Matrix Subdivision Schemes

Subdivision schemes with matrix masks are a natural extension of the well studied case of subdivision schemes with scalar masks. Such schemes arise in the analysis of multivariate scalar schemes, in subdivision processes corresponding to shift-invariant spaces generated by more than one function, in geometric modeling where each component of the curve/surface is designed by a diierent linear combination of the control points, and in the case of Hermite-type schemes, which are based on function and derivatives values simultaneously. The limit of a matrix subdivision scheme can be expressed as a combination of shifts of a reenable matrix function. It is shown that if is stable in the sense of a new stability notion for matrix valued functions, then the scheme is uniformly convergent. Also it is shown that the stability of a maximal submatrix of is related with the linear dependence of its rows, and hence of any vector valued function generated by the subdivision scheme. Finally it is shown that by proper renormalization of the process relative to the vanishing rows of , it is possible to generate vector limit functions with several components, which are the rst derivative of certain linear combinations of the other components. In case of two components this gives rise to Hermite-type schemes. The same approach allows to analyze the convergence of the scheme and the smoothness of .