F. P. PREPARATA AND M. I. SHAMOS, Computational Geometry. Texts and Monographs in Computer Science, Springer-Verlag, New York, 1985.  Home/Search   Document Not in Database   Summary   Related Articles   Check

.... 5 6(a) For n points x 1 ; x n 2 IR d , the convex hull is conv(x 1 ; x n ) n 1 x 1 : n x n fi fi fi i 0; n X i=1 i = 1 o (see Figure 6(b) Or, alternatively: The convex hull of a set of points is the smallest convex set containing these points [32]. The affine hull of n points x 1 ; x n 2 IR d is aff(x 1 ; x n ) n 1 x 1 : n x n fi fi fi n X i=1 i = 1 o : x y (a) x y (b) Figure 5: A convex (a) and a non convex set (b) 8 (a) b) Figure 6: Convex (a) and affine (b) hull of two points in ....

F.P. Preparata, M.I. Shamos: Computational Geometry, An Introduction. Texts and Monographs in Computer Science, Springer-Verlag, New-York, 1985.

....as the problem of search for the closest neighbour on the left in a bounded linear universe (cf. 15] The closest neighbour problem has been extensively studied in computational geometry. Therefore the majority of solutions are considering at least two dimensions and the continuous domain (cf. [32, 85]) A one dimensional problem can be also considered as a non overlapping interval sets problem and with this problem associated union split find problem ( 63] There is an efficient O(log log M ) data structure due to van Emde Boas et al. which solves the problem ( 34] and matches the lower ....

F.P. Preparata and M.I. Shamos. Computational Geometry. Texts and Monographs in Computer Science. Springer-Verlag, Berlin, 2 nd edition, 1985.

.... the left) and a five element subset that has to be considered (on the right) # # # # # do not consider # # # # # consider Figure 6: Bounding Heuristic Algorithms for the incremental construction of the convex hull can be found in Preparata and Shamos [6]. We call the resulting sweepline algorithm MCD SW. See figure 7. It can be combined with the first branch and bound algorithm MCD BB in order to skip more subtrees. The disadvantage of this combination, denoted by MCDSWBB (figure 8) is the necessity to compute all the d i , i 2 f1; kg, ....

Franco P. Preparata, Michael I. Shamos. Computational geometry. Texts and monographs in computer science. Springer Verlag, Corr. and expanded 2. print, 1988. ISBN 0-387-96131-3. 20

....vertex is created on each of the sides of an existing triangle, or by a 1 side split [34] 35] in which an existing triangle is split into two by inserting a line from a vertex of the triangle to a point on the opposite side. An alternative triangular partition is based on a Delaunay triangulation [36] of the image, which is constructed on an initial set of seed points , and is adapted to the image by adding extra seed points in regions of high image variance [37] 38] 39] Polygonal partitions have been constructed by recursive subdivision of an initial coarse grid by the insertion of line ....

F. P. Preparata and M. I. Shamos, Computational Geometry, Texts and Monographs in Computer Science. Springer-Verlag, New York, NY, USA, 1985.

....p m nodes n leaves oe 3 oe 4 oe 2 oe 1 oe 0 m nodes E A B C D F Fig. 3. An external segment tree based on a set of N segments, three of which, AB, EF and EF , are shown. where a list of segments is associated with each internal node. Each segment is stored in O(log N) such lists. See e.g. [5, 17] for a definition of the segment tree and the operations on it. Because a segment can be stored in O(log N) nodes, the technique sketched in section 2, where we just group the nodes in an internal version of the structure into super nodes, does not apply directly. Instead we need to change the ....

F. Preparata, M. Shamos: Computational Geometry, An Introduction. Text and Monographs in Computer Science, Springer-Verlag 1985.

....(2) uses a radial function Phi( Delta) OE(k Delta k 2 ) with OE compactly supported on [0; ae] ae IR 0 , the evaluation of s(x) for a given x 2 IR d requires quick access to all x j with kx Gamma x j k 2 ae. This is a variation of the k nearest neighbors problem of computational geometry [27]. If evaluation of s(x) for a very large number of points x of a compact set Omega ae IR d is required, a preprocessing step using a space decomposition technique should be implemented. Such things are common practice in Computer Graphics software, for instance to speed up ray tracing ....

Preparata, F. P., and Shamos, M. I., Computational Geometry, Texts and Monographs in Computer Science, D. Gries (ed.), Springer, New York, 1985.

.... Diagrams Substantially distinct skeletonization algorithms have been proposed in literature [29, 26, 24, 17, 6] The approach [15, 29, 26, 30] employed in this paper is based on the close relationship between the medial axis transform, distance transforms [41] DT) and the Voronoi diagram (VD) [34] of a shape s boundary points. Voronoi tessellation: Given a set of points (sites) Omega = fp i g in the plane, the (possibly unbounded) convex polygonal region V (i) containing only site p i , the Voronoi polygon of p i , is defined as the set of all points lying closer to p i than to any other ....

