An upper bound on the volume of the symmetric difference of a body and a congruent copy

TL;DR

This paper establishes an upper bound on the volume of the symmetric difference between a shape and its rigid motion, aiding in shape matching by ensuring the similarity measure's stability under small transformations.

Contribution

It provides sharp bounds on the symmetric difference volume for translations and rotations, linking geometric boundary measures to transformation distances, advancing shape matching techniques.

Findings

Bound is sharp for translations.

Bound is sharp for rotations with smooth boundaries.

Supports Lipschitz continuity of shape overlap functions.

Abstract

Let A be a bounded subset of IR^d. We give an upper bound on the volume of the symmetric difference of A and f(A) where f is a translation, a rotation, or the composition of both, a rigid motion. The volume is measured by the d-dimensional Hausdorff measure, which coincides with the Lebesgue measure for Lebesgue measurable sets. We bound the volume of the symmetric difference of A and f(A) in terms of the (d-1)-dimensional volume of the boundary of A and the maximal distance of a boundary point to its image under f. The boundary is measured by the (d-1)-dimensional Hausdorff measure, which matches the surface area for sufficiently nice sets. In the case of translations, our bound is sharp. In the case of rotations, we get a sharp bound under the assumption that the boundary is sufficiently nice. The motivation to study these bounds comes from shape matching. For two shapes A and B in IR^d and a class of transformations, the matching problem asks for a transformation f such that f(A) and B match optimally. The quality of the match is measured by some similarity measure, for instance the volume of overlap. Let A and B be bounded subsets of IR^d, and let F be the function that maps a rigid motion r to the volume of overlap of r(A) and B. Maximizing this function is a shape matching problem, and knowing that F is Lipschitz continuous helps to solve it. We apply our results to bound the difference |F(r) - F(s)| for rigid motions r,s that are close, implying that F is Lipschitz continuous for many metrics on the space of rigid motions. Depending on the metric, also a Lipschitz constant can be deduced from the bound.