Smoothness Properties of Lie Group Subdivision Schemes

Linear stationary subdivision rules take a sequence of input data and produce ever denser sequences of subdivided data from it. They are employed in multiresolution modeling and have intimate connections with wavelet and more general pyramid transforms. Data which naturally do not live in a vector space, but in a nonlinear geometry like a surface, symmetric space, or a Lie group (e.g. motion capture data), require different handling. One way to deal with Lie group valued data has been proposed by D. Donoho [3]: It is to employ a log-exponential analogue of a linear subdivision rule. While a comprehensive discussion of applications is given by Ur Rahman et al. in [9], this paper analyzes convergence and smoothness of such subdivision processes and show that the nonlinear schemes essentially have the same properties regarding C and C smoothness as the linear schemes they are derived from.