  We present an algorithm to compute the geometric median of shapes which is
based on the extension of median to high dimensions. The median finding problem
is formulated as an optimization over distances and it is solved directly using
the watershed method as an optimizer. We show that computing the geometric
median of shapes is robust in the presence of outliers and it is superior to
the mean shape which can easily be affected by the presence of outliers. The
geometric median shape thus faithfully represents the true central tendency of
the data, contaminated or not. Our approach can be applied to manifold and non
manifold shapes, with connected or disconnected shapes. The application of
distance transforms and watershed algorithm, two well established constructs of
image processing, lead to an algorithm that can be quickly implemented to
generate fast solutions with linear storage requirements. We demonstrate our
methods in synthetic and natural shapes and compare median and mean results
under increasing contamination by strong outliers.
