Shmuel Rippa. Minimal Roughness Property of the Delaunay Triangulation. Computer Aided Geometric Design 7(6):489--497, November 1990.  Home/Search   Document Not in Database   Summary   Related Articles   Check

.... of any element does not properly contain any other node, if each node is associated with a number, and f(x) is the linear finite element interpolation function derived from these numbers and the triangulation, then the Delaunay triangulation minimizes I = f) 2 6 This latter property [17] implies a certain optimality in solving elliptic systems with the mesh: if we solve Laplaces equation with linear finite elements, we are minimizing I with respect to the field values at the nodes, and the use of the Delaunay rather than another triangulation additionally minimizes I with ....

S. Rippa, Minimal Roughness Property of the Delaunay Triangulation, PhD thesis, Tel-Aviv University, 1990.

....Similarly, fatness may be de ned as the sum of triangle inradii. Lambert [44] pointed out that DT (S) maximizes fatness or, equivalently, the mean inradius. Triangular surfaces obtained from lifting DT (S) to 3 space minimize roughness, which is the integral of the gradient squared; see Rippa [60]. It is also known that a variant of DT (S) minimizes the minimum angle; see Eppstein [29] combinatorial properties Delaunay triangulation local improvement le: optriang date: November 15, 1999 2 The Delaunay triangulation is a special instance of regular triangulations, which are obtained ....

S.Rippa: `Minimal roughness property of the Delaunay triangulation', Computer Aided Geometric Design 7 (1990) pp. 489-497

....of the polygons creating very elongated triangles close to the borders that may bring numerical errors during rendering. This problem may be overcome [DeFlo 96a] but the result will never represent a global optimal solution like the one given by a single Delaunay triangulation of the whole domain [Rippa90]. Models where the refinement process takes into account the entire domain may not preserve the hierarchical relation among polygons belonging to distinct layers. In such models each layer covers the entire domain and it requires much storage due to the replication of triangles that does not ....

....into each facet the point p with the maximum approximation error that falls inside the facet and by inserting also the edges that connect p to the vertices of that facet. After the point insertion, a modified Delaunay triangulation is performed to minimize the roughness of the approximation [Rippa90]. The root is a degenerated node that points to the coarsest approximation of the terrain. Each layer built this way is more refined that the previous one and the most refined layer corresponds to the same approximation given by D. By construction, every facet induced by the model is a triangle. ....

- Rippa, S., "Minimal Roughness Property of the Delaunay Triangulation", Computer Aided Geometric Design 7:293-301.

....call it the slope of xyz. For a triangulation A of S define oe(A) maxfoe(xyz) j xyz a triangle of Ag, as usual. A minmax slope triangulation of S minimizes the maximum oe of any triangle. Triangulations are commonly used to compute surfaces interpolating point set data with elevations. Rippa [Rip90] recently proved that, regardless of elevations, the Delaunay triangulation minimizes the integral (over the convex hull of S) of r 2 f among all triangulations of S. See [DLR90] for other interesting optimization criteria. The five point example of Figure 6.1 again shows that the edge flipping ....

S. Rippa. Minimal roughness property of the Delaunay triangulation. Computer Aided Geometric Design 7 (1990), 489--497.

....of a Structured Grid data structures used for searching can bring the costs down close to O (NlogN ) The grid created is usually smoothed with a Laplacian filter, resulting in a grid with a high degree of regularity. Another sophisticated grid generation scheme is a Delaunay triangulation [3, 25, 34, 48, 49]. Delaunay alone is not a complete scheme, but only a method of triangulating a given set of points. One way to automate the introduction of points and create a Delaunay triangulation is first to triangulate the boundary nodes [23] The Delaunay triangulation of these points is taken as the ....

S. Rippa, Minimal Roughness Property of the Delaunay Triangulation. PhD thesis, Tel-Aviv University, 1989.

....Interpolation for Real and Distant Image Pairs 9 The joint view triangulation for two views is a pair of inter related image triangulations, one for each image, based on an underlying locally dense displacement map. The Delaunay triangulation is chosen because of its minimal roughness property [Rip90]. Triangulating in image space allows non rigid scenes to be handled. We call matched triangle (resp. unmatched triangle) a triangle which covers a region of matched (resp. unmatched) pixels in its image. Matched and unmatched triangles are separated by constrained edges. If we assume that the ....

D. Rippa. Minimal roughness property of the Delaunay triangulation. Computer Aided Geometric Design, 7: 489--497, 1990.

.... of the two dimensional Delaunay triangulation is partly due to the fact that it optimizes various quality measures, including the smallest angle [Sibs78] the largest circumscribed circle [D AS89] the largest minimum enclosing circle [D AS89, Raja91] and the integral of the gradient squares [Ripp90]. Algorithms that construct the Delaunay triangulation of a given set of n points in the plane in time O(n log n) can be found in Guibas, Stolfi [GuSt85] Fortune [Fort87] Guibas et al. GKS90] and other publications in computational geometry [PrSh85, Edel87] In practical applications it is ....

