an:02111592
Zbl 1051.42025
Daubechies, Ingrid; Runborg, Olof; Sweldens, Wim
Normal multiresolution approximation of curves
EN
Constructive Approximation 20, No. 3, 399-463 (2004).
00107421
2004
j
42C40 65D10 65D17 65T60 68U07
subdivision scheme; multiresolution approximation; plane curve; approximation of curves; convergence; regularity; stability
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.
Manfred Tasche (Rostock)