H. Edelsbrunner, D. Kirkpatrick, and R. Seidel. On the shape of a set of points in the plane. IEEE Transactions on Information Theory, 29(4):551559, 1983.  Home/Search   Document Not in Database   Summary   Related Articles   Check

.... of the actual solvent contact surface is detailed in [5] 3 Maintaining the Molecular Surface Under Quadratic Growth We call quadratic growth the scheme of growing balls that keeps the Power Diagram unaltered, and thus the topology of the union of balls is given by the corresponding ff shape [23, 24]. Under this growth we only need to maintain 12 the set of trimming curves of each patch in the surface. In particular we need to efficiently detect any topological changes (new intersections between curves, creation deletion of connected components) that occur in the trimming curves (circles and ....

H. Edelsbrunner, D. G. Kirkpatrick, and R. Seidel. On the shape of a set of points in the plane. IEEE Transactions on Information Theory, IT-29:551--559, 1983.

.... constraints can be solved with sequential quadratic programming (SQP [11] Recent statistical studies tend to suggest that the set of illuminants, which through the physics of superposition is convex, is better described by a non convex shape [3] Describing illuminant colours using alpha shapes [5, 1] provides a more accurate and smaller set of illuminants. Moreover, since alpha shapes are defined as the union of convex regions, it is a simple matter to find the best intersection within the alpha shape. We simply carry out SQP for each of the convex constituents of the alpha shape. Thus, it is ....

....programming (SQP) 11] To summarise 4 576 8 9 : 3 2 = 21) 0D 0 . Chromaticity space can be chosen so that convex sets in the prime space stay convex in chromaticity space and vice versa [6] 3. 3 Alpha shape constraint Alpha shapes [5, 1] express the intuitive notion of shape of a point set . An alpha shape is uniquely defined by the set and a parameter . A simplex belongs to the (2 dimensional) alpha shape of when there exists a circle of radius that does not contain any point of and where all the vertices of ....

H. Edelsbrunner, D. Kirkpatrick, and R. Seidel. On the shape of a set of points in the plane. IEEE Transactions on Information Theory, 29(4):551--559, 1983.

....sets in R 2, and some for point sets in R 3. Jarvis [44] was one of the first to consider the problem of computing the shape as a generalization of the convex hull of a planar point set. A mathematically rigorous definition of shape was later introduced by Edelsbrunner, Kirkpatrick, and Seidel [26]. They proposed a natural generalization of convex hulls, called a hulls, and their combinatorial variant, a shapes (see figure 1.2) Their notion of a shapes is the two dimensional analogue of the spatial notion described in chapter 2. Two dimensional a shapes are related to the dot patterns ....

....these extensions in order to avoid unnecessary complications and to be faithful to the currently available implementations. For completeness, negative values of , weighted points, and higher dimensions are now briefly discussed. Negative alpha values. This extension has been described in [26] for the two dimensional case. 3 For negative. complexes are most naturally defined as subcomplexes of the so called furthest point Delaunay triangulation of 5 (see section 1.1 or [21,67] For T C 5 and IT[ 4, the tetrahedron rT belongs to this triangulation if and only if bT contains ....

[Article contains additional citation context not shown here]

H Edelsbrunner, D G Kirkpatrick, and R Seidel. On the shape of a set of points in the plane. IEEE Transactions an Information Theory, IT-29(4):551 559, 1983.

.... variation analysis, geographic information systems, computational geometry, computational morphology, and computer vision use the underlying structure (also referred to as the skeleton or internal shape) of a set of data points revealed by means of a proximity graph (see for example [24] 20] [8], 17] 25] 27] 1] A proximity graph attempts to exhibit the relation between points in a point set. Two points are joined by an edge if they are deemed close by some proximity measure. It is the measure that determines the type of graph that results. Many different measures of proximity ....

H. Edelsbrunner, D. G. Kirkpatrick, and R. Seidel. On the Shape of a Set of Points in the Plane. IEEE Transaction on information Theory, 29, 1983, pp. 551-559.

