Polygonal Approximations of a Curve — Formulations and Algorithms

Imai, Hiroshi; Iri, Masao

doi:10.1016/b978-0-444-70467-2.50011-4

Cited by 139 publications

(161 citation statements)

References 9 publications

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“…In these methods, we compute the associated error of the O(n 2 ) possible links in respectively O(n 2 ) and O(n 3 ) time. Having these links, the former simplification is done using the general algorithm of Imai and Iri [17] and the latter is done using the algorithm presented by Bose et al [5].…”

Section: Our Resultsmentioning

confidence: 99%

“…3 Having the error of all links, there are efficient general algorithms to solve min-k version of both max-simplification and sum-simplification problems. As a general approach, Imai and Iri [17] modeled the min-k version of the max-simplification problem by a directed acyclic graph G over the vertices of the path P = p 0 , p 1 , . .…”

Section: T P I P J )) Over All Points T On the Subpath P (I J) Thimentioning

confidence: 99%

“…In distance-distortion error function, the error of a simplification is defined as the maximum distance between the initial and the simplification paths. This distance is further measured using various metrics including Hausdorff distance for L 1 , L 2 or L ∞ metrics [3,8,9,14,17,21] and Fréchet distance [4,10]. In area-distortion, the error of a simplification is defined as a function of the area created between the initial and the simplification paths [5,18,19,23,24].…”

Section: Introductionmentioning

confidence: 99%

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Efficient Observer-Dependent Simplification in Polygonal Domains

Zarei

Ghodsi

2011

Algorithmica

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In this paper, we consider a special version of the well-known linesimplification problem for simplifying the boundary of a region illuminated by a point light source q, or its visibility polygon VP(q). In this simplification approach, we should take the position of q as an essential factor into account to determine the quality of the resulting simplification. For this purpose, we redefine the known distanceand area-distortion error criteria as the main simplification criteria to take into account the distance between the observer q and the boundary of VP(q). Based on this, we propose algorithms for simplifying VP(q). More precisely, we propose simplification algorithms of O(n 2 ) and O(n 4/3+δ ) running time for observer-dependent distance-distortion simplification criterion and an O(n 3 ) simplification algorithm for observer-dependent area-distortion criterion where n is the number of vertices in VP(q). Moreover, we consider the observer-dependent distance-distortion simplification problem in the data streaming model where the vertices of VP(q) are given as a stream and only a constant amount of memory is available.

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Section: Our Resultsmentioning

confidence: 99%

Section: T P I P J )) Over All Points T On the Subpath P (I J) Thimentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Efficient Observer-Dependent Simplification in Polygonal Domains

Zarei

Ghodsi

2011

Algorithmica

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“…However, the Douglas-Peucker algorithm is only heuristic and is not guaranteed to be optimal (in terms of the number of vertices used or the error of the resulting simplification). Imai and Iri [16] solved both versions of the problem by modeling it as a shortest-path problem on directed acyclic graphs. The running time of their method was proved to be quadratic or near quadratic by Chin and Chan [7] and Melkman and O'Rouke [17].…”

Section: Introductionmentioning

confidence: 99%

“…Alt and Godau [4] proposed an algorithm to compute the Fréchet distance between two polygonal paths in quadratic time; combined with the approach of Imai and Iri [16], this can be used to compute an optimal solution to the min-δ or the min-k problem for the Fréchet distance.…”

Section: Introductionmentioning

confidence: 99%

Streaming Algorithms for Line Simplification

Abam

Berg

Hachenberger

et al. 2009

Discrete Comput Geom

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We study the following variant of the well-known line-simplification problem: we are getting a (possibly infinite) sequence of points p 0 , p 1 , p 2 , . . . in the plane defining a polygonal path, and as we receive the points, we wish to maintain a simplification of the path seen so far. We study this problem in a streaming setting, where we only have a limited amount of storage, so that we cannot store all the points. We analyze the competitive ratio of our algorithms, allowing resource augmentation: we let our algorithm maintain a simplification with 2k (internal) points and compare the error of our simplification to the error of the optimal simplification with k Comput Geom (2010) 43: 497-515 points. We obtain the algorithms with O(1) competitive ratio for three cases: convex paths, where the error is measured using the Hausdorff distance (or Fréchet distance), xy-monotone paths, where the error is measured using the Hausdorff distance (or Fréchet distance), and general paths, where the error is measured using the Fréchet distance. In the first case the algorithm needs O(k) additional storage, and in the latter two cases the algorithm needs O(k 2 ) additional storage.

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Recent progress on geometric algorithms for approximating functions: Toward applications to data analysis

Tokuyama

2006

Electron Comm Jpn Pt III

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SUMMARYData simplification is an extremely important issue in our current information-oriented society. Normally, a real-world database contains a massive amount of raw data, and when we consider the data as a distribution function, it has fluctuations due to sampling errors, outliers, and/or invalid inputs. Therefore, for data analysis technology such as data mining, it is important to approximate the input data by a simplified function. There are various approaches to function approximation, and functional analytical methods and learning-based techniques are quite popular. Apart from them, computational geometric approach based on optimization using discrete algorithms is widely studied. However, the conventional application of computational geometrical techniques is pattern matching, and to apply them to data analysis, their formulation and optimization criteria must be changed accordingly. Therefore, various difficulties and computational barriers arise, which must be eliminated or avoided. In this paper, we discuss data approximation in computational geometry and describe current trends centered on the author's latest research.

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Polygonal Approximations of a Curve — Formulations and Algorithms

Cited by 139 publications

References 9 publications

Efficient Observer-Dependent Simplification in Polygonal Domains

Efficient Observer-Dependent Simplification in Polygonal Domains

Streaming Algorithms for Line Simplification

Recent progress on geometric algorithms for approximating functions: Toward applications to data analysis

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