Barber, C. B., Dobkin, D. P., & Huhdanpaa, H. T. (1996). The Quickhull algorithm for convex hulls. ACM Trans. on Mathematical Software, 22(4),469--483.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....known in its H representation. Step 2 can be implemented by rst computing the vertices of P , that is, its V representation, from its H representation, and then computing the convex hull of P from its V representation. An algorithm for this purpose is described by Barber, Dobkin, and Huhdanpaa [1]. B ueler, Enge, and Fukuda [3] provide an in depth survey of techniques and extensive remarks concerning polytope volume computation and related matters concerning their H and V representations. As was mentioned in the introduction, other algorithms for reconstruction from surface area ....

C. B. Barber, D. P. Dobkin, and H. Huhdanpaa, The quickhull algorithm for convex hulls, ACM Trans. Math. Software 22(1996), 469-483.

....lighter pixels are more salient. Below: the 5 differential convolution kernels, at one # value, used to create measures #x. because we rely on human vision, and simple not only to implement but also to use. We allow the user to point to clusters in order to group them, and compute the convex hull [34] of each group. We identify a salient feature as all pixels within the convex hull (each cluster is included in at most one feature, and a cluster need not be included in any feature) This mode of interaction is much simpler for the user than having to identify salient features from images ab ....

C. B. Barber, D. P. Dobkin, and H. T. Huhdanpaa, "The quickhull algorithm for convex hulls," ACM Transactions of Mathematical Software, , no. 22, pp. 469--483, 1996.

....pattern is constructed by creating a line segment between each pair of points (u,v) such that there exists a circle passing through u and v that encloses no other point. This characteristic simulates roadmaps. To construct Delaunay graphs for our experiments, we used the Qhull software package [1], which generates Delaunay tringulations on a square frame with unit size of 1. Figure 2 illustrates Delaunay graph with 15 nodes. In graphs built by Delaunay triangulation nodes are connected to the nodes that are near them. However, in roadmaps which are the object of this research, sometimes ....

C. B. Barber, D. P. Dobkin, and H. Huhdanpaa. The quickhull algorithm for convex hull. ACM Trans. on Mathematical Software, 1996.

....and simple triangulation, is a fundamental task in computational geometry. In the last few years some libraries like CGAL [3] and LEDA [7] provide implementations of algorithms for these data structures, and there are also popular packages in non library format such as triangle [8] or Qhull [1]. In this abstract we compare some of these packages with our implementation. This implementation is based on a prototype of a generic graph library under development at University Trier. We discuss in the following a number of di erent versions of Delaunay ipping based implementations and ....

C. B. Barber, D. P. Dobkin, and H. Huhdanpaa. The Quickhull algorithm for convex hulls. ACM Trans. Math. Softw., 22(4):469-483, Dec. 1996.

....output polytope P is also known in its H representation. Step 2 can be implemented by first computing the vertices of P , that is, its V representation, from its and then computing the convex hull of P from its V representation. An algorithm for this purpose is described by Barber et al. [1]. Bueler et al. 3] provide an in depth survey of techniques and extensive remarks concerning polytope volume computation and related matters concerning their H V representations. As was mentioned in the Introduction, other algorithms for reconstruction from surface area measures can be ....

....on n that there is an # net U # (n,#)of O(# ) points . Clearly there is a set of O(# 1 ) points in S that form an # net. Suppose that U # (n 1,# 2) is an (# 2) net in S ) points. Let X 1 j #: j 0, ### , and Y U (n 1,# 2) X . Then X is an (# 2) net in [ 1, 1], so Y is an # net in the cylinder S [ 1, 1] containing O(# ) points. The map of this cylinder into S that takes (1, u, z) to ( # 1 , u, z) where we use cylindrical coordinates, does not increase distances between points. The image U # (n,#)of Y under this map is therefore an # net ....

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C. B. Barber, D. P. Dobkin, and H. Huhdanpaa, The quickhull algorithm for convex hulls, ACM Trans. Math. Software 22 (1996), 469--483.

