M. de Berg and K. Dobrindt. On levels of detail in terrains. In Proceedings 11th ACM Symposium on Computational Geometry, pages C26--C27, Vancouver (Canada), 1995. ACM Press.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....adapting a mesh to the needs of an application, possibly varying its resolution over different areas of the entity represented. A number of different LOD models have been proposed in the literature. Most of them have been developed for applications to terrain modeling in GISs (see, for instance, [1, 4, 9]) and to surface representation in computer graphics and virtual reality applications (see, for instance, 10, 8, 15, 7] and they are strongly characterized by the data structures and optimization techniques they adopt as well as custom tailored to perform specific operations, and to work on ....

M. de Berg and K. Dobrindt. On levels of detail in terrains. In Proceedings 11th ACM Symposium on Computational Geometry, pages C26--C27, Vancouver (Canada), 1995. ACM Press.

....from the numerical part: the geometrical part includes computing and storing the Delaunay hierarchy of meshes, whereas the numerical part involves computing and storing the coarse and detail coefficients. As stated in section 3.1, the geometrical part has been extensively worked in others papers. [2] for ex. describes a data structure that is basedon blocks, as defined in section 3.1. The size of all proposed data structures is proportional to the number of vertices, with a bigger constant if the adjacencies are stored. The computation time for the construction of the hierarchy (without ....

Mark de Berg and Katrin T. G. Dobrindt. On levels of detail in terrains. In Technical Report UU-CS-1995-12. Utrecht University, 1995.

....a terrain at a sequence of increasing resolutions. Such model does not rely on a special construction technique, and can be built by simplification as well as by refinement. Interference links are stored between pairs of consecutive triangles which have a proper intersection. The model proposed by de Berg and Dobrindt (1995) is built through iterative simplification of a Delaunay TIN: at each step, a set of independent vertices (i.e. vertices that are not endpoints of the same edge) of small degree is removed, and the holes left by those vertices are re triangulated. Interference links are maintained between the ....

De Berg M, Dobrindt K 1995 On levels of Detail in terrains. Technical Report UU-CS-1995-12. Utrecht (The Nederlands), Utrecht University.

....points are read in together with their incidence information forming a triangle mesh. ffl contour lines Contour lines are lists of coordinates associated with a certain isovalue. Terrain simplification is a well known problem and many methods have been proposed in the literature. See [GH95] dBD95] for a review of them. Several iterative techniques for producing a TIN from a digital elevation model have been proposed [PM92] The overall goal is to produce an approximation that preserves the major topographic features using 41 a greatly reduced set of points selected from the input ....

Mark de Berg and Katrin Dobrindt. On levels of detail in terrains. In Proc. 11th Annual ACM Symp. on Computational Geometry, Vancouver, B.C., June 1995. Also available as Utrecht University tech report UU-CS-1995-12, .

.... Terrain Models and Measuring Terrain Model Accuracy David Scott Andrews May 3, 1996 Abstract We describe a set of TIN simplification methods that enable the use of the triangulation hierarchy introduced by Kirkpatrick [Kir83] and modified by de Berg and Dobrindt [dBD95a, dBD95b]. This triangulation hierarchy can be used to form a terrain model combining areas with varying levels of detail. One variant of the delete simplification method formed simplifications with accuracy close to the greedy method. We also investigated different variables that can be used to measure ....

....Delaunay triangulation. The surface is continuous at each level of the hierarchy but discontinuities may appear if areas from different levels of the hierarchy are combined. The triangulation hierarchy presented with Kirkpatrick s optimal search algorithm was modified by de Berg and Dobrindt [dBD95a, dBD95b] to be used for polyhedral terrains or TINs. They present a method for displaying terrains combining areas with different levels of detail. The details of our implementation of their method are described in Section 4.5. The other hierarchical TINs presented above replace a single triangle with a ....

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Mark de Berg and Katrin Dobrindt. On levels of detail in terrains. In Proceedings of the Eleventh Annual ACM Symposium on Computational Geometry, pages C26--C27, 1995.

.... Terrain Models and Measuring Terrain Model Accuracy David Scott Andrews May 3, 1996 Abstract We describe a set of TIN simplification methods that enable the use of the triangulation hierarchy introduced by Kirkpatrick [Kir83] and modified by de Berg and Dobrindt [dBD95a, dBD95b]. This triangulation hierarchy can be used to form a terrain model combining areas with varying levels of detail. One variant of the delete simplification method formed simplifications with accuracy close to the greedy method. We also investigated different variables that can be used to measure ....

