Hyperbolic Hausdorff Distance for Medial Axis Transform

Although the Hausdorff distance is a popular device to measure the differences between sets, it is not natural for some specific classes of sets, especially for the medial axis transform which is defined as the set of all pairs of the centers and the radii of the maximal balls contained in another set. In spite of its many advantages and possible applications, the medial axis transform has one great weakness, namely its instability under the Hausdorff distance when the boundary of the original set is perturbed. Though many attempts have been made for the resolution of this phenomenon, most of them are heuristic in nature and lack precise error analysis. In this paper, we show that this instability can be remedied by introducing a new metric called the hyperbolic Hausdorff distance, which is most natural for measuring the differences between medial axis transforms. Using the hyperbolic Hausdorff distance, we obtain error bounds, which make the operation of medial axis transform almost an isometry. By various examples, we also show that the bounds obtained are sharp. In doing so, we show that bounding both the Hausdorff distance between domains and the Hausdorff distance between their boundaries is necessary and sufficient for bounding the hyperbolic Hausdorff distance between their medial axis transforms. These results drastically improve the previous results and open a new way to practically control the Hausdorff distance error of the domains under its medial axis transform error, and vice versa.