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Geometric Subdivision and Multiscale Transforms

In: Handbook of Variational Methods for Nonlinear Geometric Data

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  • Johannes Wallner

    (TU Graz)

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 discuss 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.

Suggested Citation

  • Johannes Wallner, 2020. "Geometric Subdivision and Multiscale Transforms," Springer Books, in: Philipp Grohs & Martin Holler & Andreas Weinmann (ed.), Handbook of Variational Methods for Nonlinear Geometric Data, chapter 0, pages 121-152, Springer.
    • Handle: RePEc:spr:sprchp:978-3-030-31351-7_4
    DOI: 10.1007/978-3-030-31351-7_4
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