Normal multiresolution approximation of curves.

*(English)*Zbl 1051.42025Subdivision 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.

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.

Reviewer: Manfred Tasche (Rostock)

##### 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 |