....model includes more points in rough areas Figure 4.3: Subdividing a triangle into three children and it forms a continuous surface at each level of the hierarchy. The major problem with this model is its tendency to form elongated triangles that can lead to inaccuracies in numerical interpolation [dBD95a]. In Section 4.2.1, this hierarchy is used as a method for building a TIN from a DEM. 16 These two structures are modified in an attempt to address the problems of surface discontinuities and elongated triangles. Scarlatos and Pavlidis [SP90] aim to triangulate while looking more carefully at ....

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Mark de Berg and Katrin Dobrindt. On levels of detail in terrains. Technical report, Utrecht University, 1995.

....modeling techniques for arbitrary polygonal meshes and, in particular, for digital terrains have been developed in the past. Hierarchical triangulations based on TINs have been applied to generate multiresolution models which can be used by level of detail algorithms (de Berg and Dobrint [2], de Floriani and Puppo [3] Gross et al. 8] and Voigtmann et al. 16] Regular grids have been used for multiresolution modeling (Falby et al. 6] and for real time, continuous LOD rendering (Lindstrom et al. 10] and Pajarola [12] These approaches suffer from the following limitations: ....

M. de Berg, K. Dobrindt. On Levels of Detail in Terrains. Proceedings of 11th ACM Symposium on Computational Geometry, C26-C27, 1995.

....uses Delaunay triangulation at each level of detail in order to avoid sharp triangles. However, since this method does not have a tree hierarchy, it is difficult to combine parts from different levels of detail in the same scene. A survey is given by De Floriani et al. 3] De Berg and Dobrindt [1] proposed a hierarchy of levels of detail that uses Delaunay triangulation at each level. It is the first method that uses Delaunay triangulation, and also allows different levels of detail to be combine in the same scene. This technique achieves spatial continuity of the terrain for each rendered ....

....of vertices of level i and J i the subset of I i which is removed to define I i 1 . Then I i 1 = I i Gamma J i . The critical point is to construct the levels such that the union of the triangles at every level is a Delaunay triangulation. This is achieved using the De Berg and Dobrindt method [1]. They proposed restricting the selection of vertices J to be removed from I such that if p is selected (p 2 J ) then all the adjacent vertices qk of p are not selected (qk 62 J ) Thus, all the vertices adjacent to 1234567890123 1234567890123 1234567890123 1234567890123 1234567890123 ....

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M. De Berg and K. Dobrindt. On levels of detail in terrains. In Proc. 11th Annual ACM Symp. on Computational Geometry, Vancouver, B.C., June 1995. Also available as Utrecht University tech report UU-CS-1995-12, URL=ftp://ftp.cs.ruu.nl/pub/RUU/CS/techreps/CS-1995.

....are never deleted until the complete patch is thrown out of the scene map. FIGURE 9. LOD distribution Avoiding discontinuity, between regions of different terrain complexity, is another requirement of aspect two of continuous LOD rendering. This is a hard problem for multiresolution triangulations [1,3], however, efficiently solved for grid based inputs by the RQT. Nevertheless, neatly stitching together independent terrain patches of different resolutions is another problem. In [8] discontinuities were allowed between quadtree blocks, whereas [15,16] solved the problem by consistently ....

....terrain structure. Maintaining a triangulation for every LOD does not support continuous LOD rendering and the storage costs are very high. More sophisticated multiresolution triangulation models [2] replace a number of triangles by more and smaller triangles to refine the approximation, see also [1,13]. However, extraction and incremental refinement of a triangulation are rather complex for on demand requirements. Furthermore, the topology representation is very complicated compared to the RQT. The same arguments hold for hierarchical triangulation models as described in [3] Furthermore, no ....

M. de Berg and K. Dobrindt. "On levels of detail in terrains". In 11th Symposium on Computational Geometry, pages C26--C27. ACM, 1995.

....as needed in a dynamic environment. Because the LOD changes as we get closer to some location, it should be efficient to dynamically load additional data from disk in order to refine a region that is already in core at a low resolution. This goal is difficult to achieve for surface triangulations [6, 4, 5], because different LODs in adjacent triangles may lead to cracks in the terrain, the well studied sliver polygons, unless special care is taken to neatly stitch adjacent triangles together not an easy task. In ViRGIS, the visualization uses a viewpoint centered LOD strategy as shown in Figure ....