....The points were then projected down to a low dimension (two or three) without attempting to relax the distortions thus introduced. The resulting distance matrix was too large to store in memory, so its elements were recomputed when required. Delaunay triangulation was performed using qhull [3]. Delaunay neighbors were stored in a list structure. The Epanechnikov kernel (the tip of a paraboloid, thus with finite support) 57] was used for density estimation, and the bandwidth was chosen as three times the median of the nearestneighbor distances (this dependence on the number of ....

C. B. Barber, D. P. Dobkin, and H. Huhdanpaa. The quickhull algorithm for convex hulls. ACM Trans. Math. Soft., 22(4):469--483, 1996. http://www.geom.umn.edu/locate/qhull.

....known in its 74 representation. Step 2 can be implemented by first computing the vertices of P, that is, its F representation, from its 74 representation, and then computing the convex hull of P from its F representation. An algorithm for this purpose is described by Barber, Dobkin, and Huhdanpaa [1]. Bfieler, Enge, and Fukuda [3] provide an in depth survey of techniques and extensive re marks concerning polytope volume computation and related matters concerning their 74 and F representations. As was mentioned in the introduction, other algorithms for reconstruction from surface area ....

....by induction on n that there is an z net U(n,z) of O(z ) points in S . Clearly there is a set of O(z ) points in S that form an z net. Suppose that U(n 1,z 2) is an (z 2) net in S containing O(z ) points. Let X: l j j:0, and Y = U (n 1, z 2) x X. Then X is an (z 2) net in [ 1, 1], so Y is an z net in the cylinder S x [ 1, 1] containing O(z ) points. The map of this cylinder into S that takes (1, u, z) to (1 z,u, z) where we use cylindrical coordinates, does not increase distances between points. The image U(n,z) of Y under this map is therefore an z net in S ....

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C. B. Barber, D. P. Dobkin, and H. Huhdanpaa, The quickhull algorithm for convex hulls, ACM Trans. Math. Software 22(1996), 469-483.

....on to the head controller and structure results forward to the host Sun Workstation for display. As noted above, the structure results themselves are computed on a separate pipeline. The Structure processor computes explicit structure and derives the convex hull using the Quickhull algorithm [2]. 4.2 Control and head platform The platform used is the four axis device shown in Figure 4 with pan, elevation and two vergence axes driven by geared DC motors, and capable of both high acceleration and smooth slow speed performance. Visual position and velocity demands generated at 25 Hz are ....

C. Bradford Barber, David P. Dobkin, and Hannu Huhdanpaa. The quickhull algorithm for convex hull. Technical Report GCG53, Geometry Center, University of Minnesota, July 1993. ftp from geom.umn.edu.

....from these three sub components. We use the SWIFT implementation of the Voronoi marching technique [EL01] to efficiently perform the separation distance query. It performs distance queries between non convex polyhedra by using a hierarchy of convex hulls. We use the public domain QHULL package [BDH93] for convex hull computation in 3D. QHULL is particularly efficient for dealing with a relatively small number of points, which is the case in our algorithm. We use the QSlim implementation [GH97] of the quadric error metric simplification algorithm to ensure that the intermediate nodes of the ....

B. Barber, D. Dobkin, and H. Huhdanpaa. The quickhull algorithm for convex hull. Technical Report GCG53, The Geometry Center, MN, 1993.

....Figure 10: The transputer network used to implement the stereo tracking and structure calculation. On the left are typical times for the processes and communication between processes. The Structure transputer computes explicit structure and derives its convex hull using the Quickhull algorithm [20]. The Structure transputer also communicates with the host Sun workstation and controls all the pipelines. The Sync processor performs housekeeping tasks, ensuring that the images remain synchronized and providing time stamping information for the frame grabbers. 4.3 Experimental results We ....

