Grants for Visiting Postdoc, Advanced Doctoral Students or Master’s Students to AIMS South Africa 2025 Interpolatory Subdivision and Wavelets on an Interval with General Integer Arity
Dr. Rejoyce Gavhi-Molefe
Proposed project:
Interpolatory Subdivision and Wavelets on an Interval with General Integer Arity
Description:
Wavelet decomposition and reconstruction algorithms, as widely applied in signal analysis applications, often have the drawback of being based on the assumption that the signal to be analyzed is known on the whole real line. Similarly, subdivision schemes are nearly always constructed based on the assumption that the initial control point sequence isbi-infinite. Nevertheless, various (often unsatisfactory) ad hoc methods are used in practice to overcome these difficulties. Hence there is a significant need for a systematic unified approach to construct wavelets on a bounded interval and subdivision schemes for finite initial control point sequences. Underlying the mathematical analysis of both wavelets and subdivision schemes is the concept of a refinable function, that is, a self-reproducing function that can be expressed as a linear combination of the integer translates of its own dilation by factor two.
A method of adapting the binary Dubuc-Deslauriers subdivision scheme defined for bi-infinite sequences to accommodate sequences of finite length has been studied in [1]. Based on this work thereof, this research project investigates a method of adapting the Dubuc-Deslauriers subdivision scheme with general integer arity to accommodate sequences of finite length. Two numerical examples of signature smoothing and two-dimensional feature extraction of the d-ary subdivision and wavelet algorithms will be explored.
Keywords:
Subdivision schemes, refinable function, wavelet decomposition, finite sequences, algorithms