles, the vertices must not have valence six and there are no restrictions on the initial triangles. A sequence of nested spaces or multiresolution analysis can be associated with the base triangulation. In this paper we use the lifting scheme to construct basis functions for the complementary space that captures the details that are lost when going to a coarser resolution. The subdivision scheame appears as the first lifting step or prediction step. A second lifting step, the update, is used to achieve certain properties for the complement spaces and the wavelet functions such as orthogonality and vanishing moments. Te design of the update step is based on stability considerations. We prove stability for both the scaling functions and the wavelet functions.