  The Hausdorff distance is a measure of (dis-)similarity between two sets
which is widely used in various applications. Most of the applied literature is
devoted to the computation for sets consisting of a finite number of points.
This has applications, for instance, in image processing. However, we would
like to apply the Hausdorff distance to control and evaluate optimisation
methods in level-set based shape optimisation. In this context, the involved
sets are not finite point sets but characterised by level-set or signed
distance functions. This paper discusses the computation of the Hausdorff
distance between two such sets. We recall fundamental properties of the
Hausdorff distance, including a characterisation in terms of distance
functions. In numerical applications, this result gives at least an exact lower
bound on the Hausdorff distance. We also derive an upper bound, and
consequently a precise error estimate. By giving an example, we show that our
error estimate cannot be further improved for a general situation. On the other
hand, we also show that much better accuracy can be expected for
non-pathological situations that are more likely to occur in practice. The
resulting error estimate can be improved even further if one assumes that the
grid is rotated randomly with respect to the involved sets.
