Nira Dyn, David Levin, and Shmuel Rippa. Data dependent triangulations for piecewise linear interpolation. IMA Journal on Numerical Analysis, 10(1):137--154, January 1990.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....hyperbolic paraboloid (gure ) Choosing a particular triangulation of S gives a particular piecewise linear interpolation of the surface. The following denitions are the classic ones taken from the papers concerning data dependent triangulations (especially the papers from Dyn, Levin and Rippa [DLR90b], DLR90a] or Brown [Bro91] Given an interior edge of the triangulation T , there are two triangles which have as common edge. These two triangles dene a quadrilateral Q. denition 1 An interior edge of T is locally optimal with respect to a given cost function if one of the following ....

N. Dyn, D. Levin, and S. Rippa. Data dependent triangulations for piecewise linear interpolation. IMA Journal of Numerical Analysis, 10:137154, 1990.

....be carried out before the actual surface reconstruction. Since reverse offsetting 2 Counterprobing Triangulation Figure 2: Simulated counterprobing. must start from some surface, we use a triangulation C 0 of the measured centre points. Among various data dependent triangulations (see, e.g. [DLR90], the authors have found that those minimizing total absolute curvature (cf. vDA95] yield the best approximations of the centre surface. Simulated Counterprobing The centre surface C is created through the mechanical offsetting intrinsic to tactile digitization. In principle, one can form the ....

Nira Dyn, David Levin, and Shmuel Rippa. Data dependent triangulation for piecewise linear interpolation. IMA Journal of Numerical Analysis, 10:137--154, 1990.

....if the height information (z coordinate) is not taken into consideration. Figures 3 and 4 illustrate the di erence between a non favorable and a favorable triangulation. Several data dependent triangulation schemes have been proposed in the past which take into account spatial information [3, 4]. A major criterion is curvature. In particular, a surface might have sharp edges or ridges which need special 1 treatment like demonstrated for a di erent kind of sampling by Kobbelt et al. 8] In this case a typical approach is to decompose the surface into segments which are free of those ....

Dyn, N., Levin, D., Rippa, S.: Data Dependent Triangulations for Piecewise Linear Interpolation. IMA Journal of Numerical Analysis 10 (1990) 137-154.

....this property can be transmitted to S, providing an interesting way of optimizing surface triangulations. Such optimizations are usually based on recursively swapping edges in an existing one according to some goodness criterion as described by Schumaker [21] and Dyn, Levin, and Rippa [5]. Moreover replacing P by P may have a beneficial effect on the surface approximation when it is based on piecewise polynomials on P. In Figure 17 the Delaunay triangulation P, which maximizes the minimum angle of its triangles, was computed for the u i appearing in Figure 15. The ....

Dyn N., D. Levin, S. Rippa, Data dependent triangulations for piecewise linear interpolation, IMA J. Numer. Anal. 10 (1990), 137--154.

....straight line planar graph with vertex set S. Each bounded face is a triangle, and the triangulation includes the boundary of the convex hull. Triangulations find use in areas such as finite element analysis [BeEp92, StFi73] computational geometry [PrSh85] and surface approximation [DLR90]. Applications typically require triangulations with well shaped triangles, meaning for example that triangles with very small or large angles should be avoided. Taking a worst case approach, one can define the quality of a triangulation to be the quality of its worst triangle. Interesting ....

....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 strategy does not in Edge Insertion for Optimal Triangulations 14 general compute a minmax slope triangulation. Just imagine that points a; b; c; d; e are not perturbed and thus ....

N. Dyn, D. Levin and S. Rippa. Data dependent triangulations for piecewise linear interpolation. IMA J. Numer. Anal. 10 (1990), 137--154.

....obvious choice for the smoothing prior on the coefficients given a fixed triangulation. As mentioned above, the Sobolev semi norm (10) and its connection with the Delaunay triangulation led to the creation of several techniques for measuring the smoothness of a continuous, piecewise linear surface (Dyn et al. 1990ab) One class of criteria measures how near the fit is to a plane. Typically, these are edge based, compiling a roughness penalty along edges in 4. For example, Dyn et al. 1990ab) accumulate the jump in the normal derivative across each edge. Measuring the squared differences yields a quadratic ....

....led to the creation of several techniques for measuring the smoothness of a continuous, piecewise linear surface (Dyn et al. 1990ab) One class of criteria measures how near the fit is to a plane. Typically, these are edge based, compiling a roughness penalty along edges in 4. For example, Dyn et al. 1990ab) accumulate the jump in the normal derivative across each edge. Measuring the squared differences yields a quadratic penalty on the coefficient vector fi = fi 1 ; fi K ) which can be written fi t Afi for a positive semidefinite matrix A. As constant and linear functions have zero ....

