D. G. Kirkpatrick and J. D. Radke. A framework for computational morphology. In Computational Geometry, pages 217--248, North-Holland, 1985.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....reduces the size of the training set more than Voronoi editing. Although, the resulting Gabriel editing does not preserve the original decision boundary, the changes occur mainly outside of the zones of interest. The parameterized family of neighborhood graphs, introduced by Kirkpatrick and Radke [7], is called # skeleton. Let V be a set of points in R , each pair of points (p, q) # V V with a neighborhood U p,q # R , let P be a property defined on U = U p,q (p, q) # V V , #(x, y) denotes the distance between point x and y, and B(x, r) denotes the circle centered at x with the ....

J. D. Radke. D.G.Kirkpatrick. A Framework for computational morphology. In G. T. Toussaint, editor, Computational Geometry, NorthHolland, Amsterdam, Netherlands, 1985.

....Toussaint [10] here we only recall some of the most widely studied proximity graphs. Two continuous hierarchies of proximity graphs that includes Gabriel graphs and relative neighbourhood graphs as special cases were first defined in the computational morphology context by Kirkpatrick and Radke [13]. The elements of these infinite families of proximity graphs are called circle and lune based fi skeletons and are defined by considering a continuous family of regions of influence indexed by a single real positive parameter fi. Another parameterized family of proximity graphs, known as ....

.... exist only suboptimal algorithms that compute the lune based fi skeleton when 0 fi 1: the fastest algorithm that we know for this problem requires O(n log n) time [20] For values of fi in the range (2; 1) a (suboptimal) O(n time algorithm for lune based fi skeletons is described in [13, 19]. As for circle based fi skeletons, an optimal O(n log n) time algorithm is given in [13] for all values of fi in the interval [1; 1] while the same O(n log n) time algorithm of [20] applies for values of fi in the interval [0; 1) The problem of computing fl graphs has been studied in [23] ....

[Article contains additional citation context not shown here]

D. G. Kirkpatrick and J. D. Radke. A framework for computational morphology. In G. T. Toussaint, editor, Computational Geometry, pp. 217--248. North-Holland, Amsterdam, Netherlands, 1985.

....distance of the neighbors vary locally and adapt naturally to the distribution of the data around Z. Note that these methods also automatically and implicitly assign di erent 11 weights to the nearest geometric neighbors of Z. 8 Open Problems and New Directions In 1985 Kirkpatrick and Radke [60] (see also Radke [86] proposed a generalization of the Gabriel and relative neighborhood graphs which they called skeletons, where is a parameter that determines the shape of the neighborhood of two points that must be empty of other points before the two points are joined by an edge in the ....

David G. Kirkpatrick and John D. Radke. A framework for computational morphology. In Godfried T. Toussaint, editor, Computational Geometry, pages 217-248. North Holland, Amsterdam, 1985.

.... to the dot patterns (Fairfield [32, 33] and the circle diagrams used in bivariate cluster analysis (see for example Moss [61] Different graph structures that serve similar purposes are the Gabriel graph [56] the relative neighborhood graph [76] or their parameterized version, the skeleton [48]. In Jl 3. For spatial point sets, Boissonnat [3] suggested the use of Delaunay triangulations in con nection with certain heuristics to sculpture a single connected shape of a cloud of points. The concept of three dimensional alpha shapes of section 2.1 is more general and mathematically ....

D G Kirkpatrick and J D Radke. A framework for computational morphology. In G T Toussaint, editor, Computational Geometry, pages 234 244. Elsevier North Holland, New York, 1985.

....di Informatica e Sistemistica, Universit a di Roma La Sapienza , via Salaria 113, I 00198 Roma, Italia. Department of Computer Science, Williams College, Williamstown, MA 01267. Abstract. A family of proximity drawings of graphs called open and closed drawings, rst de ned in [16], and including the Gabriel, relative neighborhood and strip drawings, are investigated. Complete characterizations of which trees admit open drawings for 0 1 or closed drawings for 0 cos(2 =5) 1 are given, as well as partial characterizations for other values ....

....the proximity drawability testing problem, i.e. the problem of deciding whether a graph has a proximity drawing with a given type of proximity region. In particular we study the proximity drawability of trees. We consider an in nite family of parametrized proximity regions, rst introduced by [16], that covers the most well known proximity regions presented in the literature. Due to space restrictions, most proofs have been omitted in this extended abstract. For full detailed proofs, we refer the reader to the technical report [3] We consider two types of proximity regions: De nition ....

[Article contains additional citation context not shown here]

D. G. Kirkpatrick and J. D. Radke. A Framework for Computational Morphology. Computational Geometry, ed. G. T. Toussaint, Elsevier, Amsterdam, 1985, pp. 217248.

.... Constraints and Representable Trees Prosenjit Bose Giuseppe Di Battista William Lenhart Giuseppe Liotta Abstract This paper examines an infinite family of proximity drawings of graphs called open and closed fi drawings, first defined by Kirkpatrick and Radke [15, 21] in the context of computational morphology. Such proximity drawings include as special cases the well known Gabriel, relative neighborhood and strip drawings. Complete characterizations of those trees that admit open fi drawings for 0 fi fi 1 or closed fi drawings for 0 fi ....

