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Topic: Hausdorff distance

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	Spatial Similarity Functions
		Intuitively, the Hausdorff distance is a measure of the mismatch between P and Q; if the Hausdorff distance is d, then every point of P is within distance d of some point of Q and vice versa.
		An improvement over Hausdorff distance might be made by incorporating more information about the distribution of points, for example, by computing the average distance between all pairs of points in the two regions.
		Addendum: in general, the Hausdorff distance between any two closed, convex (and bounded and nonempty) sets is achieved on the boundaries of those sets.
	www.alexandria.ucsb.edu /~gjanee/archive/2003/similarity.html (666 words)

	Hausdorff Distance Image Comparison
		The function h(A,B) is called the directed Hausdorff `distance' from A to B (this function is not symmetric and thus is not a true distance).
		Thus the Hausdorff distance, H(A,B), measures the degree of mismatch between two sets, as it reflects the distance of the point of A that is farthest from any point of B and vice versa.
		It is well known that the Hausdorff distance is a metric over the set of all closed, bounded sets - it obeys the properties of identity, symmetry and triangle inequality.
	www.cs.cornell.edu /Info/People/dph/hausdorff/hausdorff.html (830 words)

	PlanetMath: Hausdorff dimension
		Hausdorff dimension is easy to calculate for simple objects like the Sierpinski gasket or a Koch curve.
		Cross-references: metric, infimum, countable, function, distance, metric space, covering, approximation, triangles, Koch curve, Sierpinski gasket, objects, calculate, cover, radius, balls, subset, bounded
		This is version 12 of Hausdorff dimension, born on 2002-05-31, modified 2006-08-02.
	planetmath.org /encyclopedia/HausdorffDimension.html (160 words)

	Hausdorff distance
		The two distances h(A, B) and h(B, A) are sometimes termed as forward and backward Hausdorff distances of A to B. Although the terminology is not stable yet among authors, eq.
		Hausdorff distance gives an interesting measure of their mutual proximity, by indicating the maximal distance between any point of one polygon to the other polygon.
		One of the main application of the Hausdorff distance is image matching, used for instance in image analysis, visual navigation of robots, computer-assisted surgery, etc. Basically, the Hausdorff metric will serve to check if a template image is present in a test image ; the lower the distance value, the best the match.
	cgm.cs.mcgill.ca /~godfried/teaching/cg-projects/98/normand/main.html (1810 words)

	Mesh Approximation Review (Site not responding. Last check: 2007-11-06)
		Definition 1a : The one sided Hausdorff distance between two meshes M1 and M2 is defined H(M1, M2) = max(min(d(a,b))), for 'a' on the surface of M1, b on the surface of M2, and where d(a,b) is the L2 distance from a to b.
		The Hausdorff distance is a good approximation for comparing meshes that are very similar, guaranteeing a specific error bound between two meshes, but as the Hausdorff measure grows larger, it becomes less meaningfull.
		An algorithm for approximating the Hausdorff distance is outlined by Gueziec [].
	www.cs.sfu.ca /~aclement/personal/LitReview/litreview.html (1668 words)

	Hausdorff dimension Summary
		This number, D(A), is called the Hausdorff dimension of A. For many fractals, such as the snowflake curve, the Menger curve, Sierpinski's triangle and carpet, the Hausdorff dimension is equal to the self-similarity dimension (which is also equal to the capacity dimension).
		In mathematics, the Hausdorff dimension is an extended non-negative real number (that is a number in the closed infinite interval [0, ∞]) associated to any metric space.
		Fractals often are spaces whose Hausdorff dimension strictly exceeds the topological dimension.
	www.bookrags.com /Hausdorff_dimension (2287 words)

	CS507 Project - Computing the Fréchet distance between two polygonal curves
		The reason of this discrepancy is that the Hausdorff distance only takes into account the sets of points on both curves and does not reflect the course of the curves.
		The Fréchet distance can be illustrated as follows : suppose a man is walking his dog and that he is constrained to walk on a curve and his dog on another curve.
		As the distance between the two green points on the curves is smaller than ε, the corresponding green point in the unit square belongs to the free space.
	www.cim.mcgill.ca /~stephane/cs507/Project.html (1857 words)

	Hausdorff distance - Wikipedia, the free encyclopedia
		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.
		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).
	en.wikipedia.org /wiki/Hausdorff_distance (348 words)

	[No title]
		The Hausdorff metric or distance is used in the study of fractals to determine whether two sets (which may be quite complicated) look the same.
		To understand the Hausdorff distance between two curves, you have to define what you mean by the distance between the curves.
		Similarly, the distance between curve B and curve A is the greatest distance among all of the distances between a point B to the curve A. At first, this may seem redundant, but the following example should explain that the definition is not symmetric:
	www.rose-hulman.edu /Class/CalculusProbs/Problems/ARCH/ARCH_5_6_0.html (946 words)

	Objects distant and near
		You may or may not remember that the most customary notion of the distance is somehow related to the Pythagorean Theorem.
		Hausdorff distance gives us a measure of nearness between sets.
		Distance, the metric function, is the mathematical abstraction of measurement.
	www.cut-the-knot.org /do_you_know/far_near.shtml (785 words)