.... variation analysis, geographic information systems, computational geometry, computational morphology, and computer vision use the underlying structure (also referred to as the skeleton or internal shape) of a set of data points revealed by means of a proximity graph (see for example [23] 28] [11], 19] 31] 1] 14] A proximity graph attempts to exhibit a relation between points in a point set by connecting pairs of points that are deemed close by some proximity measure. It is the measure that determines the type of graph that results. Many different measures of proximity have been ....

H. Edelsbrunner, D. G. Kirkpatrick, and R. Seidel. On the Shape of a Set of Points in the Plane. IEEE Transaction on Information Theory, 29, 1983, pp. 551-559.

....it is not possible to correctly reconstruct a given curve from an arbitrary sample set from it. Therefore, some restrictions are needed on the sample which specify how dense a sampling has to be to guarantee a correct output of the algorithm. The first algorithms for curve reconstruction [3, 4, 9, 10, 13] imposed a uniform sampling condition as they basically demanded that the distance between any two adjacent samples must be less than some constant. This is not satisfactory as it may require a dense sampling in areas where a sparse sampling is sufficient. Amenta, Bern, and Epstein [2] introduced ....

H. Edelsbrunner, D. G. Kirkpatrick, and R. Seidel. On the shape of a set of points in the plane. IEEE Transactions on Information Theory, vol. 29 (4) (1983), 71--78.

....which provably solve the reconstruction problem under certain assumptions on and V . Figure 1 illustrates the curve reconstruction problem. If the curve is closed, smooth, and uniformly sampled, several methods are known to work ranging over minimum spanning trees [FG94] shapes [BB97, EKS83] skeletons [KR85] and r regular shapes [Att97] A survey of these techniques appears in [Ede98] The case of non uniformly sampled closed curves was rst treated successfully by Amenta, Bern and Eppstein [ABE98] and subsequently improved algorithms such as [DK99, Gol99] appeared. Open ....

H. Edelsbrunner, D.G. Kirkpatrick, and R. Seidel. On the shape of a set of points in the plane. IEEE Transactions on Information Theory, 29(4):71-78, 1983.

....object that needs to be scaled back up to maintain the same size as the original object. This is not difficult but it adds another stage to the overall simplification process. The # hull has been defined as a generalization of the convex hull of point sets by Edelsbrunner, Kirkpatrick, and Seidel [15, 14]. Given a set of points P,aballb of radius # is defined as an empty # ball if b # P = #.For0 # # #1,the# hull of P is defined as the complement of the union of all empty # balls [14] Three dimensional # shapes have been defined on the Delaunay tetrahedralization D of the input points P by ....

H. Edelsbrunner, D. G. Kirkpatrick, and R. Seidel. On the shape of a set of points in the plane. IEEE Transactions on Information Theory, IT-29(4):551--559, July 1983.

....into the common coordinates system. 3. 5 shapes to get a surface from a measured point cloud A fair amount of work has been done to establish the de nition of shape for a set of points in 2D as well as in 3D space [9, 10] A mathematically rigorous de nition was given by Edelsbrunner et al. [11]. The concept of shapes of a nite set of points for arbitrary real was introduced in their article. The shape was derived from a straightforward generalization of a convex hull in two dimensional space. Authors have also provided an optimal algorithm for the shape construction for sets of ....

H. Edelsbrunner, D. G. Kirkpatrick, and R. Seidel. On the shape of a set of points in the plane. IEEE Transaction on Information Theory, 29(4):551-559, 1983.

....to narrow down the candidates and is a research area by itself. Once the objects have been divided into clusters, the next stage can describe the individual clusters. Known methods for describing individual clusters, or dot patterns or dot figures , include external shape description method in [EKS83], internal shape description method in [Rad88] Tou88] and fuzzy logic classification in [Abe91] Here, we introduce some of the basic vocabulary that we have so far developed as part of our description language. An object is defined to be obvious , if it is the only other object that is visible ....

