B-spline normal multi-scale transforms for planar curves

Normal multi-scale transform [4] is a nonlinear multi-scale transform for representing geometric objects that has been recently investigated [1, 7, 10]. The restrictive role of the exact order of polynomial reproduction $P_e$ of the approximating subdivision operator $S$ in the analysis of the $S$ normal multi-scale transform, established in [7, Theorem 2.6], significantly disfavors the practical use of these transforms whenever $P_e\ll P$. We analyze in detail the normal multi-scale transform for planar curves based on B-spline subdivision scheme $S_p$ of degree $p\ge3$ and derive higher smoothness of the normal re-parameterization than in [7]. We show that further improvements of the smoothness factor are possible, provided the approximate normals are cleverly chosen. Following [10], we introduce a more general framework for those transforms where more than one subdivision operator can be used in the prediction step, which leads to higher detail decay rates.