ABSTRACT
To understand and integrate wavelet methods and algorithms into the subdivision “tool-box,” it requires somewhat more sophisticated mathematical training and ability to master the methods of subdivision. In addition, the traditional formulation of scaling functions and wavelets requires existence of their dual functions. Indeed, bi-orthogonal wavelets with sufficiently high orders of (integral) vanishing moments are instrumental to data analysis, particularly in signal and image processing, via wavelet decomposition. When the wavelet “decomposition-reconstruction” algorithm is integrated with subdivision schemes, it makes better sense to reverse the order, changing it to a “reconstruction-decomposition” algorithm. An ideal spline-wavelet family for curve design and editing is introduced, in which uniqueness of the synthesis wavelets and the corresponding “reconstruction-decomposition” filter sequences are determined by the requirement of minimum filter lengths. The notion of Fourier transform to study refinability and computing refinement sequences is easily understood and self-contained.
