Dubuc--Deslauriers Subdivision for Finite Sequences and Interpolation Wavelets on an Interval

Abstract

In this paper we consider a method of adapting Dubuc--Deslauriers subdivision, which is defined for bi-infinite sequences, to accommodate sequences of finite length. After deriving certain useful properties of the Dubuc--Deslauriers refinable function on ${\mathbb{R}}$, we define a multiscale finite sequence of functions on a bounded interval, which are then proved to be refinable. Using this fact, the resulting adapted interpolatory subdivision scheme for finite sequences is then shown to be convergent. Corresponding interpolation wavelets on an interval are defined, and explicit formulations of the resulting decomposition and reconstruction algorithms are calculated. Finally, we give two numerical examples on signature smoothing and two-dimensional feature extraction of the subdivision and wavelet algorithms.