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Geometric Subdivision and Multiscale Transforms

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Handbook of Variational Methods for Nonlinear Geometric Data

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.

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Notes

  1. 1.

    More precisely, for all \(a,b\in {{\mathcal {M}}} \) there is a unique midpoint x = m(a, b) defined by \({\mathit {d}_{{{\mathcal {M}}}}}(x,a)={\mathit {d}_{{{\mathcal {M}}}}}(x,b)={\mathit { d}_{{{\mathcal {M}}}}}(a,b)/2\), and for any \(a,b,c\in {{\mathcal {M}}}\) and points \(a',b',c'\in {\mathbb R}^2\) which have the same pairwise distances as a, b, c, the inequality \({\mathit {d}_{{{\mathcal {M}}}}}(c,m(a,b)) \le d_{{{\mathbb R}}^2}(c',m(a',b'))\) holds.

  2. 2.

    i.e., σ obeys the law σ([v, w]) = [σ(v), σ(w)], where in the matrix group case, the Lie bracket operation is given by [v, w] = vw − wv.

  3. 3.

    Implying convergence of the linear rule S. N is the dilation factor, S ∗ is the derived rule, cf. Example 4.5.

  4. 4.

    “Stable” means existence of constants C 1, C 2 with C 1∥p∥≤∥S ∞p∥∞≤ C 2∥p∥ for all input data where ∥p∥ :=supi∥p i∥ is bounded. Stable rules with C n limits generate polynomials of degree ≤ n, which is a property used in the proof.

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Acknowledgements

The author gratefully acknowledges the support of the Austrian Science fund through grant No. W1230, and he also wants to thank all colleagues whose work he was able to present in this survey.

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  1. TU Graz, Graz, Austria

    Johannes Wallner

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  1. Johannes Wallner
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  1. Faculty of Mathematics, University of Vienna, Wien, Austria

    Philipp Grohs

  2. Institute of Mathematics and Scientific Computing, University of Graz, Graz, Austria

    Martin Holler

  3. Department of Mathematics and Natural Sciences, Hochschule Darmstadt, Darmstadt, Germany

    Andreas Weinmann

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Wallner, J. (2020). Geometric Subdivision and Multiscale Transforms. In: Grohs, P., Holler, M., Weinmann, A. (eds) Handbook of Variational Methods for Nonlinear Geometric Data. Springer, Cham. https://doi.org/10.1007/978-3-030-31351-7_4

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