  Any procedure applied to data, and any quantity derived from data, is
required to respect the nature and symmetries of the data. This axiom applies
to refinement procedures and multiresolution transforms as well as to more
basic operations like averages. This chapter discusses different kinds of
geometric structures like metric spaces, Riemannian manifolds, and groups, and
in what way we can make elementary operations geometrically meaningful. A nice
example of this is the Riemannian metric naturally associated with the space of
positive definite matrices and the intrinsic operations on positive definite
matrices derived from it. We disucss averages first and then proceed to
refinement operations (subdivision) and multiscale transforms. In particular,
we report on the current knowledge as regards convergence and smoothness.