Barber, C.B., Dobkin, D.P. & Huhdanpaa, H. (1993) The quickhull algorithm for convex hull. Technical Report GCG53, Geometry Center, University of Minnesota. (ftp from geom.umn.edu). 15

....marked Degeneracies. Another issue relevant to two of the performance measures is the computa tion of the convex hull. Rather than write a program to compute the convex hull specifically for this set of tools, an existing program was used. The Quickhull al gorithm as implemented by Barber [4] was used. An ACTS wrapper was coded for this routine that makes a list of vertices from an ACTS part and calls qhull. The results from qhull are then converted into an ACTS body. For average input, qhull is shown to have an expected complexity of O(nlogv) for the three dimensional case, where is ....

....from an ACTS part and calls qhull. The results from qhull are then converted into an ACTS body. For average input, qhull is shown to have an expected complexity of O(nlogv) for the three dimensional case, where is the number of input vertices and v is the number of vertices on the convex hull [4]. The complexity and the fact that qhull uses floating point, rather than integer or fixed point, vertices are the reasons qhull was chosen over other programs, such as hull, which uses an incremental algorithm discussed by Clarkson et al. 12] Note that if an extension is made to account for ....

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C. B. Barber, D. P. Dobkin, and H. Huhdanpaa. The Quickhull algorithm for convex hull. Technical Report GCG53, Geometry Center, Univ. of Minnesota, July 1993.

....for mesh generation, especially adaptive mesh generation. Point insertion to a Delaunay triangulation is considered as a basic operation, which is even named as Delaunay kernel by Freitag, Borouchaki and George in [26] in the paradigm of Delaunay mesh generation. Barber, Dopkin and Huhdanpaa[4] have provided the public state of the art implementation of Delaunay triangulation (including convex hull construction) in arbitrary dimensions, known as the qhull program. 4.2. 2d tree method. This is one of the earliest proposed and most popularly used methods in mesh generation. We give the ....

C. B. Barber, D. P. Dobkin and H. Huhdanpaa, "The Quickhull algorithm for convex hulls," A CM Trans. on Mathematical Software, Dec 1996.

....are given in [12] Convex Hull: The convex hull of a set of points is the smallest convex set containing those points. A number of algorithms are known in the literature to compute the algorithms in 2 D and 3 D [12] In our application, we use the Quickhull algorithm for computing convex hulls [2]. Its robust implementation is available as part of the Qhull public donhain package. Linear Programming: Geometrically, linear programming amounts to the following: given a set of half spaces and a vector W, compute a vertex v, in the common intersection of half spaces, that mizimizes v W. If ....

B. Barber, D. Dobkin, and H. Huhdanpaa. The quickhull algorithm for convex hull. Technical Report GCG53, The Geometry Center, MN, 1993.

....on a CM5. Our CM5 partition had 32 processors with 40 MB memory per processor. All timing results are obtained in multiuser mode. The sequential code for the lower envelope problem was a plane sweep algorithm which we implemented ourselves, and for the convex hull we used the Quickhull Algorithm [2]. The observed running times are shown in Figures 1 and 2, and in Table 1. Curves (a) and (b) in Figure 1 show the local computation times and communication times, respectively, for computing the lower envelope of a random set of non intersecting line segments (we will discuss the data generation ....

C. B. Barber, D. P. Dobkin, and H. Huhdanpaa, "The quickhull algorithm for convex hull," Tech. Report No. GCG53, The Geometry Center, University of Minnesota, Minneapolis, MN 55454, USA, 1993.

....Steps 2 0 and 3 0 can be implemented by first computing the vertices of P , that is, its V representation, from its H representation, and then computing the convex hull of P from its V representation. An algorithm for this purpose is described by Barber, Dobkin, and Huhdanpaa [1]. Bueler, Enge, and Fukuda [3] provide an in depth survey of techniques for polytope volume computation and related matters concerning their H and V representations. Note that the condition that the input directions span R n is not really necessary. If these vectors span a k dimensional ....

C. B. Barber, D. P. Dobkin, and H. Huhdanpaa, The quickhull algorithm for convex hulls, ACM Trans. Math. Software 22(1996), 469--483.

