H. Alt and M. Godau. Computing the Frechet distance between two polygonal curves. Internat. J. Comput. Geom. Appl., 5:75-91, 1995.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....on shapes, see Rucklidge [13] If to be used in pattern matching, a pattern metric should be robust for the e ects that occur with discretisation, see Fig. 1(a) Many pattern matching algorithms use a metric on simple closed curves. An important example is the Fr echet distance, see Alt and Godau [3]. Other pattern metrics for boundary curves are based on turning angle, see Cohen and Guibas [5] or normalised ane arc length, see Huttenlocher and Kedem [10] The Hausdor metric is de ned on the collection of all (non empty) closed, bounded subsets of a metric space. Some algorithms are based ....

H. Alt and M. Godeau. Computing the Frechet distance between two polygonal curves. Int. J. of Computational Geometry & Applications, pages 75-91, 1995.

....group. In practical applications it is desirable that a metric is robust for various defects caused by discretisation and unreliable feature detection. Many shape recognition algorithms use a metric on simple closed curves. An important example is the Fr echet distance, see Alt and Godau [3]. Other pattern metrics for curves are based on turning angle, see Cohen and Guibas [5] or normalised ane arc length, see Huttenlocher and Kedem [11] The Hausdor metric is de ned on the collection of all (non empty) closed, bounded subsets of a metric space. Some algorithms are based on this ....

H. Alt and M. Godeau. Computing the Frechet distance between two polygonal curves. Int. J. of Computational Geometry & Applications, pages 75-91, 1995.

....that (0) 0) 0, and (1) 1) 1. The Fr echet distance is the minimum over all monotone increasing parameterizations (t) and (t) of the maximal distance d(A( t) B( t) t 2 [0; 1] see gure 8. The special case of computing the Fr echet distance for polylines is considered in [4]. A variation of the Fr echet distance is obtained by dropping the monotonicity condition of the H F Figure 8: Hausdor (H) and Fr echet (F) distance between two curves. 8 parameterization. The resulting Fr echet distance d(A; B) is a semimetric: zero distance need not mean that the objects ....

....contours are curves where the starting point and ending point are the same. However, the starting and ending point could as well lie somewhere else on the contour, without changing the shape of the contour curve. For convex contours, the Fr echet distance is equal to the Hausdor distance [4]. 3.10 Nonlinear elastic matching distance Let A = fa 1 ; am g and B = fb 1 ; b n g be two nite sets of ordered contour points, and let f be a correspondence between all points in A and all points in B such that there are no a 1 a 2 , with f(a 1 ) f(a 2 ) The stretch s(a i ....

Helmut Alt and Michael Godeau. Computing the Frechet distance between two polygonal curves. International Journal of Computational Geometry & Applications, pages 75-91, 1995.

....the Hausdor metric is only invariant for isometries. The partial Hausdor distance is a non metric variant of the Hausdor metric that is more robust for outliers and noise, see [12, 7] It is based on a parameter that estimates the amount of outliers. The Fr echet distance, see Alt and Godau [3], is a pattern metric de ned on simple closed curves. Other resemblance measures for simple curves are based on turning angle, see Cohen and Guibas [5] and normalised ane arc length, see Huttenlocher and Kedem [10] For solid patterns, i.e. patterns equal to the closure of some open set, suitable ....

H. Alt and M. Godeau. Computing the Frechet distance between two polygonal curves. Int. J. of Computational Geometry & Applications, pages 75-91, 1995.

....let their parameterizations and be continuous functions of the same parameter t 2 [0; 1] such that (0) 0) 0, and (1) 1) 1. The Fr echet distance is the minimum over all monotone increasing parameterizations (t) and (t) of the maximal distance d(A( t) B( t) t 2 [0; 1] AG95] considers the computation of the Fr echet distance for the special case of polylines. Deciding whether the Fr echet distance is smaller than a given constant, can be done in time O(mn) Based on this result, and the parametric search technique, it is derived that the computation of the ....

....changing the shape of the contour curve. Deciding whether the Fr echet distance of two contours is smaller than , irrespective the starting point, can done in time O(mn log(mn) The corresponding computation problem, computing the Fr echet distance, can be solved in time O(mn(log(mn) 2 ) AG95] For convex contours curves, the Fr echet distance is equal to the Hausdor distance, which can be computed in time O(mn log(mn) ABGW90] 6.3 Hausdor distance Given two polygons A and B, the directed Hausdor distance from A to B can be computed using the Voronoi diagram of B, which assigns ....

Helmut Alt and Michael Godeau. Computing the Frechet distance between two polygonal curves. International Journal of Computational Geometry & Applications, pages 75-91, 1995.

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H. Alt and M. Godau. Computing the Frechet distance between two polygonal curves. Internat. J. Comput. Geom. Appl., 5:75-91, 1995.

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H. Alt and M. Godau, Computing the Frechet distance between two polygonal curves, Int. J. Comput. Geom. & Appl. 5 (1995), 75-91.

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H. Alt and M. Godau. Computing the Frechet distance between two polygonal curves. Internat. J. Comput. Geom. Appl. 5 (1995), 75-91.

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H. Alt and M. Godau. Computing the Frechet distance between two polygonal curves. Internat. J. Comput. Geom. Appl. 5 (1995), 75-91.

