Geometric Subdivision and Multiscale Transforms

Abstract

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.

Non-smooth Variational Regularization for Processing Manifold-Valued Data

“…Subdivision schemes for manifold-valued data have been considered in [109,61,121,114,110]. Interpolatory wavelet transforms and subdivision are discussed in more detail in [108]. All the above approaches consider explicit schemes, i.e., the measured data is processed in a forward way using the analogues of averaging rules and differences in the manifold setting.…”