Hausdorff metrics are used in geometric settings for measuring the distance between sets of points. They have been used extensively in areas such as computer vision, pattern recognition and computational chemistry. While computing the distance between a single pair of sets under the Hausdorff metric has been well studied, no results were known for the Nearest Neighbor problem under Hausdorff me...

An ultrafilter U is Hausdorff if for any two functions f, g ∈ ω ω , f (U) = g(U) iff f ↾X = g↾X for some X ∈ U. We will show that it is consistent that there are no Hausdorff ultrafilters.

A method is proposed for detecting whether two CCITT group 4 images were scanned from the same document. Features are extracted from rectangular patches of text and compared with a modified Hausdorff distance measure. Two images are said to be ‘‘equivalent’’ (i.e., they were scanned from the same document) if the Hausdorff measure finds that a specified number of features are located within a g...

For a closed subset K of a compact metric space A possessing an α-regular measure μ with μ(K) > 0, we prove that whenever s > α, any sequence of weighted minimal Riesz s-energy configurations ωN = {x i,N} N i=1 on K (for ‘nice’ weights) is quasi-uniform in the sense that the ratios of its mesh norm to separation distance remain bounded as N grows large. Furthermore, if K is an α-rectifiable com...

We prove that the 1-dimensional Hausdorff measure restricted to a simple real analytic curve γ : R → R , N ≥ 2, is locally 1-monotone.

Using the Chu-construction, we define a group algebra for topological Hausdorff groups. Furthermore, for isometric, weakly continuous representations of a subgroup H of a Hausdorff group G induced representations are constructed.

A topological space (X, T) with property R is maximal R (minimal R) if T is a maximal (minimal) element in the set R(X) of all topologies on the set X having property R with the partial ordering of set inclusions. The properties of maximal topologies for compactness, countable compactness, sequential compactness, Bolzano-Weierstrass compactness, and Lindelöf are investigated and the relations b...

This paper proposes a new recognition algorithm of mandatory traffic sings using the Hausdorff distance. This algorithm has been designed to detect arrows on traffic signs, especially direction signs. The arrows on these mandatory signs appear in a multitude of different forms and positions. Due to this variety this algorithm uses a structural approach to recognize the arrows. First the sign is...

Template matching face detection systems are used very often as a previous step in several biometric applications. These biometric applications, like face recognition or video surveillance systems, need the face detection step to be efficient and robust enough to achieve better results. One of many template matching face detection methods uses Hausdorff distance in order to search the part of t...