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#### One-sided Stability of Medial Axis Transform

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##### Citation

Choi, S. W., & Seidel, H.-P. (2001). One-sided Stability of Medial Axis Transform.
In B. Radig, & S. Florczyk (*Proceedings of Pattern
Recognition, 23rd DAGM Symposium* (pp. 132-139). Vienna, Austria: Springer.

Cite as: http://hdl.handle.net/11858/00-001M-0000-000F-32B6-A

##### 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.