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The Hausdorff Distance
The basis of our methodology is the median Hausdorff distance which is a
similarity measure between two arbitrary point sets. The classical Hausdorff
distance between two (finite) sets of points,
and ,
is defined as

(1) 
Here, ,
the directed distance from
to ,
will be small when every point of
is near some point of .
This distance is too fragile for practical tasks:
for example, a single point in
that is far from anything in
will cause
to be large. A natural way to take care of this problem is to
replace equation 1 with

(2) 
where
denotes the th quantile value of over the set ,
for some value of
between zero and one.
When
we get the modified median Hausdorff distance
which we use in our method.
Let
be a point set representing the reference shape,
the measured point
set. As already mentioned,
we assume that
is complete, dense and precise, while
may be incomplete,
sparse and noisy.
is obtained either analytically, or as the
result of a careful, highresolution, offline measurement.
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Dmitry Chetverikov
19981116