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On the application of the Wasserstein metric to 2D curves classification

Agnieszka Kaliszewska, Monika Syga

TL;DR

This work advances the use of the Wasserstein distance for 2D curve classification by introducing discrete weight distributions that concentrate the similarity measure on prescribed curve fragments. It formalizes both standard and partial optimal transport variants, including dual formulations, and proposes a suite of fragment-focused weight schemes (uniform, binomial, coordinate-based, and preselected supports). The authors demonstrate through archaeology-derived datasets that these weighted and partial transports yield robust clustering performance, with results illustrating the advantages of focusing on informative curve regions. The approach offers a flexible, geometry-aware framework for shape similarity that can be extended to other domains such as signals and image contours.

Abstract

In this work we analyse a number of variants of the Wasserstein distance which allow to focus the classification on the prescribed parts (fragments) of classified 2D curves. These variants are based on the use of a number of discrete probability measures which reflect the importance of given fragments of curves. The performance of this approach is tested through a series of experiments related to the clustering analysis of 2D curves performed on data coming from the field of archaeology.

On the application of the Wasserstein metric to 2D curves classification

TL;DR

This work advances the use of the Wasserstein distance for 2D curve classification by introducing discrete weight distributions that concentrate the similarity measure on prescribed curve fragments. It formalizes both standard and partial optimal transport variants, including dual formulations, and proposes a suite of fragment-focused weight schemes (uniform, binomial, coordinate-based, and preselected supports). The authors demonstrate through archaeology-derived datasets that these weighted and partial transports yield robust clustering performance, with results illustrating the advantages of focusing on informative curve regions. The approach offers a flexible, geometry-aware framework for shape similarity that can be extended to other domains such as signals and image contours.

Abstract

In this work we analyse a number of variants of the Wasserstein distance which allow to focus the classification on the prescribed parts (fragments) of classified 2D curves. These variants are based on the use of a number of discrete probability measures which reflect the importance of given fragments of curves. The performance of this approach is tested through a series of experiments related to the clustering analysis of 2D curves performed on data coming from the field of archaeology.
Paper Structure (22 sections, 1 theorem, 26 equations, 51 figures, 1 table)

This paper contains 22 sections, 1 theorem, 26 equations, 51 figures, 1 table.

Table of Contents

  1. Introduction
  2. The Wasserstein distance
  3. Wasserstein distance for different distributions
  4. Standard distributions
  5. Other relevant distributions
  6. Partial optimal transport
  7. Experiments
  8. The data used in experiments
  9. Experiments: The Wasserstein distance-basic formulation
  10. Processing of the data for the experiments
  11. Experiments on constrained data
  12. Experiments: discussion of results
  13. Conclusions
  14. Figures
  15. Data sets
  16. ...and 7 more sections

Key Result

Theorem 1

[c.f.Theorem 1,Cuturi] The optimal transport value $OT(x,y;C)$ given by eq_ot is a distance on $\Sigma_{k}$, i.e. for $x,y\in \Sigma_{k}$.

Figures (51)

  • Figure 1: The process of generating of a curve from a cross section.
  • Figure 2: Results of clustering of Set 1 according to Formula \ref{['WD']}, with weights chosen according to \ref{['waga_uniform']}.
  • Figure 3: Curves, representing contours of objects 4 and 5 plotted in $x,y$ coordinates: 4 -blue; 5 -orange (Note: the curves are plotted upside down to facilitate calculation).
  • Figure 4: Curves, representing contours of objects 2 and 9 plotted in $x,y$ coordinates: 2 -blue; 9 -orange (Note: the curves are plotted upside down to facilitate calculation).
  • Figure 5: Clustering result using the Procrustes analysis.
  • ...and 46 more figures

Theorems & Definitions (2)

  • Theorem 1
  • Example 1