H. Imai and M. Iri, Polygonal approximations of a curve-formulations and algorithms, Computational Morphology, 71-86, NorthHolland, Amsterdam, 1988.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....or as the sum of all these distances. Another group of algorithms, optimal polygonal approximation methods for particular distance measures, was implemented. The generic character regarding the base data type and the kernel type is still available, but these methods use graph search techniques [5] to compute the polygonal approximation with the Euclidean distance error assessment only. After an introduction about the purpose of this study, in the second section are presented some theoretical elements about polygonal approximation methods and the optimal and sub optimal paradigms are ....

.... Squared Euclidean Distances, Path Hull implementation [1] Dynamic Programming, 8] O (MN 3) General implementation Dynamic Programming, new implementation O(N 3) General implementation Dynamic Programming O(MN ) Sum of Euclidean Distances, Incremental technique [4] Graph Search Method, [5] O(NlogN) Maximum of Euclidean Distances, measured to line support Graph Search Method, O(N 3) Sum of Euclidean Distances, measured to line support 11 To understand the difference in running time of the several kinds of algorithms, in table 1 (page 12) are presented the time complexity limits ....

H. Imai, M. Iri, "Polygonal Approximation of a Curve - Formulation and Algorithms", Computational Morphology, (ed. G.T. Toussaint), pp. 71-86;

.... in the sense that the interior of the approximated free space is assured to be free, or it can be designed to be accurate in a least squared sense, so that for a given number of vertices in the approximation the discrepancy between the polygonal model and the actual environment is minimized [34,28]. The paper is structured as follows. In Section 2 we present an overview of previous work in mapping, localization and multi robot applications. In Section Ioannis Rekleitis et al. Multi Robot Collaboration for Robust Exploration 3 3 we present the fundamental ideas in our approach of ....

H. Imai and Masao Iri. Polygonal approximations of a curve { formulations and algorithms. In G. T. Toussaint, editor, Computational Morphology, pages 71-86. Elsevier Science Publishers B. V., New York, N.Y., 1988.

....ed boundary. Figure 1: Example of topological errors arising in simpli cation of maps. In GIS practice, most often the polygonal chains are considered in isolation, each being simpli ed using a favorite line simpli cation algorithm, such as that of Douglas and Peucker [8, 12, 14] see also [4, 5, 16, 17, 19]. Not only does this local approach lead to possible crossings between pairs of simpli ed chains, it can also lead to self intersection (nonsimplicity) of a single chain. In this paper, we investigate the problem of map simpli cation with topological constraints. The input to our problem is a set ....

....Guibas et al. [11] prove that computing a minimum link simple polygon of a given homotopy type is NP complete, as is computing a minimum link simple polygonal subdivision that is homeomorphic to an input subdivision, within a polygonal domain. de Berg et al. [6, 7] have shown how the methods of [4, 16, 19] can be applied, in conjunction with constraints to guarantee topological consistency, to obtain in O(n(n m) log n) time a minimum size simpli cation of an x monotone chain (having n vertices) that is within an approximation error and homotopically consistent with the input, with respect to a ....

H. Imai and M. Iri. Polygonal approximations of a curve-formulations and algorithms. In G. T. Toussaint, editor, Computational Morphology, pages 71-86. North-Holland, Amsterdam, Netherlands, 1988.

....version P 0 of P , and the parameter controls the closeness of P 0 to P (under a certain error criterion) Actually, there is a trade o between the two parameters m and : The smaller is, the larger m tends to be, and vice versa. Based on this relation between m and , Imai and Iri [12,14] considered two versions of optimization problems on approximating polygonal curves in the plane: i) Given , minimize m (called the min # problem) and (ii) given m, minimize (called the min problem) In this paper, we study both the min # and min problems in 2 D space. Curve ....