	BioMed Central \| Full text \| Structural similarity of loops in protein families: toward the understanding of protein ...
		Hausdorff distance can be calculated under the assumption that the Cα atoms for both structures are in a common coordinate frame which is defined by the structural alignment between two domains.
		The cluster analysis was done using the complete linkage clustering [20] where the distance between two clusters was measured as a maximum distance between a point in one cluster and a point in another cluster.
		In contrast to the RMSD cluster tree, distance matrices based on the loop (dis)similarity measure correctly cluster the representatives of the five major groups of the Ribonuclease family as per Rosenberg et al [23].
	www.biomedcentral.com /1471-2148/5/10 (3431 words)

	[No title]
		The hausdorff method is theoretically better for this application because it tolerates differences in the bitmap shapes near the foreground/background boundary.
		Each of those distances is called the directed hausdorff distance from one image to another, and the hausdorff distance is the maximum of the directed distances.
		It is relatively fast because for the hausdorff distance it uses a morphologically based matching algorithm, and for both correlation and hausdorff it does the measurement once where the centroids are aligned.
	www.leptonica.com /jbig2.html (2539 words)

	Hausdorff-Based Matching (Site not responding. Last check: 2007-11-06)
		The generalized Hausdorff measure provides a means of determining the resemblance of one point set to another, by examining the fraction of points in one set that lie near points in the other set (and perhaps vice versa).
		Hausdorff-based distance measures differ from correspondence-based matching techniques, such as point matching methods and binary correlation, because there is no pairing of points in the two sets being compared.
		A C implementation of Hausdorff matching (for matching with translation or with translation and scaling) is available as a tar file via ftp.
	www.cs.cornell.edu /vision/hausdorff/hausmatch.html (410 words)

	[No title] (Site not responding. Last check: 2007-11-06)
		Subject: Re: Hausdorff distance and fractal dimension Date: Mon, 17 Jan 2000 12:09:41 +0100 Newsgroups: sci.math.research Summary: [missing] I seriously doubt this as there are nowhere differentiable functions arbitrarily close to analytic ones; think of Weierstrass' example.
		The Hausdorff distance between the point and the ball is e.
		Subject: Re: Hausdorff distance and fractal dimension Date: Mon, 17 Jan 2000 14:04:02 +0000 (GMT) Newsgroups: sci.math.research Adam Bryant wrote: : Let A and B be compact subsets of Euclidean space.
	www.math.niu.edu /~rusin/known-math/00_incoming/haus_dim (747 words)

	[No title]
		It is relatively fast because for the hausdorff distance it uses a morphologically based matching algorithm, and for both correlation and hausdorff it does the measurement where the centroids are aligned.
		Given a distance function and a threshold, each image is sequentially placed either into an existing class or starts a new class, depending on the distance of that image from the representatives of each existing class.
		This distance is related to the radius of the resulting clusters, which is related to the maximum number of clusters that will be found in the RGB space.
	www.leptonica.com /applications.html (9263 words)

	Hausdorff Distance
		Felix Hausdorff (1868 -1942) devised a metric function between subsets of a metric space.
		two sets are within Hausdorff distance r from each other iff any point of one set is within distance r from some point of the other set.
		The Hausdorff metric h(A,B) is defined in terms of the neighborhoods.
	www.cut-the-knot.org /do_you_know/Hausdorff.shtml (687 words)

	Springer Online Reference Works (Site not responding. Last check: 2007-11-06)
		Hausdorff in 1914 (see [1]); one of his most important results is as follows: The space of closed subsets of a compact set is also compact (P.S. Urysohn arrived independently at this theorem in 1921–1922, see [2]).
		Generally, the Hausdorff metric is defined on the space of bounded closed sets of a metric space
		The Hausdorff metric topology and the exponential topology (see also Hyperspace) then coincide on the space
	eom.springer.de /h/h046710.htm (143 words)

	Hausdorff distance - Slider
		Hausdorff distance measures how far two compact subsets of a metric space are from each other.
		Hausdorff distance can be defined the same way for closed non-compact subsets of M, but in this case the distance may take infinite value and the topology of F(M) starts to depend on particular metric on M (not only on its topology).
		The Hausdorff distance between not closed subsets can be defined as the Hausdorff distance between its closures.
	enc.slider.com /Enc/Hausdorff_distance (230 words)

	Axel Mosig
		This motivates the introduction of relational distance measures, examples of which are the Hausdorff- and the bottleneck distance.
		These results generalize to a whole class of non-relational distance measures that evolve from relational distance measures by exchanging a certain norm that is part of many relational distance measures.
		Furthermore, we show that the discrete Frechet distance is related to the Dynamic Time Warping Distance (studied in the context of speech processing, computer vision and handwriting recognition) in the same way as the root-mean-square Hausdorff distance is related to the Hausdorff distance.
	www.bioinf.uni-leipzig.de /~axel/abs_phd.html (808 words)