S. Rippa. Minimal roughness property of the Delaunay triangulation. Computer Aided Geometric Design 7 (1990), 489--497.

....shown that among all triangulations of a given point set, the Delaunay triangulation optimizes various criteria. These include the maxmin angle [Sib78] the minmax circumscribed circle [D AS89] the minmax smallest enclosing circle [D AS89, Raj91] and the minimum integral of the gradient squared [Rip90]. Efficient algorithms for constructing Delaunay triangulations are abundant in the literature and based on such diverse algorithmic paradigms as edge flipping [Laws72, Laws77] divide andconquer [ShHo75, GuSt85] geometric transformation [Brow79] plane sweep [For87] and randomized ....

S. Rippa. Minimal roughness property of the Delaunay triangulation. Computer Aided Geometric Design 7 (1990), 489--497.

.... [GKS92] Among all triangulations of a given point set S, the Delaunay triangulation optimizes criteria such as the max min angle [Sibs78] the min max circumscribed circle [D AS89] the min max smallest enclosing circle [D AS89, Raja91] and the minimum integral of the gradient squared [Ripp90]. As a graph structure, the Delaunay triangulation of S, D(S) is the straight line dual of the so called Voronoi diagram [Voro07, Voro08, Aure91] Various subgraphs of D(S) have been studied in the literature. Three of those subgraphs, satisfying mst(S) rng(S) gg(S) D(S) will now be ....

S. Rippa. Minimal roughness property of the Delaunay triangulation. Comput. Aided Geom. Design 7 (1990), 489--497.

....Triangulating in image space allows non rigid scenes to be handled. We call matched triangle (resp. unmatched triangle) a triangle which approximately covers a region of matched (resp. unmatched) pixels in its image. The Delaunay triangulation is chosen because of its good uniformity properties [15]. Matched and unmatched triangles are separated by constrained edges, which are forced to be part of the triangulation. If we assume that the displacement map is such that half occluded areas coincide with unmatched ones (ideal matching case) the condition 2 is satisfied. The contours are the ....

D. Rippa. Minimal roughness property of the Delaunay triangulation. Computer Aided Geometric Design, volume 7, pages 489--497, 1990.

....reversed quadrilaterals must be Delaunay. Thus Delaunay triangulation maximizes the minimum angle, along with optimizing a number of more esoteric quality measures, such as maximum circumcircle radius, maximum enclosing circle radius, and roughness of a piecewise linear interpolating surface [105]. 20 e b Ear a c Figure 16. Edge insertion retriangulates holes by removing sufficiently good ears. Dotted lines indicate the old triangulation. As mentioned in Section 5.5, edge flipping can also be used as a general optimization heuristic. For example, edge flipping works reasonably well ....

S. Rippa. Minimal roughness property of the Delaunay triangulation. Computer Aided Geometric Design, 7:489--497, 1990.

....piecewise linear surface defined by linear interpolation over each triangle. Over all possible triangulations, the Delaunay triangulation minimizes the roughness of the resulting surface, where 9 roughness is the L 2 norm squared of the gradient of the surface, integrated over the triangulation[27]. 6 Further reading Survey papers by Aurenhammer[5] and Fortune[15] cover many aspects of Delaunay triangulations and Voronoi diagrams. The book by Okabe, Boots, and Sugihara[24] is entirely devoted to Voronoi diagrams, and has an extensive discussion of applications. Basic reference for ....

S. Rippa, Minimal roughness property of the Delaunay triangulation, Comp. Aided. Design 7:489--497, 1990.

....The Delaunay triangulation optimizes many triangulation measures. These include ffl maximizing the minimum angle [20] ffl minimizing the maximum circumscribed circle [5] ffl minimizing the maximum smallest enclosing circle 1 [5, 17] ffl minimizing the integral of the gradient squared [18, 16], Little [14] and Schumaker [19] have proposed that the triangles in a good triangulation should have large inradii. I will prove that the Delaunay triangulation maximizes the sum of the inradii (and hence the arithmetic mean) 2 Preliminaries I use R(ABC) to denote the circumradius of the ....

Samuel Rippa. Minimal roughness property of the Delaunay triangulation. Computer Aided Geometric Design, 7:489--497, 1990.

....be a powerful tool of efficiently representing and restructuring individual image or range data. However, to our knowledge, no one has yet tried to deal with the similar representation for multiple views. The triangulation in each image will be Delaunay because of its minimal roughness properties [19]. The Delaunay triangulation will be necessarily constrained as we want to separate the matched regions from the unmatched ones. The boundaries of the connected components of the matched planar patches of the image must appear in both images, therefore are the constraints for each Delaunay ....

