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The Hausdorff distance is a measure of (dis-)similarity between two setswhich is widely used in various applications. Most of the applied literature isdevoted to the computation for sets consisting of a finite number of points.This has applications, for instance, in image processing. However, we wouldlike to apply the Hausdorff distance to control and evaluate optimisationmethods in level-set based shape optimisation. In this context, the involvedsets are not finite point sets but characterised by level-set or signeddistance functions. This paper discusses the computation of the Hausdorffdistance between two such sets. We recall fundamental properties of theHausdorff distance, including a characterisation in terms of distancefunctions. In numerical applications, this result gives at least an exact lowerbound on the Hausdorff distance. We also derive an upper bound, andconsequently a precise error estimate. By giving an example, we show that ourerror estimate cannot be further improved for a general situation. On the otherhand, we also show that much better accuracy can be expected fornon-pathological situations that are more likely to occur in practice. Theresulting error estimate can be improved even further if one assumes that thegrid is rotated randomly with respect to the involved sets.

Computing the Hausdorff Distance of Two Sets from Their Signed Distance Functions