Hausdorff distance and image processing

Mathematical methods for image processing make use of function spaces which are usually Banach spaces with integral $L_p$ norms. The corresponding mathematical models of the images are functions in these spaces. There are discussions here involving the value of $p$ for which the distance between two functions is most natural when they represent images, or the metric in which our eyes measure the distance between the images. In this paper we argue that the Hausdorff distance is more natural to measure the distance (difference) between images than any $L_p$ norm.