	Hausdorff - Wikipedia, the free encyclopedia
		Felix Hausdorff, the German mathematician that Hausdorff spaces are named after.
		Hausdorff dimension, a measure theoretic concept of dimension.
		The Hausdorff distance, or Hausdorff metric, which measures how far two compact non-empty subsets of a metric space are from each other.
	en.wikipedia.org /wiki/Hausdorff (125 words)

	Perceptually Motivated Measures for Capturing Proximity of Web Page Elements: Towards Automated Evaluation of Web Page ...
		For example, the point may be the centroid of the element and the centroid-to-centroid distance be taken as a measure of the proximity between the two elements.
		While the Hausdorff distance captures the proximity well in many situations, it is dependent on the size of the elements - two large elements which are adjacent to each other have a large Hausdorff distance.
		The fundamental reason for the disconnect between the visual notion of proximity and the mathematical notion of distance is that the human visual system is highly non-linear and the notion of proximity is dependent (rather than independent) on the size and geometry of the elements.
	www2002.org /CDROM/alternate/688 (3206 words)

	Hausdorff Distance under Translation for Points, Disks, and Balls (Site not responding. Last check: 2007-11-06)
		Hausdorff Distance under Translation for Points, Disks, and Balls
		We measure similarity between A and B by computing the minimum Hausdorff distance between A+t and B, where the minimum is taken either over all vectors t in R
		Since Hausdorff distance is sensitive to out-liers, we also propose efficient approximation algorithms for computing the minimum root-mean-square(rms) Hausdorff distance, under translation, between two point sets.
	valis.cs.uiuc.edu /~sariel/research/papers/02/hausdorff (177 words)

	CS20c Progress Report
		First of all the partial one-directional Hausdorff distance between two sets A and B is the maximum over each element a of A of the minimum over each element b of B of the distance from a to b.
		More simply, take the first element a of A and find the minimum distance to the set B. Do that again for the second element of A, and so on until you've done it for every element of A, and the Hausdorff distance is defined as the maximum of each of these minimum distances.
		In our algorithm the partial one-directional Hausdorff distance is used between an edge in one image and all the edges in the other images to measure their similarity.
	www.cs.caltech.edu /~nowell/ProgressReport.html (658 words)

	IP-01724-2005: A Generalized Algorithm for Different types of Distance Transformations in Graylevel Rock Images (Site not responding. Last check: 2007-11-06)
		A distance transform of a graylevel binary rock image specifies the distance from each pixel to the nearest non-zero pixel.
		Distance transforms play a central role in the comparison of graylevel binary rock images, particularly for the images resulting from local feature detection techniques such as edge or corner detection.
		This algorithm may be used to compute the Euclidean distance, cityblock distance, chessboard distance, chamfer distance, octagon distance & linear distance and also calculate the Hausdorff distance.
	www.ewh.ieee.org /soc/sps/tip/abstracts_all/abst_ip01724-2005.html (353 words)

	Real-Time Hausdorff-Fraction Motion Tracking on a Clustered Workstation (Site not responding. Last check: 2007-11-06)
		Given a model transformation, the measure of similarity between the model and the scene is the Hausdorff Distance between the scene (or a portion of it) and the model.
		The Hausdorff distance is actually defined for two point sets representing the portion of the scene S and the model M respectively.
		In practice, it is more straightforward to fix the Hausdorff distance to some small value tau and use this value as the radius of dilation for the point sets S and M*.
	www.eecs.harvard.edu /~mdw/proj/old/rt-hausdorff/rt-hausdorff.html (4220 words)

	[CG UIUC] Hausdorff distance under translation for points, disks, and balls (Site not responding. Last check: 2007-11-06)
		Firday, 9/27/02 14:00-14:30 DCL 2501 Title: Hausdorff distance under translation for points, disks, and balls.
		We measure similarity between $\A$ and $\B$ by computing the minimum Hausdorff distance between $\A+t$ and $\B$, where the minimum is taken either over all vectors $t \in \reals^d$ or over the vectors $t$ such that $\A+t$ and $\B$ do not intersect.
		Since Hausdorff distance is sensitive to out-liers, we also propose efficient approximation algorithms for computing the minimum summed Hausdorff distance, under translation, between two point sets.
	valis.cs.uiuc.edu /pipermail/cg/2002-September/000018.html (186 words)

	Comparing Images Using the Hausdorff Distance - Huttenlocher, Klauderman, Rucklidge (ResearchIndex) (Site not responding. Last check: 2007-11-06)
		Abstract: The Hausdorff distance measures the extent to which each point of a `model' set lies near some point of an `image' set and vice versa.
		Thus this distance can be used to determine the degree of resemblance between two objects that are superimposed on one another.
		In this paper we provide efficient algorithms for computing the Hausdorff distance between all possible relative positions of a binary image and a model.
	citeseer.ist.psu.edu /huttenlocher93comparing.html (548 words)

	The Strange World of the Hausdorff Metric Geometry
		Other applications of the Hausdorff metric can be found in [8, 18], 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(
	faculty.gvsu.edu /schlicks/HausdorffGeometry/H3.htm (191 words)

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