Herbert Edelsbrunner, David G. Kirkpatrick, and Raimund Seidel. On the shape of a set of points in the plane. IEEE transactions on information theory, 1983.

....Research supported in part by a DST grant, 1997, Government of India and Max Planck Institut f ur Informatik, Germany. y Max Planck Institut f ur Informatik, D 66123 Saarbr ucken, Germany. E mail: fmehlhorn,ramosg mpi sb.mpg.de. Partially supported by esprit ltr project 28155 (GALIA) 1 [3, 7], skeleton [11] and r regular shapes [2] A survey on these techniques appears in [6] The case of non uniformly sampled closed curves was rst treated successfully by Amenta, Bern and Eppstein [1] and subsequently improved algorithms such as [5, 9, 10] appeared. a) b) c) d) e) f) ....

H. Edelsbrunner, D. G. Kirkpatrick, and R. Seidel. On the shape of a set of points in the plane. IEEE Transactions on Information Theory, vol. 29 (4) (1983), 71-78.

....the molecule has two or more sub parts connected by only two overlapping spheres. Also, it cannot generate the interior cavities of a molecule. In computational geometry, the ff hull has been defined as a generalization of the convex hull of point sets by Edelsbrunner, Kirkpatrick, and Seidel [6] [8]. For ff 0, the ff hull of a set of points P in two dimensions is defined to be the intersection of all closed complements of discs with radius ff that contain all points of P . If we generalize this notion of ff hulls over point sets to the corresponding hulls over spheres of unequal radii in ....

....ff that contain all points of P . If we generalize this notion of ff hulls over point sets to the corresponding hulls over spheres of unequal radii in three dimensions, we would get the molecular surface (along with the surface defining the interior cavities of the molecule) It has been shown in [8] that it is possible to compute the ff hulls from the Voronoi diagram of the points of P . For ff = 1 the ff hull over the set of points P is the same as their convex hull. Richards [16] had also suggested computing the molecular surface by computing a 3D Voronoi diagram first and then using its ....

H. Edelsbrunner, D. G. Kirkpatrick, and R. Seidel. On the shape of a set of points in the plane. IEEE Transactions on Information Theory, IT-29(4):551--559, 1983.

.... its application in computer vision, image processing and pattern recognition has drawn a lot of attention from researchers over the last three decades [15, 16, 17] If the curve is closed and uniformly sampled, a number of methods is known to work ranging over minimum spanning tree [8] shapes [3, 7], skeleton [11] and r regular shapes [2] A survey on these techniques appears in [6] The case of non uniformly sampled closed curves was rst treated successfully by Amenta, Bern and Eppstein [1] and subsequently improved algorithms such as [5, 9, 10] appeared. We need the following de ....

H. Edelsbrunner, D. G. Kirkpatrick, and R. Seidel. On the shape of a set of points in the plane. IEEE Transactions on Information Theory, vol. 29 (4) (1983), 71-78.

....of points by choosing a subset of the Delaunay triangulation so as to optimize the approximate medial axis of the resulting polygon. A successful earlier computational geometric approach to defining the shape of a set of points is the ff shape, introduced by Edelsbrunner, Kirkpatrick and Seidel [EKS83], and studied extensively by Edelsbrunner and others. The ff shape is a simplicial complex defined on a set of points in arbitrary dimension d; each k d 1 points are connected into a (k Gamma 1) simplex if they touch an empty ball of radius Gamma1=ff. The ff shape tends to work well for ....

Edelsbrunner, H., Kirkpatrick D.G., and Seidel, R., On the shape of a set of points in the plane, IEEE Transactions on Information Theory 29 (4) (1983), pp. 551-559.

....approximation problem. The ff approximation and the ffl approximation are both quite general problems and are useful in areas other than molecular modeling. The ff approximation problem can be used to compute a detail hierarchy of ff hulls and ff shapes [Edelsbrunner Mucke 94, Edelsbrunner 92, Edelsbrunner et al. 83] that are useful in such diverse fields as astronomy, biochemistry, statistics, and computer graphics. The ffl approximation problem is of immense value in a three dimensional computer graphics setting in simplifying complex polygonal models under the constraints of topological consistency and ....

