Wavelets from the Loop scheme

Bin Han and Zuowei Shen


A new wavelet-based geometric mesh compression algorithm was developed recently in the area of computer graphics by Khodakovsky, Schr\"oder, and Sweldens in their interesting paper \cite{KSS}. The new wavelets used in \cite{KSS} were designed from the Loop scheme by using ideas and methods of \cite{RiS, RiS:prewavelet}, where orthogonal wavelets with exponential decay and pre-wavelets with compact support were constructed. The wavelets have the same smoothness order as that of the basis function of the Loop scheme around the regular vertices which has a continuous second derivative; the wavelets also have smaller supports than those wavelets obtained by constructions in \cite{RiS, RiS:prewavelet} or any other compactly supported biorthogonal wavelets derived from the Loop scheme (e.g. \cite{Han:bw, Han:proj}). Hence, the wavelets used in \cite{KSS} have a good time frequency localization. This leads to a very efficient geometric mesh compression algorithm as proposed in \cite{KSS}. As a result, the algorithm in \cite{KSS} outperforms several available geometric mesh compression schemes used in the area of computer graphics. However, it remains open whether the shifts and dilations of the wavelets form a Riesz basis of $L_2(\RR^2)$. Riesz property plays an important role in any wavelet-based compression algorithm and is critical for the stability of any wavelet-based numerical algorithms. We confirm here that the shifts and dilations of the wavelets used in \cite{KSS} for the regular mesh, as expected, do indeed form a Riesz basis of $L_2(\RR^2)$ by applying the more general theory established in this paper.

Back to Preprints and Publications