Multidimensional interpolatory subdivision schemes

This paper presents a general construction of multidimensional interpolatory subdivision schemes. In particular, we provide a concrete method for the construction of bivariate interpolatory subdivision schemes of increasing smoothness by finding an appropriate mask to convolve with the mask of a three-direction box spline B-r,B-r,B-r of equal multiplicities. The resulting mask for the interpolatory subdivision exhibits all the symmetries of the three-direction box spline and with this increased symmetry comes increased smoothness. Several examples are computed (for r = 2,...,8). Regularity criteria in terms of the refinement mask are established and applied to the examples to estimate their smoothness.