The Strange World of the Hausdorff Metric Geometry

III. Why the Hausdorff Metric?

The Hausdorff metric has many applications. For example, a recent proposal [24] approved by the United States Army Research Office (ARO) Small Business Technology Transfer (STTR) Program has as its objective to “Develop software for terrain representation by smooth (C1-smooth or more) piecewise polynomials on irregular triangulations using nontraditional metrics.” The proposal explains that terrain is often represented by “piecewise planar surfaces on triangulated irregular networks (TINs)”. This method is used because of hardware constraints and because these models do not exhibit excessive oscillations. One problem with this method is that TINs create unrealistic flat surfaces within the triangles and non-smooth transitions or sharp discontinuities when moving from one triangle to another. It requires a large number of triangles (and a large amount of data) to minimize these problems. Other methods, for example smooth-surface techniques such as conventional polynomial and rational splines, radial basis functions, and wavelets, “require too much data, too much computing time, too much human interaction and/or do not preserve shape well.” The proposal states that nontraditional metrics, including the Hausdorff metric, “can increase the accuracy and/or efficiency of both piecewise planar surfaces and smooth piecewise polynomials on irregular triangulations.” The United States military has also used the Hausdorff distance in target recognition procedures [18].

Among the many other areas that have found applications of the Hausdorff metric are

· image matching and visual recognition by robots [9, 10] and the Computer Vision Group at Cornell University http://www.cs.cornell.edu/vision/hausdorff/hausmatch.html.

· medicine [13]

· image analysis [22, 23]

· astronomy [19]

· roots of polynomials [21]

· graph theory [4].

Other applications of the Hausdorff metric can be found in [8, 20], and the metric also plays an important role in fractal geometry [1]. In most applications the Hausdorff distance is used to compare what is seen with pre-programmed or recognized patterns -- the smaller the distance the better the match. In the remainder of this paper we investigate the geometry the Hausdorff metric imposes on the space H(Rⁿ).