Using Laurent polynomial representation for the analysis of non-uniform binary subdivision schemes

Starting with values {f 0 j }j∈Z assigned to the integers, a binary subdivision scheme defines recursively values {fk j }j∈Z, respectively assigned to the binary points {2−kj}j∈Z. The purpose of subdivision analysis is to study the convergence of such processes and to establish the existence of a limit function on R and its smoothness class. A general treatment of uniform subdivision can be found in [1–3,7,9,16,17]. Level-dependent subdivision schemes, where the scheme may vary from one refinement level to the other, are discussed in [13]. In the present work we analyze nonuniform binary subdivision schemes in which the scheme for defining the points may vary from point to point and from level to level. To present the problem for nonuniform schemes we first review one way of analyzing uniform binary subdivision schemes using a Laurent polynomial representation. A uniform binary subdivision scheme, with a finite mask {pi}i=−m, is defined by