....we study the proximity drawability testing problem: the problem of deciding whether a graph has a proximity drawing with a given type of proximity region. In particular we study the proximitydrawability of trees. We consider an infinite parametrized family of proximity regions, first introduced in [15, 21], that includes several of the most well known proximity regions from the literature. We consider two types of proximity region: Definition 1 Given a pair x; y of points in the plane, the open fi region of x and y, and the closed fi region of x and y, denoted by R(x; y; fi) and R[x; y; fi] ....

[Article contains additional citation context not shown here]

D. G. Kirkpatrick and J. D. Radke. A Framework for Computational Morphology. Computational Geometry, ed. G. T. Toussaint, Elsevier, Amsterdam, 1985, pp. 217-248.

....e Sistemistica, Universit a di Roma La Sapienza , via Salaria 113, I 00198 Roma, Italia. This work has been done when this author was visiting the School of Computer Science of McGill University. liotta infokit.ing.uniroma1.it Other proximity graphs that have been studied are the fi skeleton [12], the sphere of influence graph [25] and the fl neighborhood graph [27] An extensive survey on the current research in proximity graphs can be found in Jaromczyk and Toussaint [14] Much attention has been given over the past several years to developing algorithms for drawing graphs in the ....

D. G. Kirkpatrick and J. D. Radke. A Framework for Computational Morphology. Computational Geometry, G. T. Toussaint, Elsevier, Amsterdam, 1985, pp. 217-248.

.... 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 defined. For example, ....

....in the three dimensional space. Edelsbrunner and Tan [10] exploited the properties of relative neighborhood graphs on the plane to give a quadratic time algorithm for the minmax edge length triangulation problem. Many other proximity graphs have been studied. For example, Kirkpatrick and Radke [14] defined and studied the skeleton graphs that include both the modified gabriel and the relative neighborhood graph of a point set. Paterson and Yao [21] derived several combinatorial properties of k nearest neighborhood graphs and gave bounds on the size of such graphs. Toussaint [28] ....

D. G. Kirkpatrick and J. D. Radke. A Framework for Computational Morphology. Computational Geometry, ed. G. T. Toussaint, Elsevier, Amsterdam, 1985, pp. 217-248.

....on this function and reconstructs it with an appropriate algorithm. Our Contribution If the curve is closed, smooth, and uniformly sampled, several methods for the curve reconstruction problem are known to work ranging over minimum spanning trees [dFdMG95] # 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 first treated successfully by Amenta, Bern and Eppstein [ABE98] and subsequently improved algorithms such as [DK99, Gol99] appeared. Open non uniformly sampled ....

....we will be only concerned with reconstruction algorithms that come with a guarantee, i.e. under certain assumptions on # and the sample set S taken from #, they output the correct reconstruction as defined in section 1.1. The first algorithms like [dFdMG95] # shapes [BB97, EKS83] # skeleton [KR85], and r regular shapes [Att97] are known to work if the curve is closed, smooth and uniformly sampled, i.e. the distance between two adjacent samples must be less than some constant, which is determined by the most detailed area of the curve. A survey on these techniques appears in [Ede98] The ....

D. G. Kirkpatrick and J. D. Radke. A framework for computational morphology. In G. T. Toussaint, editor, Computational Geometry, pages 217--248. NorthHolland, Amsterdam, Netherlands, 1985.

....of radius 1=ff containing X; for ff 0, the intersection of all sets containing X which are complements of an open disk of radius Gamma1=ff. The authors derive from this construction some mathematical features related to the Delaunay triangulation of a finite cluster of points. See also [9] for an extension of this analysis. In fact, the ff hull is a translation invariant closing, and for ff 6= 0 it is an example of structural T closing, a concept that we will introduce in Section 2. The reader should consult Chapter 4 of [20] and optionally Chapters 17 and 18 of [22] for a ....

D.G. Kirkpatrick, J.D. Radke (1985). A framework for computational morphology, Computational Geometry, G.T. Toussaint ed., Elsevier Science Publ. B.V., Amsterdam, pp. 217--248.

....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 ....

D. G. Kirkpatrick and J. D. Radke. A framework for computational morphology. In "Computational Geometry", edited by G. Toussaint. North Holland, 217--248.

....of S are unavoidable. The number of unavoidable edges does not exceed 2n 2, see Xu [67] and usually is very small. Only in recent years, several less trivial subgraphs of MWT (S) have been identi ed. One of them arises from a class of empty neighborhood graphs introduced by Kirkpatrick and Radke [41]. Let p; q 2 S and 1. The edge pq is included in the skeleton, S) if the two discs of diameter jpqj and passing through both p and q are empty of points in S. It is not hard to see that (S) always is a subgraph of the Delaunay triangulation DT (S) In fact, S) can be constructed from ....

