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Computing the Hausdorff distance of two sets from their distance functions. (English) Zbl 07309478
In the paper, the author studies the problem of computing the Hausdorff distance between two sets using their distance functions. Throughout the paper, the author assumes that the considered sets are compact sets of an \(n\)-dimensional real space. In the beginning, the author shows a representation of the Hausdorff distance in terms of the distance functions. Then, he considers the situation on a rectangular, bounded grid in \(\mathbb{R}^n\) with uniform spacing \(h\) in each dimension and derives an upper bound on the distance. Using an example, the author shows that the derived estimates are sharp when there are no additional constraints on the involved sets. But when the sets are a little more specific, much better estimates are achieved. In such a case, the approximation error converges quadratically with the grid size. Moreover, the author also shows that even super-quadratic convergence rates can be expected if the orientation of the grid is not related to the two considered sets in two-dimensional space.
MSC:
68U05 Computer graphics; computational geometry (digital and algorithmic aspects)
Software:
DLMF; Octave
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