.... # ## # # ##### # ## # # ######### # ## # # #### # # # ### # # # # ## # # ### # # # ### # # # # ## # # ##### (5) where without loss of generality ## # # # # # and ## # # # # #, then we can readily show that all worthwhile algorithms and parameters fall on the ### dimensional convex hull [1]. Known parameters # and # of the convex fitness function define an iso fitness plane through the ## # dimensional fitness cost space such that all points on that plane have equal overall fitness. For example in the 2 D case of Figure 1 a fitness function ###### # # ### would define a horizontal ....

C. Bradford Barber, D. P. Dobkin, and H. Huhdanpaa. The Quickhull algorithm for convex hulls. ACM Trans. Mathematical Software, 22:469--483, 1996.

....to the constraints that z lie in the polytope formed by the intersection of all the hyperplanes found in (i) Any such solution z is a vertex of the cell, and since the w s are essentially random, the 130 solutions should include all the vertices of the cell. iii) The convex hull program QHULL [1], 15] was used to find the convex hull of these vertices. iv) The XGOBI program [26] for displaying multi dimensional data was used to help visualize the cells. To compute the volumes and second moments of the cells we decomposed the cells into simplices and used the formulae in [7] and [11, ....

C. B. Barber, D. P. Dobkin and H. T. Huhdanpaa, The Quickhull algorithm for convex hulls, ACM Trans. Mathematical Software, 22 (1996), 469--483.

....for the different sets of classifiers. We decided not to show all classifiers, only the most relevant classifiers with respect to the performance evaluation are shown. Provost has shown that a classifier is potentially optimal if and only if it lies on the northwest boundary of the convex hull [Barber, Dobkin and Huhdanpaa 1993] of the set of points in the ROC curve. From the figure we can see that the set of classifiers with the entropy based and cost based discretization method are potentially optimal. From a visual interpretation, we can rank the methods for the region with a FP rate lower than 35 as follows: entropy, ....

Barber C., Dobkin D., and Huhdanpaa H. (1993). The quickhull algorithm for convex hull. Technical Report GCG53, University of Minesota.

....t 1 25 5 5 25 58 The interface specification of the SHARC read cycle is shown in 2.5.17. Once the Figure 2.5.16 Timing diagram of the SHARC read cycle. timing parameter range (ns) t SACKC (C) 6, t HACKC (C) 1, t SSDATI (C) 3, t HSDATI (C) 2, t DAAK (C) 10] t DADRO (D) [0, 8] t HADRO (D) 0, t DRWL (D) 8, 13] t DRDO (D) 1, 4] t DARL (R) 2, t RW (R) 13, t RWR (R) 6, Table 2.5.3. Timing parameters for the 40 MHz version of the SHARC DSP (C: timing constraint; D: propagation delay; R: correlation data) t SACKC t HACKC t SSDATI t HSDAT I t DAAK ....

....of the SHARC read cycle is shown in 2.5.17. Once the Figure 2.5.16 Timing diagram of the SHARC read cycle. timing parameter range (ns) t SACKC (C) 6, t HACKC (C) 1, t SSDATI (C) 3, t HSDATI (C) 2, t DAAK (C) 10] t DADRO (D) 0, 8] t HADRO (D) 0, t DRWL (D) [8, 13] t DRDO (D) 1, 4] t DARL (R) 2, t RW (R) 13, t RWR (R) 6, Table 2.5.3. Timing parameters for the 40 MHz version of the SHARC DSP (C: timing constraint; D: propagation delay; R: correlation data) t SACKC t HACKC t SSDATI t HSDAT I t DAAK t DARL t RW t RWR t DADRO t HADRO ....

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C. B. Barber, D. P. Dobkin, and H. Huhdanpaa, "The quickhull algorithm for convex hulls", Technical report GCC53, The Geometry Center, Minnesota, U.S.A., 1993.

....bit rates obtained over all techniques described in Appendix A. The minimum rate and mean square error (MSE) together de ne a point in the rate MSE plane for each of the sixty candidate parameter combinations for each image. The convex hull of these points was determined using the qhull program [2] for each original image, and the parameter value combinations that appeared most frequently among vertices of the facets facing the origin were adopted for use in testing the lossless stage. The (n r ; n d ; ft m g) triples found in this way were: 5,3,3) 4,3,1) 3,3,2) 3,3,3) and (2,3,3) ....