D. Rippa. Minimal roughness property of the delaunay triangulation. CAGD, 7, 1990.

....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 parameter ff might be tricky. Figure 1: ff shapes Although a large amount of work has been done on this topic, few theoretical results [9] exist mainly due to the difficulty formalizing the problem. In most papers, no mathematical definition of the searched mesh is given. In section 2, we try to fill this deficiency by formulating the problem more precisely. In particular, the searched solution is defined as a particular mesh of the ....

S. Rippa. Minimal roughness property of the delaunay triangulation. Computer Aided Geometric Design, 7:489--497, 1990.

....gives in general a good approximation because it connects the points by 2 This assumes the data points are in general position, otherwise some extra edges have to be added. Figure 3: Perspective view of a triangular irregular network and the triangulation of its data points. proximity. Rippa [Rip90] showed that when the Delaunay triangulation is used to interpolate a bivariate function it minimizes the roughness of the interpolation. In some cases specific data dependent triangulation can give better results, though. Dyn et al. DLR90] examine different optimization criteria. Scarlatos and ....

S. Rippa. Minimal roughness property of the Delaunay triangulation. Comput. Aided Geom. Design, 7:489--497, 1990.

....over a suitable space of functions. One example of such functional is the roughness, defined by the Sobolev semi norm of the function. Surprisingly, the surface with minimum roughness is always given by the Delaunay triangulation of projected points, independently of the z values of its vertices [Rip90]. When the data set al..so includes a set of line segments, it is also important that such segments appear as either edges, or chains of edges in the TIN. In this case, either constrained [Lee86, Sch87] or conforming [Ede93, Saa91] triangulations are used. A constrained triangulation T of a set V of ....

....data points by computing a triangulation having vertices at data points. In case raw data also include line segments, a constrained or a conforming triangulation is computed. Thus, the basis for TIN construction are algorithms for building a Delaunay triangulation, a data dependent triangulation [Dyn90, Rip90], a constrained Delaunay [Lee86] and a conforming Delaunay triangulation [Ede93, Saa91] The interested reader is referred to Chapter 9 , and to [Aur91] for further details. 9 REFER HERE THE CHAPTER BY KLEIN AND AURENHAMMER ON VORONOI DIAGRAMS 5. TIN from RSG. This conversion is usually ....

Rippa, S., Minimal roughness property of Delaunay triangulation, Computer Aided Geometric Design, 7, 1990, pp.489-497.

....for encoding them, and the algorithms for their construction and analysis are very general, and can be applied to build different hierarchical models for specific applications. The Delaunay triangulation, which we adopted in our model, has some desirable properties (e.g. minimal roughness [Rip90]) that make it suitable for many applications. However, some researchers recommend using data dependent triangulations for surface representation, since they are more adaptive to sampled data [Rip92] a model based on such triangulations can be obtained by simply modifying the update procedure in ....

Rippa, S., "Minimal roughness property of Delaunay triangulation", Computer Aided Geometric Design, 7, 1990, pp.489-497.

.... main problem in parametric interpolation [Franke (1982) Schumaker (1976) even though structuring is not entirely trivial because some kind of triangulation of the domain must be chosen (see Figure 10) The challenge is to find triangulations on which good interpolants can be based [Dyn Levin Rippa (1990), Brown (1991) for instance, Delaunay triangulations are optimal for piecewise linear interpolants, with respect to several natural criteria [Omohundro (1990) Rippa (1990) the triangulation of the domain in Figure 10 is a Delaunay triangulation. Parametric and non parametric scattered data ....

.... must be chosen (see Figure 10) The challenge is to find triangulations on which good interpolants can be based [Dyn Levin Rippa (1990) Brown (1991) for instance, Delaunay triangulations are optimal for piecewise linear interpolants, with respect to several natural criteria [Omohundro (1990) Rippa (1990)] the triangulation of the domain in Figure 10 is a Delaunay triangulation. Parametric and non parametric scattered data interpolation further differ in their density requirements for samples: samples need to be denser and more uniformly distributed for non parametric interpolation than for ....

S. Rippa, Minimal roughness property of the Delaunay triangulation, Computer Aided Geometric Design 7 (1990) 489--497.

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Shmuel Rippa. Minimal Roughness Property of the Delaunay Triangulation. Computer Aided Geometric Design 7(6):489--497, November 1990.

No context found.

D. Rippa. Minimal roughness property of the delaunay triangulation. Computer Aided Geometric Design, 7:489--497, 1990.

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S. Rippa, "Minimal Roughness Property of the Delaunay Triangulation," Computer Aided Geometric Design, vol. 7, pp. 489--497, 1990.

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Rippa, S., "Minimal Roughness Property of the Delaunay Triangulation", CAGD, Vol. 7, No. 6., 1990, pp-489--497.

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Samuel Rippa. Minimal roughness property of the Delaunay triangulation. Comput. Aided Geom. Design, 7:489--497, 1990.

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