....measure. Other metrics such as the L 1 metric (Manhattan distance) or L1 distance (Chessboard metric) could be used as well. Throughout this paper, however, only the Euclidean distance will be employed. z A detailed discussion of further properties of Voronoi diagrams can be found in references [34] or [9] fig.anchor.ps 70 Theta 56 mm p i p j w ij e Voronoi diagram SIZ( B U boundary point B L SIZ( B U B L Fig. 1. Computing a saliency measure from boundary distances. The distance w ij between the generators of Voronoi edge e, p i and p j , defines a simple significance ....

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F. P. Preparata and M. I. Shamos, Computational Geometry, Texts and Monographs in Computer Science, Springer-Verlag, New York, second edition, 1990.

....step d, the last point d is added to . Since d 2 I , I . A note on the running time: although the usual complexity of a convex hull algorithm is O(d log d) for d points in the plane, the complexity is smaller when those points are ordered like ours: i; r(i) Compare with Theorem 4. 12 in [28]) Algorithm 2 has a running time of O(d) operations (including a xed number of transcendental operations) Indeed, each point is added to the list precisely one time. It can be discarded only once, so the interior while loop is executed at most d 1 times in one execution of the algorithm. 4 ....

F. P. Preparata and M. I. Shamos, Computational geometry, Texts and Monographs in Computer Science, Springer-Verlag, New York, 1985. An introduction.

....problem posed by Atallah et al. 1 Introduction Triangulations belong to the most useful structures in computational geometry. Several different kinds of triangulations in the plane and higher dimensions have been studied. For many of them efficient polynomial time algorithms have been derived [2, 15]. Since triangulations, and, in particular, planar triangulations are so useful it is natural to ask whether fast and processor feasible parallel algorithms for their construction are available. Often, by a fast and processor feasible algorithm one means an NC algorithm, i.e. an algorithm running ....

....Lund, Sweden. y Department of Computer Science, Memorial University of Newfoundland, St. John s, Canada A1C 5S7. z It shouldn t be mixed with the known sweep line triangulation method which consists in decomposing the input configuration into monotone polygons which are triangulated separately [15]. form a simple polygon (i.e. a polygon without holes) the sweep line triangulation could be constructed by an NC algorithm. In this paper, we consider among others one of the classical triangulations, called the greedy triangulation. It is obtained by repeatedly inserting a shortest diagonal ....

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F.P. Preparata and M.I. Shamos. Computational Geometry, An Introduction. Texts and Monographs in Computer Science, Springer Verlag, New York, 1985.

....the feature space to a finite interval, E uniform e not necessarily increases. In a naive implementation, the complexity for computing E uniform e given a set of k feature vectors is O(k 2 ) However, there are algorithms which find the closest pair of feature vectors in O(k log k) steps [ Preparata and Shamos, 1985 ] Due to the min function, E uniform e is not a continuous differentiable criterion. For each weight update, only the closest pair of feature vectors is considered. However, for determining the closest pair, all elements of the sample have to be evaluated (encoded) This seems a very ....

F. P. Preparata and M. I. Shamos. Computational Geometry, An Introduction. Texts and Monographs in Computer Science. Springer-Verlag, 1985.

....If the query point is a member of the given set then it will be the solution, and if two or more points are of equal distance from the query point we choose one of them arbitrarily. This problem arises in many areas such as modeling of robot arm movements and integrated circuits layouts (cf. [26]) The problem has been heavily studied in the IR 2 contiguous domain where it is solved using Voronoi diagrams (cf. 29] Furthermore, the problem can be generalized by considering the points as multidimensional records in which individual fields are drawn from an ordered domain (cf. 22] In ....

....a data structure may change. 2. 1 The Literature Background Finding the closest element in a set to a query element is an important problem arising in many subfields of computer science, including computational geometry, pattern recognition, VLSI design, data compression and learning theory (cf. [11, 22, 26]) As noted, there are several versions of the problem. Clearly, the number of dimensions, d, the distance norm, typically L 2 , L 1 or L1 , and model of computation impact the appropriate choice of methods. First consider continuous searching space (domain) IR. In a one dimensional case all ....

F.P. Preparata and M.I. Shamos. Computational Geometry. Texts and Monographs in Computer Science. Springer-Verlag, Berlin, 2 nd edition, 1985.

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F. P. PREPARATA AND M. I. SHAMOS, Computational Geometry. Texts and Monographs in Computer Science, Springer-Verlag, New York, 1985.

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F. P. Preparata and M. I. Shamos, Computational Geometry, Texts and Monographs in Computer Science, Springer-Verlag, New York, second edition, (1990).

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F. P. Preparata and M. I. Shamos, Computational Geometry, Texts and Monographs in Computer Science, Springer-Verlag, New York, 1990.

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F. P. Preparata and M. I. Shamos. Computational Geometry. Texts and Monographs in Computer Science. Springer Verlag, 1985.

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F. P. Preparata and M. I. Shamos, Computational Geometry, Texts and Monographs in Computer Science, Springer-Verlag, New York, second edition, (1990).

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F. P. Preparata and M. I. Shamos, Computational Geometry, Texts and Monographs in Computer Science, Springer-Verlag, New York, second edition, (1990).

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