....and approximating objects. Di erent error criteria have been used in solving various polygonal curve approximation problems (e.g. see [1, 2, 5, 7 18, 21 28] In this paper, we consider two commonly used error criteria for studying polygonal curve approximations: The error criterion used in [5, 12,14,18], which we call the tolerance zone criterion, and the criterion used in [9, 14,24] which we call the in nite beam criterion (the in nite beam criterion is also called the parallel strip criterion in [9, 14, 24] Under the tolerance zone criterion, the approximation error between a segment p j p ....

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H. Imai and M. Iri. Polygonal approximations of a curve-formulations and algorithms. Computational Morphology, pages 71-86, North{Holland, Amsterdam, 1988.

....on the two resulting subchains. This algorithm is simple but has a worst case complexity of O(n 2 ) time and may not yield optimal approximations. However, the computational complexity can be improved using computational geometric techniques to O(n log n) time [32] In 1985 Iri and Imai [33] proposed an elegant algorithm that finds the approximation that minimizes m subject to the two other constraints. Their approach is very general and elegant and can be applied in a wide variety of contexts. Their paper is still the best place to start for an introduction to this problem and its ....

Hiroshi Imai and Masao Iri. Polygonal approximations of a curve: formulations and algorithms. In Godfried T. Toussaint, editor, Computational Morphology, pages 71--86. North-Holland, Amsterdam, Netherlands, 1988.

....of finding an approximation polygon with the minimal number of vertices and with the error within a given bound. The min ffl problem is the problem of finding an approximation polygon with the minimal error and a given number of vertices. Algorithms for both optimality problems are discussed by [12]. A sequence of successively more detailed approximations requires the iterative application of these algorithms. However, in general this yields no hierarchy of approximations, and it is computationally expensive. The iterative end point fit method for approximating plane polylines is described ....

H. Imai and M. Iri (1988). Polygonal approximations of a curve -- formulations and algorithms. In Toussaint [20], pp. 71 -- 86.

....that i 1 i m , and that the subpaths P 1;i 1 Gamma1 and P i m 1;n are completely contained in the balls centered at P i 1 and P i m , respectively. We explain later in the introduction how to reduce the general problem to the restricted version of it. 1. 1 Related Previous Work Imai and Iri [17, 18] and Melkman and O Rourke [22] study the 2 dimensional min # and min polygonal path approximation problems. Using the tolerance zone measure of error, as defined above, they achieve algorithms whose running times are O(n 2 log n) and O(n 2 log 2 n) respectively, and using O(n 2 ) space, ....

....and Peucker [8] which 5 is popular in GIS, does extend to curves in 3 dimensions in O(n 2 ) worst case time, but solves neither the min # nor the min problems. There has been other less related, but still significant, work done on other metrics for comparing 2 dimensional polygonal paths [18] that do not necessarily use the same set of vertices. Arkin et al. 2] Alt and Godau [1] and Rote [25] describe several metrics for comparing polygonal curves. Guibas et al. 15] study different error measures for the min # problem, with an eye toward those that could be implemented in O(n log ....

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H. Imai and M. Iri, Polygonal approximations of a curve-formulations and algorithms, in Computational Morphology, North--Holland, Amsterdam, 1988, pp. 71--86.

....to polynomial. The L 2 optimal approximation to a function y(x) can be found in O(mn 2 ) time, worst case, using dynamic programming. Remarkably, a slight variation in the error metric permits a much faster algorithm: the L# optimal approximation to a function can be found in O(n) time [63], using visibility techniques (see also [123, 91] When the problem is generalized from functions to planar curves, the complexity of the best L# optimal algorithms we know of jumps to O(n 2 log n) 63] These methods use shortest path graph algorithms or convex hulls. For space curves ....

.... faster algorithm: the L# optimal approximation to a function can be found in O(n) time [63] using visibility techniques (see also [123, 91] When the problem is generalized from functions to planar curves, the complexity of the best L# optimal algorithms we know of jumps to O(n 2 log n) [63]. These methods use shortest path graph algorithms or convex hulls. For space curves (curves in 3 D) there are O(n 3 log m) L# optimal algorithms [62] Asymptotic Approximation. In related work, McClure and de Boor analyzed the error when approximating a highly continuous function y(x) using ....

