Linear Multiscale Transforms Based on Even-Reversible Subdivision Operators

Multiscale transforms for real-valued data, based on interpolatory subdivision operators, have been studied in recent years. They are easy to define and can be extended to other types of data, for example, to manifold-valued data. In this chapter, we define linear multiscale transforms, based on certain linear, non-interpolatory subdivision operators, termed “even-reversible.” For such operators, we prove, using Wiener’s lemma, the existence of an inverse to the linear operator defined by the even part of the subdivision mask and term it “even-inverse.” We show that the non-interpolatory subdivision operators, with spline or pseudo-spline masks, are even-reversible and derive explicitly, for the quadratic and cubic spline subdivision operators, the symbols of the corresponding even-inverse operators. We also analyze properties of the multiscale transforms based on even-reversible subdivision operators, in particular, their stability and the rate of decay of the details.