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H. Alt and M. Godau. Computing the Frechet distance between two polygonal curves. Internat. J. Comput. Geom. Appl., 5:75-91, 1995.

....contrast to other common distance measures like the Hausdor distance, the Fr echet distance respects the one dimensional structure of the curves and doesn t just treat them as a point set. The study of the Fr echet distance from a computational point of view has been initiated by Alt and Godau [1]. The decision problem is the problem to decide, for a given , whether the Fr echet distance between two curves is at most . The optimization problem is the problem to nd the optimal , i.e. to compute the Fr echet distance. Alt and Godau [1] treated the case of two polygonal curves. For two ....

....point of view has been initiated by Alt and Godau [1] The decision problem is the problem to decide, for a given , whether the Fr echet distance between two curves is at most . The optimization problem is the problem to nd the optimal , i.e. to compute the Fr echet distance. Alt and Godau [1] treated the case of two polygonal curves. For two curves of m and n pieces, respectively, they showed how to solve the decision problem in O(mn) time and the optimization problem in O(mn log(mn) time. Some related problems have also been considered, like minimizing the Fr echet distance under ....

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H. Alt and M. Godau, Computing the Frechet distance between two polygonal curves, Internat. J. Comput. Geom. Appl. 5 (1995), 75-91.

...., such that P is completely contained in the neighborhood of Q, and vice versa. It is somehow the most straightforward and natural distance measure between curves or other compact sets. But since it does not consider the course of curves there are examples where it is not appropriate (see [4]) The Fr echet distance deals with this problem. As a popular illustration of it imagine a man is walking his dog, he is walking on one curve, the dog on the other. Both are allowed to control their speed but are not allowed to go backwards. Then the Fr echet distance of the curves is the minimal ....

....Figure 2: Two curves with small weak Fr echet distance having a large Fr echet distance. For given polygonal curves P; Q with n and m vertices, respectively, one can compute H (P; Q) in O (m n) log(m n) time, see [2] and F (P; Q) as well as F (P; Q) in O(mn log(m n) time, see [4]. The following result from [3] see also [8] shows that for certain classes of curves the three distance measures are closely related, so we can do better than mn) when we want to compute the Fr echet distance. Theorem 1. For any pair of convex closed curves P and Q, H (P; Q) F (P; Q) ....

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H. Alt and M. Godau. Computing the Frechet distance between two polygonal curves. Internat. J. Comput. Geom. Appl., 5:75-91, 1995.

....166 backwards. Then the Fr echet distance of the curves is the minimal length of a leash that is necessary. For given polygonal curves P; Q with m and n vertices, respectively, one can compute H (P; Q) in O (m n) log(m n) time, see [1] and F (P; Q) in O(mn log(m n) time, see [3]. Here, a polygonal curve is a curve P : 0; n] R 2 with n 2 N, such that for all i 2 f0; 1; n 1g each P i : P j [i;i 1] is ane, i.e. P (i ) 1 )P (i) P (i 1) for all 2 [0; 1] The following result from [2] see also [4] shows that for certain classes of curves the two ....

....increasing chords of [5] where 2 =3. 2 The upper bound Throughout the rest of the paper let P : 0; m] R 2 and Q : 0; n] R 2 be polygonal curves. Unless stated otherwise 0 is a xed real parameter. In the sequel we will use the notion of a free space which was introduced in [3]: De nition (Free space, 3] The set F (P; Q) f(s; t) 2 [0; m] 0; n] j jjP (s) Q(t)jj g, or F for short, denotes the free space of P and Q. Sometimes we refer to [0; m] 0; n] as the free space diagram; the feasible points p 2 F will be called white and the infeasible points ....

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H. Alt and M. Godau. Computing the Frechet distance between two polygonal curves. Internat. J. Comput. Geom. Appl., 5:75-91, 1995.

....are also di erent kinds of transformations that are allowed to match them, see [5] for a survey. We will focus here on the Fr echet distance F for polygonal curves, and we will search for a translation which, when applied to the rst curve, minimizes the Fr echet distance to the second one. In [4] it is shown how to compute the Fr echet distance for two polygonal curves. The only algorithm we know of that decides whether there is a transformation that, when applied to the rst curve, results in a Fr echet distance less or equal than some given parameter (this is called the decision ....

....the Deutsche Forschungsgemeinschaft under Grant No. AL 253 4 3. in a xed direction. But to our knowledge there is no algorithm which actually computes the Fr echet distance under a non trivial class of transformations 1 . In the following we will adopt some basic de nitions and results from [4] on which we will subsequently build up. De nition 1 (Polygonal curve) A continuous mapping f : a; b] R 2 with a; b 2 R and a b is called a curve. A polygonal curve is a curve P : 0; n] R 2 with n 2 N, such that for all i 2 f0; 1; n 1g each P i : P j [i;i 1] is ane, i.e. P (i ....

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H. Alt and M. Godau. Computing the Frechet distance between two polygonal curves. Internat. J. Comput. Geom. Appl., 5:75-91, 1995.

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H. Alt and M. Godau. Computing the Frechet distance between two polygonal curves. ############# ## ## ############# ######## ### ############, 5:75-91, 1995.

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H. Alt and M. Godau. Computing the Frechet distance between two polygonal curves. Internat. J. Comput. Geom. Appl., 5:75-91, 1995.

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