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Hiroshi Imai and Masao Iri. Polygonal approximations of a curve -- formulations and algorithms. In G. T. Toussaint, editor, Computational Morphology, pages 71--86. Elsevier Science, 1988.

....the approximation error, or dissimilarity, d(A; A k ) Given a polyline and an error bound , construct an approximation polyline A with dissimilarity d(A; A ) minimizing the number of vertices. Both approximations can be computed in O(n 2 log n) time for various error measures [II88] However, these optimal approximations are not suitable for constructing a hierarchy of approximations, in the sense that each segment at one level may be re ned at the next level of approximation. Approximating polygons at various levels allows the hierarchical processing of curves [Vel98] ....

Hiroshi Imai and Masa Iri. Polygonal Approximation of a Curve { Formulations and Algorithms, pages 71-86. North-Holland, 1988.

....problem, for d 3, in this paper. One issue that immediately arises in the path approximation problem is that one must formalize the error between the original path P and the approximation P 0 . We believe that one of the most natural definitions is the tolerance zone error measure [3, 17, 18, 22]. Define the tolerance zone (for 0) of a line segment pq to be the region of space (or zone ) that is the union of all radius balls centered at points along the segment pq (see Figure 1) This definition is, of course, parameterized by the metric used to define radius balls. We ....

....i 1 i m , and that the subpaths P 1;i 1 Gamma1 and P i m 1;n are completely contained in the balls centered at P i 1 and P i m , respectively. We explain later in the introduction how to reduce the general problem to the restricted version of it. 1. 1 Related Previous Work Imai and Iri [17,18] and Melkman and O Rourke [22] study the 2 D min # and min polygonal path approximation problems. Using the tolerance zone measure of error, as defined above, they achieve algorithms whose running times are O(n 2 log n) and O(n 2 log 2 n) respectively, and using O(n 2 ) space, for the ....

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H. Imai and M. Iri, Polygonal approximations of a curve-formulations and algorithms, Computational Morphology, 71--86, North--Holland, Amsterdam, 1988.

....abstraction of f . Therefore the time points of the corners of p k thus the non differentiable points of p k must also be elements of T . Figure 1 illustrates this. The piecewise linear approximation of a function or a set of data points has been investigated by many researchers, e.g. (Imai and Iri, 1988, Baines, 1994, M.T.Goodrich, 1995) In (Hakimi and Schmeichel, 1991) an O(n) algorithm is introduced, which finds within a given error ffl a piecewise linear function y 1 1 2 3 3 t Interesting time points Class A: Class B: ymax ymin f (t) j threshold y p (t) 1 p (t) 2 p (t) 3 p (t) 4 ....

Imai, H. and Iri, M. (1988). Polygonal Approximation of a Curve -- Formulations and Algorithms. In G.T.Toussaint, editor, Computational Morphology, pages 71--86.

....of the approximation is farther than ffl distance away from some point on the actual curve. ffl Min ffl Approximations: Given the number of vertices desired in the output approximation curve, minimize the error between the approximation curve and the input curve. Imai and Iri [Imai Iri 86] Imai Iri 88] have shown that the min # approximation to a piecewise linear function can be accomplished in an optimal O(n) time. The min ffl problem for the piecewise linear functions is much harder to solve and no efficient algorithms are known that solve it optimally for the general case. The vertices of ....

....; p i(m) 1 = i(1) i(2) i(m) n, which are a subset of p 1 ; p 2 ; p n . This is shown in Figure 3.5. Min ffl as well as the min # approximation problems have been solved for this approximation scheme, using different measures of the approximation error criterion [Imai Iri 88] Let the error of approximating the curve between vertices p i ; p i 1 ; p i 2 ; p j by the line segment p i p j be given by the maximum of the distance between the segment p i p j and the points p k (i k j) For this error measure, the min # problem can be solved in time O(n 2 log ....