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Dyn, N., Levin, D. and Rippa, S. (1990a) Data dependent triangulations for piecewise linear interpolation.

....on the data in order to understand the underlying topology of the interpolating surface (object) So the first step in evaluating a surface is to obtain a triangulation of the interpolating points. As the triangulation strongly influences the quality of the resulting (approximative) surface [DLR90, QS90], the first initial triangulation is then optimised . Afterwards the data can be further examined in order to obtain a suitable interpolant. In the literature a number of typical schemes are reported for elaboration of data in surface modelling. In principle, they can be characterised by the ....

....of explicitly revealing the proximated relationships among points on the surface of the object [Boi84] A good triangulation may help to solve many problems. These problems are not limited by only smooth interpolation over the data, but also concern the definition of the shape of the object [Zah71, SH79, Boi84, Car87, DLR90], or, the other way around, the reduction of the number of points without much damaging the actual shape of the object [FHMB84] the control of the automatic processing of surfaces [DM83] or the estimates of some geometric properties, such as area, volume, axes of inertia, or extraction of ....

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N. Dyn, D. Levin, and S. Rippa. Data dependent triangulations for piecewise linear interpolation. IMA Journal of Numerical Analysis, 10:137--154, 1990.

....of the data set. In the light of shape preserving approximation it is highly desirable to start a reconstruction from this convex triangulation, however [4] This is one of the reasons that several articles have appeared investigating criteria determining a best triangulation, see e.g. [5,7]. As a further complication, the functional problem does not cover all possible applications. For example, in terrain modelling or tomography, one requires genuine 3D triangulation methods and to obtain such a method is essentially more complicated (see [10] for an overview) In this article we ....

....cases (as far as we could judge at least) the global minimum was found. As an alternative, we can minimise the mean curvature [1] which also has its analogue in polyhedral approximation theory, for defining good triangulations. It slightly di#ers from one method proposed by Dyn, Levin and Rippa [5]. This method also generates the convex triangulation if the data is convex. Further extensions are to incorporate boundary conditions. In [1] we show that we can generalise our algorithm for non closed objects: for example, it easily picks out the proper triangulation starting with data from the ....

Dyn, N., D. Levin, and S. Rippa, Data dependent triangulations for piecewise linear interpolation, IMA Journal of Numerical Analysis, 10:137--154, 1990.

....identified as the connections for the anisotropic mesh in the x coordinate plane. This technique for selecting incidences was shown to be optimal for the model problem of x 2 with a fixed vertex set by D Azevedo and Simpson in [6] and has been used as a heuristic for data fitting by Dyn et al. in [10]. Calculations for the compressible Navier Stokes flow around a two element airfoil are reported in [14] for free stream Mach number of 0:5 and Reynold s number 5 2 10 3 . The adapted mesh covers recirculation zones, boundary layers and the wake region with 2 2 10 5 triangles and ....

N Dyn, D Levin, and S Rippa. Data dependent triangulations for piecewise linear interpolation. IMA Journal of Numerical Analysis, pages 137--154, 1990. 16

.... account the z values of points in V instead of simply their x y coordinates: the idea is to either maximize or minimize some quantity that expresses certain properties of the resulting surface (e.g. the roughness or the thin plate energy , or the maximum jump between adjacent patches; see Dyn et al. (1990) for a survey) 3 Visibility Structures for Terrains Measuring visibility requires computing visibility for (portions of) the surface itself or for objects located on the surface (representing, for example, facilities such as towers, buildings, radio transmitters, etc) The problem of testing ....

Dyn N, Levin D, Rippa S 1990 Data Dependent Triangulations for Piecewise Linear Interpolation. IMA Journal of Numerical Analysis 10(5): 137-154.

....the quality of triangulations objectively, smoothness criteria have to be defined. Here, the max min criterion and the total absolute curvature criterion will be introduced. 1. 2 Characterizing the Quality of Triangulations In the literature, several triangulation quality criteria can be found [1, 2, 3]. Looking at a shaded triangulation, it is very easy to distinguish between smooth and jagged surfaces. Typically, in computer graphics, two objective quality definitions for triangulations are used: triangle based criteria and edge based criteria. Triangle based criteria follow the rule of ....

