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Normal multiresolution approximation of curves. (English) Zbl 1051.42025
Subdivision is a powerful procedure for iteratively creating smooth curves and surfaces. Combined with wavelet methods, subdivision can be used to approximate functions, curves, and surfaces. The authors discuss the multiresolution approximation of a plane curve $$\Gamma$$ in detail. A multiresolution approximation of $$\Gamma$$ is called normal if all wavelet detail vectors align with a locally defined normal direction which only depends on the coarser levels. Here normal direction means a normal onto an approximation of $$\Gamma$$. This notion is very useful for compression applications.
In this interesting paper, the authors study properties as convergence, speed of convergence, regularity, and stability of a normal multiresolution approximation of $$\Gamma$$. It is shown that these properties critically depend on the underlying subdivision scheme and that in general the convergence of normal multiresolution approximation of $$\Gamma$$ equals the convergence of the underlying subdivision scheme. The central idea is to study the normal multiresolution approximation of $$\Gamma$$ as a perturbation of a linear subdivision scheme.

##### MSC:
 42C40 Nontrigonometric harmonic analysis involving wavelets and other special systems 65D10 Numerical smoothing, curve fitting 65D17 Computer-aided design (modeling of curves and surfaces) 65T60 Numerical methods for wavelets 68U07 Computer science aspects of computer-aided design
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