# Geometric subdivision and multiscale transforms

## TL;DR

This paper explores how to perform subdivision and multiscale transforms on data with geometric structures, ensuring operations respect data symmetries and intrinsic properties.

## Contribution

It provides a framework for geometric subdivision and multiscale transforms on various structured spaces, emphasizing intrinsic operations and convergence properties.

## Key findings

- Analysis of geometric structures like metric spaces and manifolds.
- Discussion of intrinsic averaging and refinement operations.
- Current understanding of convergence and smoothness in 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.