....effects of the solvent into the overall potential energy computations during the interactive modifications of a molecule on a computer. 1. 3 ff Approximation of Molecules The ff hull has been defined as a generalization of the convex hull of point sets by Edelsbrunner, Kirkpatrick, and Seidel [Edelsbrunner et al. 83] Given a set of points P , a ball b of radius ff is called an empty ff ball if b P = OE. For 0 ff 1, the ff hull of P is defined as the surface of the complement of the union of all empty ff balls [Edelsbrunner 92] The smooth molecular surface (as defined by Richards [Richards 77] for a ....

[Article contains additional citation context not shown here]

H. Edelsbrunner, D. G. Kirkpatrick, and R. Seidel. On the shape of a set of points in the plane. IEEE Transactions on Information Theory, IT-29(4):551--559, July 1983.

....conditions are satisfied by the input. Amenta, Bern and Eppstein [1] proposed a framework based on local feature size under which they show two graphs, crust and fi skeleton, coincide with G if the points are sufficiently sampled. Some of the other effective approaches include ff shapes by [6] which is analyzed later by [3] r regular shapes by [2] A shapes by [7] and a Delaunay based method by [4] A survey of these methods appear in [5] In this paper we show that a modified nearest neighbor graph also coincides with G. The algorithm and its analysis are simple. Nevertheless, it ....

H. Edelsbrunner, D. G. Kirkpatrick, and R. Seidel. On the shape of a set of points in the plane. IEEE Transactions on Information Theory, vol. 29 (4) (1983), 71--78.

....allowed and consequently, it is impossible to get a surface formed of several connected components or having holes. The Voronoi graph is sometimes used to get additional information on the skeleton of the object [4, 5, 6] More complex graphs have also been introduced like ff hulls and ff shapes [7, 8]. ff shapes are a generalization of the convex hull of a point set. An ff shape is a polytope surrounding the set of points. The parameter ff controls the maximum curvature of any cavity of the polytope. Several ff shapes with different values of ff are presented in figure 1. The choice of the ....

H. Edelsbrunner, D. G. Kirkpatrick, and R. Seidel. On the shape of a set of points in the plane. IEEE Transactions on Information Theory, 29(4):551--559, 1983.

....2 ) One cannot use linear time median finding or randomization to improve the running time. If one assumes that the input data is such that farthest vertices are found near the middle of chains, then one expects O(n log n) behavior. The second implementation uses a path hull data structure [2, 4] to maintain a dynamic convex hull of the polygonal chain V . Using Melkman s convex hull algorithm [7] we compute two convex hulls from the middle of the chain outward. We can find a farthest vertex v f from a line by locating two extreme points on each hull using binary search. When we split V ....

....An efficient algorithm for finding the CSG representation of a simple polygon. Algorithmica, 10:1 23, 1993. 3] D. H. Douglas and T. K. Peucker. Algorithms for the reduction of the number of points required to represent a line or its caricature. The Canadian Cartographer, 10(2) 112 122, 1973. [4] J. Hershberger and J. Snoeyink. Speeding up the Douglas Peucker line simplification algorithm. In Proc. 5th Intl. Symp. Spatial Data Handling, pages 134 143. IGU Commission on GIS, 1992. 5] R. B. McMaster. A statistical analysis of mathematical measures for linear simplification. Amer. Cartog. ....

[Article contains additional citation context not shown here]

H. Edelsbrunner, D. G. Kirkpatrick, and R. Seidel. On the shape of a set of points in the plane. IEEE Transactions on Information Theory, IT-29(4):551-- 559, 1983.

....narrow down the candidates and is a research area by itself. Once the objects have been divided into clusters, the next stage can describe the individual clusters. Known methods for describing individual clusters, or dot patterns or dot figures, include the external shape description method in [14], internal shape description method in [60] 72] and a fuzzy logic classification in [2] Describing a set of points by humans seems to involve dividing and organizing the points into clusters or gestalts . Many cluster detecting algorithms have been developed. These methods are generally ....