M. de Berg and K. Dobrindt. On levels of detail in terrains. In 11th ACM Symposium on Computational Geometry, pages C26--C27. ACM, June 5 - 7 1995.

....refined as necessary. Greene, Kass, and Miller [14] partition a polygonal scene with an oct tree. The polygons inside an oct tree cube are not rendered if the projection of the cube falls inside a quad tree node that is covered by a closer object. In computational geometry, de Berg and Dobrindt [15] use a Kirkpatrick mesh decimation technique [16] to generate a hierarchy of different levels of terrain detail which fit together smoothly. Cole and Sharir [17] preprocess a terrain to answer ray shooting queries from a fixed point and from a fixed vertical line. Bern, Dobkin, Eppstein, and ....

M. de Berg and K. Dobrindt, "On levels of detail in terrains", in Proc. 11th Annu. ACM Sympos. Comput. Geom., 1995, pp. C26--C27.

....has received a significant amount of attention in recent years. In this chapter, we give a brief critical survey of this work. We broadly classify these schemes into three categories: 1) the tree based schemes of De Floriani et al. 17, 19, 18] 2) the DAG based scheme of Dobrindt and de Berg [16] and Puppo [58] and (3) the Progressive Mesh scheme of Hoppe [38] 8.1 Tree Based Schemes The approach to multiresolution modelling taken in [19, 18] is to generate a mesh consisting of nested triangles, and to arrange these triangles into a tree shaped hierarchy (see Figure 8.1) At the top ....

....to numerical instabilities in computations, and aliasing effects in graphics applications. Methods which avoid these elongated triangles by splitting long edges are described in [18] but these can lead to vertical faces (discontinuities) in the TIN. 8. 2 DAG Based Schemes Dobrindt and De Berg [16], and De Floriani [17] address the problem of the thin triangles which occur in tree based schemes by using a very different approach. By repeatedly deleting an independent set of TIN vertices, and retriangulating the resulting holes, a hierarchy of triangulations which has a depth of O(log n) and ....

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M. de Berg and K. Dobrindt. On levels of detail in terrains. In ACM Symposium on Computational Geometry, pages ??--??, 1995.

.... The horizon has complexity O(n ff(n) 5 and can be computed in optimal time O(n log n) 18] 5 ff(n) is the extremely slowly growing inverse of Ackermann s function and can be considered constant for any 6 A hierarchical model can represent the terrain at various resolutions [19] 20] [21]. For this multiresolution model, De Floriani and Magillo present an algorithm to compute the horizon for a single point at different resolutions in time O(n log n) 22] Their algorithm allows the horizon to be updated efficiently with changing levels of detail. Cole and Sharir [23] describe how ....

M. de Berg and K. Dobrindt, "On levels of detail in terrains", in Proceedings of the 11th Annual ACM Symposium on Computational Geometry, 1995, pp. C26--C27.

.... (TINs) A number of different approaches have been developed to create TINs from height fields using Delaunay and other triangulations [10, 11, 27] and hierarchical triangulation representations have beenproposed that lend themselves to usage in multiresolution level of detail algorithms [4, 5, 26]. Regular grid surface polygonalizations have also been implemented as terrain and general surface approximations [3] Such a gridded terrain surface representation is used in VGIS and is described in [20] Other surface approximation representations include techniques such as wavelet transforms ....

DE BERG, M. and DOBRINDT, K. T. G. On Levels of Detail in Terrains. In 11th ACM Symposium on ComputationalGeometry, June 1995.

....structures on top of TINs [27, 103] and on techniques to improve the quality of TIN meshes [101] Scarlatos dissertation [100] gives a good survey of terrain modeling and representation. A very recent approach to building hierarchical models of terrains is given by de Berg and Dobrindt [26], who apply a hierarchical refinement of the Delaunay triangulation to represent terrain TINs at many levels of detail. 56, 57] describe the drop heuristic and provides comparisons with other methods. Common to all these methods is the need to have a complete starting triangulation that is ....

....features or of edge features, requiring that the surface approximation include these points and segments in the output TIN. In top down algorithms, such requirements can be incorporated using constraints; for example, line segments can be preserved using constrained Delaunay triangulation (e.g. [26]) In our bottom up algorithm, we can incorporate such constraints directly, at low cost, within the test for triangle feasibility: A triangle T 0 is not feasible if its projection, T , contains a point feature on its interior or boundary, except at a vertex, or intersects an edge feature, ....