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H. Imai and M. Iri. Polygonal approximations of a curve -- Formulations and Algorithms. In G. T. Toussaint, editor, Computational Morphology, pages 71--86. North-Holland, Amsterdam, Netherlands, 1988.

.... information systems [Buttenfield 85, Cromley 88, Douglas Peucker 73, Hershberger Snoeyink 92, Li Openshaw 92, McMaster 87] digital image analysis [Asano Katoh 93, Hobby 93, Kurozumi Davis 82] and computational geometry [Chan Chin 92, Eu Toussaint 94, Guibas et al. 93, Imai Iri 88, Melkman O Rourke 88] Often the input is a polygonal chain and a maximum allowed error ffl, and methods are described to obtain another polygonal chain with fewer vertices that lies at distance at most ffl from the original polygonal chain. Some methods yield chains of which all vertices are ....

....a polygonal chain is reduced in complexity, the output polygonal chain must be a simple polygonal chain. Several of the line simplification methods described before don t satisfy this constraint [Chan Chin 92, Cromley 88, Douglas Peucker 73, Eu Toussaint 94, Hershberger Snoeyink 92, Imai Iri 88, Li Openshaw 92, Melkman O Rourke 88] The second condition that need be satisfied is that the output chain does not intersect any other polygonal chain in the subdivision. In other words, the simplification method must respect the fact that the polygonal chain to be simplified has a ....

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H. Imai and M. Iri, Polygonal approximations of a curve -- formulations and algorithms. In: G.T. Toussaint (Ed.), Computational Morphology, Elsevier Science Publishers, 1988, pp. 71--86.

....display. Though the properties of the individual algorithms used are characterized and classified, the properties of these heuristic combinations are not. Criteria much like our fattening [5, 26, 29] may then be used a posteriori to test the quality of the resulting approximations. Imai and Iri [16, 17, 18] and other researchers [2, 4, 8, 13, 23, 25, 31] have chosen mathematical criterion for the approximations and then sought efficient algorithms to find best approximations. The algorithms they have developed, however, have quadratic or greater running times especially for those that use original ....

....in O(n 2 ) time and linear space. For definition 4, the time increases to O(n 2 log n) and the objects must be constant size polygons or equal radius circles. All three restrictions on turns or vertices are supported. This should be compared to the general graph based approach of Imai and Iri [18], which, in our terminology, would create a graph with an edge (j; k) if there is an ordered stabber from O j through O k and then search the graph for the shortest path. Our dynamic programming method shares the problem of a super quadratic running time, but saves a factor of O(n) in space by ....

Hiroshi Imai and Masao Iri. Polygonal approximations of a curve---formulations and algorithms. In Godfried T. Toussaint, editor, Computational Morphology. North Holland, 1988.

.... proposed recently [Her97] The GIS literature is full of other heuristics proposed for the same purpose, which are usually compared empirically (see [McM87] for a survey and experimantal comparisons; Li92, Per92] for more recent examples) A rigorous approach to the problem is due to Imai and Iri [Ima88] who first gave a formalization of line simplification as an optimization problem. Namely, given an input curve and some metric to measure the error in approximating it, they outline two basic problems: i) minimizing the number of vertices of the output chain for a given threshold error; ii) ....

Imai, H., Iri, M., Polygonal approximation of a curve - formulations and algorithms, Computational Morphology, G.T. Toussaint (Ed.), Machine Intelligence and Pattern Recognition, 6, North-Holland, 1988, pp.71-86.