.... tree (MST) relative neighborhood graph (RNG) Gabriel graph (GG) Delaunay triangle (DT) 60] and sphere of influence graph (SIG) 72] These methods are considered to describe the internal structure of the point sets, whereas the convex hull or ff hull methods describe the external shapes [14]. A NNG is formed by simply connecting the vertices to produce all nearest neighbor pairs. MST connects every node to some other nodes 59 in such a way that the total length of the edges is minimized and all members of the point set are connected. Zahn [76] uses MST in a graph theoretical way to ....

H. Edelsbrunner, D. G. Kirkpatrick, and R. Seidel. On the shape of a set of points in the plane. IEEE Transactions on Information Theory, IT29 (4):pp:551--559, July 1983.

....10,000 points in a matter of minutes. The main difficulty, both in theory and in practice, is the reconstruction of sharp edges. 2 Related work The idea of using Voronoi diagrams and Delaunay triangulations in surface reconstruction is not new. The well known ff shape of Edelsbrunner et al. [9, 10] is a parameterized construction that associates a polyhedral shape with an unorganized set of points. A simplex (edge, triangle, or tetrahedron) is included in the ff shape if it has some circumsphere with interior empty of sample points, of radius at most ff (a circumsphere of a simplex has the ....

H. Edelsbrunner, D.G. Kirkpatrick, and R. Seidel. On the shape of a set of points in the plane, IEEE Transactions on Information Theory 29:551-559, (1983).

....shape, which is the macro structure of the boundary, and internal shape, which is the micro structure of the interior [Kirkpatrick Radke (1985) Some simple computational descriptions of external shape are: the bounding box, the bounding rectangle, the bounding circle, and the convex hull. Edelsbrunner, Kirkpatrick and Seidel (1983) defined a parametrized family of external shape descriptors, called ff hulls, which include bounding circles and convex hulls. Unlike the simple descriptors listed above, ff hulls are able to describe shapes with holes and multiple components. Moreover, discrete versions of ff hulls, called ....

H. Edelsbrunner, D. G. Kirkpatrick and R. Seidel, On the shape of a set of points in the plane, IEEE Transactions on Information Theory 29 (1983) 551--559.

....in IR 2 , and some for point sets in IR 3 . Jarvis [23] was one of the first to consider the problem of computing the shape as a generalization of the convex hull of a planar point set. A mathematically rigorous definition of shape was later introduced by Edelsbrunner, Kirkpatrick, and Seidel [14]. Their notion of ff shapes is the twodimensional analogue of the spatial notion described in this paper. Two dimensional ff shapes are related to the dot patterns of Fairfield [17, 18] and the circle diagrams used in bivariate cluster analysis (see for example Moss [33] Different graph ....

....to avoid unnecessary complications and to be faithful to the currently available implementation of the concepts in this paper. For completeness, negative values of ff, weighted points, and higher dimensions are now briefly discussed. Negative Alpha Values. This extension has been described in [14] for the two dimensional case. 5 For negative ff, ff complexes are most naturally defined as subcomplexes of the so called furthest point Delaunay triangulation of S (see for example [11, 35] For T S and jT j = 4, the tetrahedron oe T belongs to this triangulation iff b T contains all points ....

[Article contains additional citation context not shown here]

H Edelsbrunner, D G Kirkpatrick, and R Seidel. On the shape of a set of points in the plane. IEEE Transactions on Information Theory, IT-29(4):551--559, 1983.

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H. Edelsbrunner, D. Kirkpatrick, and R. Seidel. On the shape of a set of points in the plane. IEEE Transactions on Information Theory, 29(4):551559, 1983.

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Edelsbrunner et al., Shape of points in the plane, IEEE Transactions on Information Theory, IT-29, 550-559 (1983).

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H. Edelsbrunner, D.G. Kirkpatrick, and R. Seidel. On the shape of a set of points in the plane. IEEE Transactions on Information Theory, 29(4):71-78, 1983.

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