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Abstract:
Medial axis transform (MAT) is very sensitive to the noise,
in the sense that, even if a shape is perturbed only slightly,
the Hausdorff distance between the MATs of the original shape and
the perturbed one may be large. But it turns out that MAT is stable,
if we view this phenomenon with the one-sided Hausdorff distance,
rather than with the two-sided Hausdorff distance. In this paper,
we show that, if the original domain is weakly injective,
which means that the MAT of the domain has no end point which
is the center of an inscribed circle osculating the boundary at
only one point, the one-sided Hausdorff distance of the original
domain's MAT with respect to that of the perturbed one is bounded
linearly with the Hausdorff distance of the perturbation.
We also show by example that the linearity of this bound cannot be
achieved for the domains which are not weakly injective. In particular,
these results apply to the domains with the sharp corners, which
were excluded in the past. One consequence of these results is that
we can clarify theoretically the notion of extracting ``the essential
part of the MAT'', which is the heart of the existing pruning methods.
