Hausdorff distance
From Wikipedia, the free encyclopedia
The Hausdorff distance, or Hausdorff metric, measures how far two compact non-empty subsets of a metric space are from each other. It is named after Felix Hausdorff.
Informally, the Hausdorff distance between two sets of points, A and B, is the longest distance you would have take from an arbitrary point in A to the corresponding nearest point in B (choose a point in A, and the distance to the closest point in B is equal to or less than the Hausdorff distance).
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[edit] Definitions
Let X and Y be two compact subsets of a metric space M. The Hausdorff distance dH(X,Y) is the minimal number r such that the closed r-neighborhood of X contains Y and the closed r-neighborhood of Y contains X. In other words, if d(x, y) denotes the distance in M, then
- <math> d_{\mathrm H}(X,Y) = \max\{\,\sup_{x \in X} \inf_{y \in Y} d(x,y),\, \sup_{y \in Y} \inf_{x \in X} d(x,y)\,\}\mbox{.} \! </math>
This distance function turns the set of all non-empty compact subsets of M into a metric space, say F(M). The topology of F(M) depends only on the topology of M. If M is compact, then so is F(M). If M is complete, then so is F(M).
Hausdorff distance can be defined in essentially the same way for non-empty closed and bounded subsets of M, but taking an infimum over r instead of a minimum. (In a general metric space, closed and bounded subsets are more general than compact subsets. For instance in the space of continuous real-valued functions on [0,1] metrized with the sup-norm, the closed unit ball is closed and bounded but not compact.) If we allow closed unbounded subsets then the distance may take infinite values. When working with non-compact subsets, the topology of F(M) depends on the metric on M and not only on its topology. The Hausdorff distance between general subsets can be defined as the Hausdorff distance between their closures. It gives a pre-metric (or pseudometric) on the set of all subsets of M (the Hausdorff distance between any two sets with the same closure is zero).
In Euclidean geometry, one often uses an analog, Hausdorff distance up to isometry. Namely, let X and Y be two compact figures in a Euclidean space; then DH(X,Y) is the minimum of dH(I(X),Y) along all isometries I of Euclidean space. This distance measures how far X and Y are from being isometric.
[edit] Applications
In computer vision, the Hausdorff distance can be used to find a given template in an arbitrary target image. The template and image are often pre-processed via an edge detector giving a binary image. Next, each 1 (activated) point in the binary image of the template is treated as a point in a set, the "shape" of the template. Similarly, an area of the binary target image is treated as a set of points. The algorithm then tries to minimize the Hausdorff distance between the template and some area of the target image. The area in the target image with the minimal Hausdorff distance to the template, can be considered the best candidate for locating the template in the target.[1]
[edit] See also
[edit] References
- Munkres, James; Topology, Prentice Hall; 2nd edition (December 28, 1999). ISBN 0131816292.
[edit] External links
- http://planetmath.org/encyclopedia/HausdorffMetric.html
- Completeness and Total Boundedness of the Hausdorff Metric (pdf)
- http://cgm.cs.mcgill.ca/~godfried/teaching/cg-projects/98/normand/main.htmlbg:Хаусдорфова мярка
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