....with high level geographic information. A statistical approach describing the benefits of geometric simplification in geography can be found in [15] Similar techniques appear in computer graphics when approximating digitized images by polygons [11, 19, 20] In computational geometry, Imai and Iri [8, 9, 10] and Natarajan [17] gave linear time algorithms for the problem of approximating a piecewise linear function with a minimum number of line segments according to a given tolerance . Recent surveys of approximation techniques can be found in [4, 5, 6, 7] In this paper, we study a new variant of ....

....is bounded by k, the time complexity can further be reduced to O(N kn) Notice that in the latter, the time and space complexity does not depend on the range m of the function. A similar problem, the approximation of a given polygonal chain by another simpler chain has been studied by Imai and Iri [9, 10]. Their algorithm runs in optimal O(n) time. 2 Preliminaries At the beginning, we are given a continuous function f : 0; n Gamma 1] 0; m Gamma 1] x 7 f(x) on the domain D : 0; n Gamma 1] Theta [0; m Gamma 1] which is covered by a regular grid G : f(x; y) j x = 0; n Gamma ....

H. Imai and M. Iri, Polygonal approximations of a curve -- formulations and algorithms, in G.T. Toussaint (Ed.), Computational Morphology, Elsevier Science Publishers, pp 71--86, 1988

....or C oriented link distance. A min link s t path is a polygonal path from s to t that achieves the link distance. In many problems, the link distance provides a more natural measure of path complexity than the Euclidean length. The link distance also has applications to curve simplification [187, 222, 295]. Since this handbook contains a chapter by Maheshwari and Sack [271] devoted entirely to the subject of link distance, we refer the reader to that survey for further information. 4.3 The Weighted Region Metric In the weighted region problem , we are given a piecewise constant function, f : ....

H. Imai and M. Iri. Polygonal approximations of a curve-formulations and algorithms. In G. T. Toussaint, editor, Computational Morphology, pages 71--86. North-Holland, Amsterdam, Netherlands, 1988.

....and data compression. The advantages of the simplification process include removing unnecessary cluttering due to excessive detail, saving disk and memory space, and reducing the rendering time. The basic framework for computing a simplification of a polygonal chain C is provided by Imai and Iri [5]. Under the uniform metric criterion, their algorithm ran in O(n 2 log n) time. Chin et al. 2] improve the running time of their algorithm to quadratic. Recently Agarwal and Varadarajan [1] improve the running time to O(n 4=3 ffi ) for certain error criterion. Since near linear time ....

Imai, H., and Iri, M. Polygonal approximations of a curve-formulations and algorit hms. In Computational Morphology, G. T. Toussaint, Ed. North-Holland, Amsterdam, Netherlands, 1988, pp. 71--86. 2

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H. Imai and M. Iri, Polygonal approximations of a curve-formulations and algorithms, Computational Morphology, 71-86, NorthHolland, Amsterdam, 1988.

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H. Imai and M. Iri. Polygonal approximations of a curve-formulations and algorithms. In G. T. Toussaint, editor, Computational Morphology, pages 71-86. North-Holland, Amsterdam, Netherlands, 1988.

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Imai H., and Iri M., Polygonal Approximations of a Curve-Formulations and Algorithms. Computational Morphology, editor:G. T. Toussaint, North Holland, 1988.

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Hiroshi Imai and Masao Iri. Polygonal approximations of a curve -- formulations and algorithms. In G. T. Toussaint, editor, Computational Morphology,pages 71--86. Elsevier Science, 1988.

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H. Imai and M. Iri, Polygonal approximations of a curve --- formulations and algorithms, Computational Morphology --- A Computational Geometric Approach to the Analysis of Form (G. T. Toussaint, ed.), North-Holland, Amsterdam-New-York, 1988, pp. 71-86.

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H. Imai and M. Iri. Polygonal approximations of a curve-formulation and algorithms. In Computational Morphology, G. Toussaint ed., North Holland, Amsterdam, pages 71--86, 1988.

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H. Imai and M. Iri. Polygonal approximations of a curve---formulations and algorithms. In G. T. Toussaint, editor, Computational Morphology. North Holland, 1988.

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