Multi-dimensional Average-Interpolating Refinement on Arbitrary Lattices

Multi-dimensional datasets containing local averages of a function arise in many applications such as processing of CCD captures and medical images. Motivated by this fact we introduce multi-dimensional average-interpolating refinement on arbitrary lattices in arbitrary dimensions. Our refinement algorithm results in smooth scaling functions of compact support. This method forms a basis for multi-dimensional multi-resolution analysis and subdivision on datasets obtained by locally averaging a smooth function. As an example, we present two-dimensional polynomial average-interpolating subdivision on the quincunx lattice and show that the resulting scaling functions are highly regular in the sense of Sobolev.