Distance de Hausdorff
La Distance de Hausdorff est un concept de mathematics that quantifies how far apart two subsets of a metric space are from each other. It is particularly useful in various fields such as vision par ordinateur, traitement d'image, and shape analysis.
Formally, given two non-empty subsets A and B of a metric space (which often refers to a space equipped with a fonction de distance), la Distance de Hausdorff, notée d_H(A, B), est définie comme :
d_H(A, B) = max(h(A, B), h(B, A))
Où :
- h(A, B) = maxa ∈ A minb ∈ B d(a, b) – This measures the greatest distance from any point in set A to the nearest point in set B.
- h(B, A) = maxb ∈ B mina ∈ A d(b, a) – This measures the greatest distance from any point in set B to the nearest point in set A.
The overall Hausdorff Distance thus captures the maximum of these two measures, providing a comprehensive measurement de la séparation entre les deux ensembles.
Une caractéristique clé de la Distance de Hausdorff est its ability to handle non-convex shapes and irregular boundaries effectively. In practical applications, such as comparing shapes in image recognition, the Hausdorff Distance helps to determine how similar or different two shapes are based on their geometric properties.
En résumé, la Distance de Hausdorff est un outil précieux tant en mathématiques théoriques qu’appliquées, permettant de comparer des formes et des ensembles de manière rigoureuse.