Computing the Hausdorff Distance of Two Sets from Their Distance Functions

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.

Operations on Signed Distance Functions

We present a theoretical overview of signed distance functions and analyze how this representation changes when applying an offset transformation. First, we analyze the properties of signed distance and the sets they describe. Second, we introduce our main theorem regarding the distance to an offset set in (X,||.||) strictly normed Banach spaces. An offset set of D in X is the set of points equidistant to D. We show when such a set can be represented by f(x)-c=0, where c denotes the radius of the offset. Finally, we explain these results for applications that offset signed distance functions.