A Middle Curve Based on Discrete Fréchet Distance (original) (raw)

Middle curves based on discrete Fréchet distance

Computational Geometry, 2020

Given a set of polygonal curves, we present algorithms for computing a middle curve that serves as a representative for the entire set of curves. We require that the middle curve consists of vertices of the input curves and that it minimizes the maximum discrete Fréchet distance to all input curves. We consider three dierent variants of a middle curve depending on in which order vertices of the input curves may occur on the middle curve, and provide algorithms for computing each variant.

Approximately matching polygonal curves with respect to the Fréchet distance

Computational Geometry, 2005

Matching Polygonal Curves with Respect to the Fréchet Distance

Lecture Notes in Computer Science, 2001

In this paper we present approximate algorithms for matching two polygonal curves with respect to the Fréchet distance. We define a discrete version of the Fréchet distance as a distance measure between polygonal curves and show that this discrete version is bounded by the continuous version of the Fréchet distance. For the task of matching with respect to the discrete Fréchet distance, we develop an algorithm that is based on intersecting certain subsets of the transformation group under consideration. Our algorithm for matching two point sequences of lengths m and n under the group of rigid motions has a time complexity of O(m 2 n 2) for a certain approximation. Based on the same idea of intersecting subsets of the transformation group, we propose algorithms for matching subcurves and closed polygonal curves, as well as for finding longest common subcurves.

Exact Algorithms for Partial Curve Matching via the Fréchet Distance

Proceedings of the Twentieth Annual ACM-SIAM Symposium on Discrete Algorithms, 2009

Curve matching is a fundamental problem that occurs in many applications. In this paper, we study the problem of measuring partial similarity between curves. Specifically, given two curves, we wish to maximize the total length of subcurves that are close to each other, where closeness is measured by the Fréchet distance, a common distance measure for curves. The resulting maximal length is called the partial Fréchet similarity between the two input curves.

Exact algorithms for partial curve matching via the Fr�chet distance

Soda, 2009

Computational-geometric methods for polygonal approximations of a curve

1986

In cartography, computer graphics, pattern recognition, etc., we often encounter the problem of approximating a given finer piecewise linear curve by another coarser piecewise linear curve consisting of fewer line segments. In connection with this problem, a number of papers have been published, but it seems that the problem itself has not been well modelled from the standpoint of specific applications, nor has a nice algorithm, nice from the computational-geometric viewpoint, been proposed. In the present paper, we first consider (i) the problem of approximating a piecewise linear curve by another whose vertices are a subset of the vertices of the former, and show that an optimum solution of this problem can be found in a polynomial time. We also mention recent results on related problems by several researchers including the authors themselves. We then pose (ii) a problem of covering a sequence of n points by a minimum number of rectangles with a given width, and present an O(n log n)-time algorithm by making use of some fundamental established techniques in computational geometry. Furthermore, an O(mn(log n)2)-time algorithm is presented for finding the minimum width w such that a sequence of n points can be covered by at most m rectangles with width w. Finally, (iii) several related problems are discussed. 9

Computational Geometry, 2014

The Fréchet distance is a well studied and commonly used measure to capture the similarity of polygonal curves. Unfortunately, it exhibits a high sensitivity to the presence of outliers. Since the presence of outliers is a frequently occurring phenomenon in practice, a robust variant of Fréchet distance is required which absorbs outliers. We study such a variant here. In this modified variant, our objective is to minimize the length of subcurves of two polygonal curves that need to be ignored (MinEx problem), or alternately, maximize the length of subcurves that are preserved (MaxIn problem), to achieve a given Fréchet distance. An exact solution to one problem would imply an exact solution to the other problem. However, we show that these problems are not solvable by radicals over Q and that the degree of the polynomial equations involved is unbounded in general. This motivates the search for approximate solutions. We present an algorithm, which approximates, for a given input parameter δ, optimal solutions for the MinEx and MaxIn problems up to an additive approximation error δ times the length of the input curves. The resulting running time is upper bounded by O n 3 δ log n δ , where n is the complexity of the input polygonal curves.

Staying Close to a Curve

Given a point set S and a polygonal curve P in R d , we study the problem of finding a polygonal curve through S, which has a minimum Fréchet distance to P. We present an efficient algorithm to solve the decision version of this problem in O(nk 2) time, where n and k represent the sizes of P and S, respectively. A curve minimizing the Fréchet distance can be computed in O(nk 2 log(nk)) time. As a by-product, we improve the map matching algorithm of Alt et al. by a log k factor for the case when the map is a complete graph.

Polygonal approximation of closed discrete curves

Pattern Recognition, 2007

Optimal polygonal approximation of closed curves differs from the case of open curve in the sense that the location of the starting point must also be determined. Straightforward exhaustive search would take N times more time than the corresponding optimal algorithm for an open curve, because there are N possible points to be considered as the starting point. Faster sub-optimal solution can be found by iterating the search and heuristically selecting different starting point at each iteration. In this paper, we propose to find the optimal approximation of a cyclically extended closed curve of double size, and to select the best possible starting point by search in the extended search space for the curve. The proposed approach provides solution very close to the optimal one using at most twice as much time as required by the optimal algorithm for the corresponding open curve.

APPROXIMATION OF POLYGONAL CURVES WITH MINIMUM NUMBER OF LINE SEGMENTS OR MINIMUM ERROR

International Journal of Computational Geometry & Applications, 1996

An algorithm for approximating a given open polygonal curve with a minimum number of biarcs is introduced. In computer-aided manufacturing environments, the paths of cutting tools are usually described with circular arcs and straight line segments. Greedy algorithms for approximating a polygonal curve with curves of higher order can be found in the literature. Without theoretical bounds it is difficult to prove anything about the quality of these algorithms. Also, in robotics motion planning, smoothing polygonal paths will increase robot performance, because robots usually have difficulties dealing with sharp turns. We present an algorithm which allows us to build a directed graph of all possible biarcs and look for the shortest path from the start point to the end point of the polygonal curve. We can prove a runtime of O(n 2 log n) for an original polygonal chain with n vertices with given tangent directions.

A Middle Curve Based on Discrete Fréchet Distance (original) (raw)

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