Uniform Powell–Sabin spline wavelets

• Published in: Journal of computational and applied mathematics Vol. 154; no. 1; pp. 125 - 142
• Main Authors: Windmolders, Joris, Vanraes, Evelyne, Dierckx, Paul, Bultheel, Adhemar
• Format: Journal Article
• Language: English
• Published: Amsterdam Elsevier B.V 01-05-2003; Elsevier
• Subjects: Applied sciences; B-splines; Computer aided design; Computer science; control theory; systems; Exact sciences and technology; Fourier analysis; Lifting; Mathematical analysis; Mathematics; Numerical analysis; Numerical analysis. Scientific computation; Numerical approximation; Powell–Sabin splines; Sciences and techniques of general use; Software; Subdivision; Wavelets; Wavelets; 42C40; 65D17; 68U07; Powell–Sabin splines; Subdivision; 65T60; Lifting; 65D07; B-splines; B spline; Bézier coordinate; Triangulation; Segmentation; Multiresolution analysis; Spline; Spline approximation; Surface; Numerical approximation; Scaling function; Powell Sabin spline wavelet; Bézier curve; Wavelet transformation; Thresholding; Sequence space
• ISSN: 0377-0427; 1879-1778
• DOI: 10.1016/S0377-0427(02)00817-8

This paper discusses how the subdivision scheme for uniform Powell–Sabin spline surfaces makes it possible to place those surfaces in a multiresolution context. We first show that the basis functions are translates and dilates of one vector of scaling functions. This defines a sequence of nested spaces. We then use the subdivision scheme as the prediction step in the lifting scheme and add an update step to construct wavelets that describe a sequence of complement spaces. Finally, as an example application, we use the new wavelet transform to reduce noise on a uniform Powell–Sabin spline surface.