C. Barber, D. Dobkin, and H. Huhdanpaa. The quickhull algorithm for convex hulls. ACM Trans. on Mathematical Software, December 1996. http://www.geom.umn.edu/software/qhull.

....complexes. In Section 3 we present a brief summary of the bulldozer algorithm, which is described more fully in a companion report [6] In Section 4 we evaluate the performance of the bulldozer algorithm on a variety of data sets, and compare it to the performance of the Minnesota Quickhull [2] implementation. In Section 5 we evaluate an implementation of Garland and Heckbert s terrain modeling algorithm [11] based on our representation of surfaces. 2 Surfaces Both the convex hull algorithm and the terrain modeling algorithm presented in later sections construct a two dimensional ....

....how the exterior point is chosen, and how the set of exterior points is maintained during the construction. During construction most of these algorithms maintain, for each exterior point, one face that is visible to that point. Clarkson and Shor s algorithm [8] the Minnesota Quickhull algorithm [2], the Motwani and Raghavan algorithm [15] and the Bulldozer algorithm described here, all maintain 7 such information. The motivation for keeping this association is that, since the set of faces that are visible to any one point is connected, knowing one visible face allows the algorithm to walk ....

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C. B. Barber, D. P. Dobkin, and H. T. Huhdanpaa. The quickhull algorithm for convex hulls. ACM Trans. on Mathematical Software, December 1996.

....and the difference is even more important on data set ellipse2 which is representative of real applications. 5.5. 3 Comparison with other software We have compared with some Delaunay softwares available on the WWW: ffl qhull by Bradford Barber and Hannu Huhdanpaa, duality with 3D convex hull [BDH93] (available at http: www.geom.umn.edu locate qhull) ffl div conquer by Jonathan Shewchuk, divide and conquer [She96] ffl sweep by Jonathan Shewchuk, plane sweep ffl incremental by Jonathan Shewchuk, incremental with Mcke et al. localization. These three codes supports exact arithmetic on ....

C. B. Barber, D. P. Dobkin, and H. Huhdanpaa. The Quickhull algorithm for convex hull. Technical Report GCG53, Geometry Center, Univ. of Minnesota, July 1993.

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Barber, C. B., Dobkin, D. P., & Huhdanpaa, H. T. (1996). The Quickhull algorithm for convex hulls. ACM Trans. on Mathematical Software, 22(4),469--483.

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C. B. Barber, D. P. Dobkin, and H. Huhdanpaa, "The quickhull algorithm for convex hull," Tech. Report No. GCG53, The Geometry Center, University of Minnesota, Minneapolis, MN 55454, USA, 1993.

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BARBER C., DOBKIN, D., HUHDANPAA, H., 1993. The Quickhull algorithm for convex hulls, ACM Transactions on Mathematical Software, 22, 4, 469-483.

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Barber, C. B., Dobkin, D. P., & Huhdanpaa, H. T. (1996). The quickhull algorithm for convex hulls. Acm trans. on mathematical software, Dec.

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C.B. Barber, D.P. Dobkin, and H.T. Huhdanpaa. The quickhull algorithm for convex hulls. ACM Trans. on Mathematical Software, 22(4):469--483, 1996. Hanna Kurniawati and David Hsu

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Barber, C.B., D.P. Dobkin, H.T. Huhdanpaa. The Quickhull Algorithm for Convex Hull. In ACM Trans. on Mathematical Software, no.22, pp.469-483, 1996.

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C.B. Barber, D.P. Dobkin, and H. Huhdanpaa. The quickhull algorithm for convex hull. Technical Report GCG53, The Geometry Center, University of Minnesota, Minneapolis, 1993.

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Barber, C. B., Dobkin, D. P., & Huhdanpaa, H. T. (1996). The Quickhull algorithm for convex hulls. ACM Trans. on Mathematical Software, 22(4),469--483.

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Barber, C. B., Dobkin, D. P., and Huhdanpaa, H. The Quickhull algorithm for convex hulls. ACM Trans. Math. Softw. 22, 4 (Dec. 1996), 469--483.