....described above can be used to yield this triangulation. An edge exchange operator in the 2D space has been defined to serve as a mutation operator. It flips an edge within a convex polygon of four points. The polygon has to meet the condition that three of the four points are not collinear [1, 3]. The advantage of the application of a 2D operator lies in the fact that special cases can be ignored which appear when edges are flipped in 3D space. Hence, a simple projection of the digitized 3D points into 2D space has to be performed. The mutation operator flips an edge by randomly choosing ....

N. Dyn, D. Levin, and S. Rippa. Data Dependent Triangulations for Piecewise Linear Interpolation. In: IMA Journal of Numerical Analysis, 10:137--154, 1990.

....where S is the set of vertices of a simple polygon and the triangulation is restricted to within the polygon. Various criteria that can be used to define optimal triangulations arise in areas such as finite element analysis [StFi73] computational geometry [PrSh85] and surface approximation [DLR90]. Many of these criteria are defined as maxmin (short for maximizes the minimum) or minmax of some triangle or edge measure. The first quantifier is over all triangulations of the same point set and the second is over all triangles or edges of a triangulation. Two example criteria are maxmin area ....

N. Dyn, D. Levin and S. Rippa. Data dependent triangulations for piecewise linear interpolation. IMA J. Numer. Anal. 10 (1990), 137--154.

....than simply having triangles close to equilateral ones [Nadl85] This requirement is an example of a data dependent criterion. In general, a data dependent criteria is one that takes into account more than just the locations of the vertices. Work done on data dependent criteria can be found in [DLR90b, QuSc90, Ripp92, RiSc90]. We sample only two such criteria for interpolating z = f(x; y) They are the min max AABNV, short for acute angle between normal vectors [DLR90a, DLR90b] and the min max slope [WaPh84, page 218] To understand these, we imagine that a triangulation on points (x i ; y i ) in the plane is ....

....one that takes into account more than just the locations of the vertices. Work done on data dependent criteria can be found in [DLR90b, QuSc90, Ripp92, RiSc90] We sample only two such criteria for interpolating z = f(x; y) They are the min max AABNV, short for acute angle between normal vectors [DLR90a, DLR90b], and the min max slope [WaPh84, page 218] To understand these, we imagine that a triangulation on points (x i ; y i ) in the plane is actually a spatial triangulation on (x i ; y i ; z i ) in space, i.e. a terrain formed by triangular faces. The intrinsic dimension of this triangulation is ....

N. Dyn, D. Levin and S. Rippa. Data dependent triangulations for piecewise linear interpolation. IMA J. Numer. Anal. 10 (1990), 137--154.

....criteria. The following section reviews triangulations in general and the Delaunay triangulation in particular. 2.4. 1 Triangulations A triangulation of a set of vertices V is a set of closed triangles T formed by edges connecting pairs of vertices in V that satisfy the following conditions [DLR90]. ffl The set of all vertices of triangles in T is V . ffl Each edge of a triangle in T contains only two vertices (its endpoints) in V . ffl The union of all triangles in T is equal to the convex hull of V . ffl The intersection of the interiors of any two distinct triangles in T is empty. ....

....procedures are listed below. ffl minimize the maximum or sum of perimeter 2 =area of the two triangles [Law72] 6 Figure 2. 6: Swapping a diagonal in a triangulation ffl maximize the minimum angle in the two triangles [LS80] ffl minimize the angle between the normals of the two triangles [DLR90] A local cost function on each pair of neighbouring triangles can be used as an optimization procedure. A global cost function for the triangulation can be formed by using either the maximum or the sum of the local cost functions. Lawson proved that any triangulation of a finite vertex set can be ....

Nira Dyn, David Levin, and Samuel Rippa. Data dependent triangulations for piecewise linear interpolation. IMA Journal of Numerical Analysis, 10(1):137--154, 1990.

....use Delaunay triangulations [ 9 ] which possess many interesting theoretical properties and with which the complexity estimation of our adaptive thinning algorithm is easy. Yet it seems that for reducing the approximation error, a better strategy would be that of data dependent triangulations [ 4 ]. This will be checked in future work. The outline of the paper is as follows: In Section 2 we present in details the notion of adaptive thinning, and formulate our thinning algorithm. The two theoretical results on our adaptive thinning algorithm, mentioned above, are investigated in Section 3. ....

....of the points in lR , data dependent triangulation takes into account the values of f at these points, and aims to reduce the error between the piecewise linear interpolant on the triangulation and the function f . Various strategies for data dependent triangulation are investigated in [ 4 ]. However, for datadependent triangulation, the topological changes required for decremental triangulation are not guaranteed to be local. Thus, we leave the study of the use of data dependent triangulation strategies to future work. 4. In case the Delaunay triangulation is used, it is possible ....

