Issue |
ESAIM: COCV
Volume 30, 2024
|
|
---|---|---|
Article Number | 64 | |
Number of page(s) | 34 | |
DOI | https://doi.org/10.1051/cocv/2024008 | |
Published online | 12 September 2024 |
Approximation of splines in Wasserstein spaces
1
Institute for Numerical Simulation, University of Bonn, Endenicher Allee 60, 53115 Bonn, Germany
2
Fakultät für Mathematik, Universität Bielefeld, Postfach 100131, 33501 Bielefeld, Germany
* Corresponding author: jorge.justiniano@ins.uni-bonn.de
Received:
30
December
2022
Accepted:
19
January
2024
This paper investigates a time discrete variational model for splines in Wasserstein spaces to interpolate probability measures. Cubic splines in Euclidean space are known to minimize the integrated squared acceleration subject to a set of interpolation constraints. As generalization on the space of probability measures the integral of the squared acceleration is considered as a spline energy and regularized by addition of the usual action functional. Both energies are then discretized in time using local Wasserstein-2 distances and the generalized Wasserstein barycenter. The existence of time discrete regularized splines for given interpolation conditions is established. On the subspace of Gaussian distributions, the spline interpolation problem is solved explicitly and consistency in the discrete to continuous limit is shown. The computation of time discrete splines is implemented numerically, based on entropy regularization and the Sinkhorn algorithm. A variant of Nesterov’s accelerated gradient descent algorithm is applied for the minimization of the fully discrete functional. A variety of numerical examples demonstrate the robustness of the approach and show striking characteristics of the method. As a particular application the spline interpolation for synthesized textures is presented.
Mathematics Subject Classification: 53B20 / 65D07 / 35Q49 / 65K10 / 68U10
Key words: Optimal transport / spline interpolation / texture synthesis / Sinkhorn algorithm / Nesterov’s accelerated gradient descent method
© The authors. Published by EDP Sciences, SMAI 2024
This is an Open Access article distributed under the terms of the Creative Commons Attribution License (https://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Current usage metrics show cumulative count of Article Views (full-text article views including HTML views, PDF and ePub downloads, according to the available data) and Abstracts Views on Vision4Press platform.
Data correspond to usage on the plateform after 2015. The current usage metrics is available 48-96 hours after online publication and is updated daily on week days.
Initial download of the metrics may take a while.