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C. B. Barber, D. P. Dobkin, and H. Huhdanpaa, "The quickhull algorithm for convex hull," Tech. Report No. GCG53, The Geometry Center, University of Minnesota, Minneapolis, MN 55454, USA, 1993.

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C. B. Barber, D. P. Dobkin, and H. Huhdanpaa, "The quickhull algorithm for convex hull," Tech. Report No. GCG53, The Geometry Center, University of Minnesota, Minneapolis, MN 55454, USA, 1993.

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C. B. Barber, D. P. Dobkin, and H. Huhdanpaa. The Quickhull algorithm for convex hulls. ACM Trans. Math. Softw., 22(4):469--483, 1996.

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C. B. Barber, D. P. Dobkin, and H. Huhdanpaa. The Quickhull algorithm for convex hull. Technical Report GCG53, Geometry Center, Univ. of Minnesota, July 1993.

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C. Bradford Barber, David P. Dobkin, and Hannu Huhdanpaa. The quickhull algorithm for convex hulls. ACM Trans. Math. Softw., 22 (4):469--483, December 1996.

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C. B. Barber, D. P. Dobkin, and H. Huhdanpaa. The Quickhull algorithm for convex hull. Technical Report GCG53, Geometry Center, Univ. of Minnesota, July 1993.

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Barber, C. B., Dobkin, D. P., and Huhdanpaa, H. The Quickhull algorithm for convex hulls. ACM Trans. Math. Softw. 22, 4 (Dec. 1996), 469--483.

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C. B. Barber, D. P. Dobkin, and H. P. Huhdanpaa, "The quickhull algorithm for convex hulls," ACM Trans. Mathematical Software, vol. 22, no. 4, pp. 469--483, 1996.

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Barber C., Dobkin D., and Huhdanpaa H. (1993). The quickhull algorithm for convex hull. Technical Report GCG53, University of Minesota.

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C.B. Barber, D.P. Dobkin, H.T. Huhdanpaa, The quickhull algorithm for convex hulls, ACM Trans. on Mathematical Software 22 (1996) 469-483. URL: http://www.geom.umn.edu/software/qhull/.

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C. B. Barber, D. P. Dobkin, and H. Huhdanpaa. The Quickhull algorithm for convex hulls. ACM Trans. Math. Softw., 22(4):469--483, 1996.

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C. B. Barber, D. P. Dobkin, and H. Huhdanpaa, "The quickhull algorithm for convex hull," Tech. Report No. GCG53, The Geometry Center, University of Minnesota, Minneapolis, MN 55454, USA, 1993.

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B. Barber, D. Dobkin, and H. Huhdanpaa. The quickhull algorithm for convex hull. Technical Report GCG53, The Geometry Center MN, 1993.

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Barber, C. B., Dobkin, D. P., and Huhdanpaa, H. The Quickhull algorithm for convex hulls. ACM Trans. Math. Softw. 22, 4 (Dec. 1996), 469--483.

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C. B. Barber, D. P. Dobkin, and H. Huhdanpaa. The quickhull algorithm for convex hulls. ACM Trans. Math. Softw., 22(4):469--483, Dec. 1996.

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C.B. Barber, D.P Dobkin, H. Huhdanpaa, "The quickhull algorithm for convex hulls", ACM Transactions on mathematical software, 22(4):469-483, December 1996.

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Barber, C. B., Dobkin, D. P., and Huhdanpaa, H. (1996). The Quickhull Algorithm for Convex Hulls. ACM Transactions on Mathematical Software, 22(4):469-483. (Technical Report GCG53, University of Minesota).

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C.B. Barber, D.P. Dobkin, and H.T Huhdanpaa. The quickhull algorithm for convex hulls. In ACM Transactions on Mathematical Software, 1996.

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C. B. Barber, D. P. Dobkin, and H. Huhdanpaa. The Quickhull algorithm for convex hulls. ACM Trans. Math. Softw. 22(4):469-483, Dec. 1996.

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