euclidean-hausdorff 1.2.3

quick approximation of the Gromov–Hausdorff distance restricted to Euclidean isometries

euclidean-hausdorff

Given the coordinates of 2- or 3-dimensional point clouds $A, B \subset \mathbb{R}^k$ (where $k \in {2, 3}$), estimates their Euclidean–Hausdorff distance (which itself is a relaxation and an upper bound of the Gromov–Hausdorff distance)

$$d_\text{EH}(X, Y) = \inf_{T:E(k)} d_\text{H}(T(A), B),$$

where the infimum is taken over all $k$-dimensional Euclidean isometries and $d_\text{H}$ is the Hausdorff distance in $\mathbb{R}^k$.

The distance is estimated from above by discretizing the compact feasible region (of the above minimization) into a search grid, whose vertices each represent a combination of some translation, rotation, and reflection.