Log-exponential analogues of univariate subdivision schemes in Lie groups and their smoothness properties

The necessity to process data which live in nonlinear ge- ometries (e.g. motion capture data, unit vectors, subspaces, positive definite matrices) has led to some recent developments in nonlinear multiscale representation and subdivision algorithms. The present paper analyzes convergence and C 1 and C 2 smoothness of subdivi- sion schemes which operate in matrix groups or general Lie groups, and which are defined by the so-called log-exponential analogy. It is shown that a large class of such schemes has essentially the same smoothness as the linear schemes they are derived from. This work extends previous work on Lie group subdivision schemes - we con- sider alternative definitions of analogous schemes, arbitrary dilation factors, and symmetry of the nonlinear scheme.