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

## TL;DR

This paper presents a method to compute the Hausdorff distance between sets characterized by signed distance functions, providing bounds and error estimates useful for shape optimization and image analysis.

## Contribution

It introduces a novel approach to calculate the Hausdorff distance for level-set based sets, including bounds and error estimates, extending applications beyond finite point sets.

## Key findings

- Provides an exact lower bound for the Hausdorff distance using signed distance functions.
- Derives an upper bound and a precise error estimate for the computation.
- Shows that practical scenarios can achieve higher accuracy than the worst-case bounds.

## Abstract

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.