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Wavelets from the Loop scheme

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Bin Han and Zuowei Shen

## Abstract

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.

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