D.G.Kirkpatrick, J.D.Radke: `A framework for computational morphology', G.T.Toussaint (ed.): Computational Geometry, Elsevier, Amsterdam (1985) pp. 217-248

....of the two disks of radius d(u; v) one centered at u and the other centered at v does not contain any other element of P . An infinite family of proximity graphs, called fi skeleton graphs and including the Gabriel graph and the relative neighbourhood graph as special cases is defined in [6]. The fi skeleton of a set of points P is a geometric graph in which pairs of points are adjacent if and only if their fi neighbourhood is empty, i.e. it does not contain any other point of P . To each value of fi there corresponds a different fi neighbourhood; the lune based and circle based ....

....a set P of points is known, it is possible to scan all the fi skeletons of P by varying the value of fi from 0 to 1 and removing all edges (u; v) for which fi (u; v) is larger than the current value of fi. An O(n 3 ) time algorithm for computing the fi spectrum of a set of n points is given in [6]; the algorithm can be used both for the case that the fi neighbourhood is lune based and case that it is circle based. Very recently, an O(n 2 log n k) time algorithm for computing the lune based fi spectrum has been presented in [12] however, k is a parameter that depends upon the geometry ....

D. G. Kirkpatrick and J. D. Radke. A framework for computational morphology. In G. T. Toussaint, editor, Computational Geometry, pages 217--248. North-Holland, Amsterdam, Netherlands, 1985.

....problems at the moment. Moreover due to the angular conditions we can not expect the heuristic methods such as edge flipping and greedy methods work well for these new classes. On the other hand, recent research has revealed promising ways to determine large subgraphs, namely, the fi skeleton [12, 13] and the LMT skeleton [4, 5, 6] of the minimum weight triangulation. The experimental results show that these subgraphs are well connected for most of the point sets having relatively small sizes that are generated from uniform random distributions. Therefore they are useful for the design of ....

D. Kirkpatrick and J. Radke, "A framework for computational morphology", in G. Toussaint ed, Computational Geometry , pp. 217-248, Elsevier Science Publishers, 1985.

No context found.

D. G. Kirkpatrick and J. D. Radke. A framework for computational morphology. In G. T. Toussaint, editor, Computational Geometry, pages 217-248. North-Holland, 1985.

No context found.

D. Kirkpatrick and J. Radke, A framework for computational morphology, Compu- tational Geometry, ed. G. T. Toussaint (1985) 217-248.

....exists a point set whose minimum spanning tree is not a k spanner. At the other end of the spectrum, a Delaunay triangulation is guaranteed to be a 2:42spanner [11] For proximity graphs inbetween these two extremes, such as Gabriel graphs[8] relative neighborhood graphs[16] and skeletons[12] with 2 [0; 2] some interesting questions arise. We show that the spanning ratio for Gabriel graphs (which are skeletons with = 1) is ( in the worst case. For all skeletons with 2 [0; 1] we prove that the spanning ratio is at most O(n ) where = 1 log 2 (1 ) 2. For ....

....It is conjectured that the spanning ratio of the Delaunay triangulation is =2. The complete graph has S = 1, but is less interesting because the number of edges is not linear but quadratic in n. In this paper, we concentrate on the parameterized family of proximity graphs known as skeletons [12] with in the interval [0; 2] The family of graphs contains certain well known proximity graphs such as the Gabriel graph [8] when = 1 and the relative neighborhood graph [16] when = 2. As graphs become sparser, their spanning ratios increase. For example, it is trivial to show that there ....

[Article contains additional citation context not shown here]

D. G. Kirkpatrick and J. D. Radke. A framework for computational morphology. Computational Geometry, G. T. Toussaint, Elsevier, Amsterdam, 217-248, 1985.

No context found.

D. Kirkpatrick and J. Radke, A framework for computational morphology, Computational Geometry, ed. G. T. Toussaint (1985) 217-248.

No context found.

D. G. Kirkpatrick and J. D. Radke. A framework for computational morphology. In Computational Geometry, pages 217--248, North-Holland, 1985.

No context found.

David G. Kirkpatrick and John D. Radke. A framework for computational morphology. In Godfried T. Toussaint, editor, Computational Geometry, pages 217-248. North Holland, Amsterdam, 1985.

No context found.

D. G. Kirkpatrick, J. D. and Radke. A framework for computational morphology. Computational Geometry, G. Toussaint, ed., Elsevier, pp. 217-248.

No context found.

D. G. Kirkpatrick and J. D. Radke. A Framework for Computational Morphology. Computational Geometry, ed. G. T. Toussaint, Elsevier, Amsterdam, 1985, pp. 217248.

No context found.

D. G. Kirkpatrick and J. D. Radke. A framework for computational morphology. In G. Toussaint, editor, Computational Geometry, Elsvier, pages 217-248, 1985.

No context found.

D. G. Kirkpatrick, J. D. and Radke. A framework for computational morphology. Computational Geometry, G. Toussaint, ed., Elsevier, pp. 217-248.

No context found.

D. Kirkpatrick and J. Radke, "A framework for computational morphology", in G. Toussaint ed, Computational Geometry (Elsevier Science Publishers, 1985) 217-248.

First 50 documents  Next 50

Online articles have much greater impact   More about CiteSeer.IST   Add search form to your site   Submit documents   Feedback  

CiteSeer.IST - Copyright Penn State and NEC