Global divergences between measures: from Hausdorff distance to Optimal Transport

Jean Feydy (1, 2) , Alain Trouvé (1)

1 CMLA - Centre de Mathématiques et de Leurs Applications
2 DMA - Département de Mathématiques et Applications - ENS Paris

The data attachment formula is a key component of shape registration pipelines: computed at every step, its gradient is the vector field that drives a deformed model towards its target. Unfortunately, most classical formulas are at most semi-local: their gradients saturate and stop being informative above some given distance, with appalling consequences on the robustness of shape analysis pipelines. In this paper, we provide a unified view of three fidelities between measures that alleviate this problem: the Energy Distance from statistics; the (weighted) Hausdorff distance from computer graphics; the Wasserstein distance from Optimal Transport theory. Provided with efficient GPU routines and theoretical guarantees, these tools should allow the shape analyst to handle large deformations without hassle.

Domaines: Vision par ordinateur et reconnaissance de formes [cs.CV]; Traitement des images [eess.IV]; Mathématiques [math]

Jean Feydy, Alain Trouvé. Global divergences between measures: from Hausdorff distance to Optimal Transport. 2018. ⟨hal-01827184v1⟩