N. Dyn, D. Levin and S. Rippa, Data dependent triangulations for piecewise linear interpolation, IMA J. Numer. Anal. 10 (1990), 137-154.

....use Delaunay triangulations [ 9 ] which possess many interesting theoretical properties and with which the complexity estimation of our adaptive thinning algorithm is easy. Yet it seems that for reducing the approximation error, a better strategy would be that of data dependent triangulations [ 4 ]. This will be checked in future work. The outline of the paper is as follows: In Section 2 we present in details the notion of adaptive thinning, and formulate our thinning algorithm. The two theoretical results on our adaptive thinning algorithm, mentioned above, are investigated in Section 3. ....

....of the points in lR 2 , data dependent triangulation takes into account the values of f at these points, and aims to reduce the error between the piecewise linear interpolant on the triangulation and the function f . Various strategies for data dependent triangulation are investigated in [ 4 ]. However, for datadependent triangulation, the topological changes required for decremental triangulation are not guaranteed to be local. Thus, we leave the study of the use of data dependent triangulation strategies to future work. 4. In case the Delaunay triangulation is used, it is possible ....

N. Dyn, D. Levin and S. Rippa, Data dependent triangulations for piecewise linear interpolation, IMA J. Numer. Anal. 10 (1990), 137--154.

....4 has been tested, and the results do not differ much from the global method. 4. Towards data dependent approximants The procedures discussed above are linear, and they do not use any information about the approximated function. We know that data dependent approximations can be much better, see [7] for example. With the approximation scheme presented here we have a convenient way of introducing a good data dependent scheme. The essential point in data dependent approximation in IR d is to use more information from directions in which the variations in the function are small, and less ....

N. Dyn, D. Levin and S. Rippa, 1990 Data dependent triangulation for piecewise linear interpolation, IMA J. Numer. Anal. 10 137-154.

....use Delaunay triangulations [ 8 ] which possess many interesting theoretical properties and with which the complexity estimation of our adaptive thinning algorithm is easy. Yet it seems that for reducing the approximation error, a better strategy would be that of data dependent triangulations [ 4 ]. This will be checked in future work. The outline of the paper is as follows: In Section 2 we present in details the notion of adaptive thinning, and formulate our thinning algorithm. The two theoretical results on our adaptive thinning algorithm, mentioned above, are investigated in Section 3. ....

....of the points in lR 2 , data dependent triangulation take into account the values of f at these points, and aims to reduce the error between the piecewise linear interpolant on the triangulation and the function f . Various strategies for data dependent triangulation are investigated in [ 4 ]. However, for data dependent triangulation, the topological changes required for decremental triangulation are not guaranteed to be local. Thus, we leave the study of the use of data dependent triangulation strategies to future work. 4. In case the Delaunay triangulation is used, it is possible ....

N. Dyn, D. Levin and S. Rippa, Data dependent triangulations for piecewise linear interpolation, IMA J. Numer. Anal. 10 (1990), 137-154.

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Nira Dyn, David Levin, and Shmuel Rippa. Data dependent triangulations for piecewise linear interpolation. IMA Journal on Numerical Analysis, 10(1):137--154, January 1990.

No context found.

N. Dyn, D. Levin, S. Rippa, Data Dependent Triangulation for Piecewise Linear Interpolation, IMAJ. Numerical Analysis, vol.10 (1990), pages 137-154

No context found.

N. Dyn, D. Levin, and S. Rippa. Data dependent triangulations for piecewise linear interpolation. IMA Journal of Numerical Analysis, 10(1):137--154, January 1990.

No context found.

Nira Dyn, David Levin, and Samuel Rippa. Data dependent triangulations for piecewise linear interpolation. IMA Journal of Numerical Analysis, 10:137--154, 1990.

No context found.

N. Dyn, D. Levin, and S. Rippa. Data dependent triangulations for piecewise linear interpolation. IMA Journal of Numerical Analysis, 10:137-154, 1990.

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Nira, D., Levin, D., Rippa, S.,"Data Dependent Triangulations for Piecewise Linear Interpolation", J. Numer. Anal., Vol. 10, No. 1, 1990, pp. 137-154.

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N. Dyn, D. Levin, and S. Rippa. Data dependent triangulations for piecewise linear interpolation. IMA J. Numer. Anal., 10:137--154, 1990.

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