M. de Berg and K. Dobrindt. On levels of detail in terrains. In Proc. 11th Annu. ACM Sympos. Comput. Geom., pages C26--C27, 1995.

....as needed in our VR environment. Because the LOD changes as we get closer to some location, it should be efficient to dynamically load additional data from disk in order to refine a region that is already in core at a low resolution. This goal is difficult to achieve for surface triangulations [6, 4, 5], because different LODs in adjacent triangles may lead to holes in the terrain, the well studied sliver polygons, unless special care is taken to neatly stitch adjacent triangles together not an easy task. In ViRGIS, the visualization uses a viewpoint centered LOD strategy as shown in Figure ....

M. de Berg and K. Dobrindt. On levels of detail in terrains. In 11th ACM Symposium on Computational Geometry, pages C26--C27. ACM, June 5 - 7 1995.

.... on creating hierarchical structures on top of TINs [7, 20] and on techniques to improve the quality of TIN meshes [21] For a survey, see Scarlatos dissertation [18] and the survey by Heckbert [12] A recent approach to building hierarchical models of terrains is given by de Berg and Dobrindt [6], who apply a hierarchical refinement of the Delaunay triangulation to represent terrain TINs at many levels of detail. See also [13, 14] for an approach called the drop heuristic and its comparison with other methods. Common to all these previous methods is a necessity to have a complete ....

M. de Berg and K. Dobrindt. On levels of detail in terrains. In Proc. 11th Annu. ACM Sympos. Comput. Geom., pages C26--C27, 1995.

....adapting a mesh to the needs of an application, possibly varying its resolution over different zones of the entity represented. A number of different LOD models have been proposed in the literature. Most of them have been developed for applications to terrain modeling in GISs (see, for instance, dBD95, DFP95, LKR 96] and to surface representation in computer graphics and virtual reality applications (see, for instance, LE97, Hop97, XESV97, GTLH98] and they are strongly characterized by the data structures and optimization techniques they adopt, as well as custom tailored to perform ....

M. de Berg and K. Dobrindt. On levels of detail in terrains. In Proceedings 11th ACM Symposium on Computational Geometry, pages C26--C27, Vancouver (Canada), 1995. ACM Press.

....One can use the triangulation as presented in Figure 1, or if it is possible to Figure 1: A Delaunay triangulation in 3 D of a terrain given as a DEM. The terrain here is a grid that has 32 by 24 points. use a reduced model, the overall computation will be performed faster. Mark de Berg [8] proposed a system that starts with a Delaunay triangulation of a terrain and simplifies the model to a desired level of detail preserving the property of the Delaunay triangulation. The interesting part of his approach is that it allows combinations of different levels of detail in the same ....

....presented here runs in polynomial time. A complexity analysis for each step in the algorithm is presented below with a pointer to more detailed references. ffl Step 1: 1. Delaunay triangulation: O(nlog(n) see Preparata and Shamos [15] 2. The hierarchical representation: O(n) see de Berg [8]. ffl Step 2: 5 coloring: O(n) see Chiba [3] Note Figure 7: Final placement of observers given by step 3 18 observers are kept on the terrain. Note that this final step was approximated by hand (the complete algorithm is currently being implemented) X X X X 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 C ....

de Berg, M., and Dobrindt, K. T. G. On levels of detail in terrains. Tech. Rep. UU-CS-199512, Utrecht University, April 1995.

....to the coarser stage (i.e. the previous next stage in the case of a top down bottom up construction) Relation is defined between any two components of which interfere and are made of simplexes coming from two consecutive complexes (see Fig. 2. 5 (c) The model proposed by de Berg and Dobrindt [7] represents a special case of a two dimensional pyramidal model (see Fig. 2.5) Such model is built through iterative simplification of a complex representing a two dimensional domain at full resolution: at each step, a set of independent vertices of small degree is removed, and the holes left ....

....components of the model are also stored. In [1] each component is represented by simply listing all its simplexes. The structure in [2] represents global topology, by linking a simplex to the list of all its adjacent simplexes. Data structures which represent interference links between components [7, 15] are based on the auxiliary concept of . Intuitively, a is defined by two connected sets of simplexes (called the and the of the bubble) which provide two descriptions of the same portion of the domain at two consecutive resolutions (see Figure 4.2 (a) In a generic MSM, we have a bubble for ....

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M. De Berg, M., K.T.G. Dobrindt, 1995, On the levels of detail in terrains, , Vancouver, BC (Canada), c26--c27. Also published in extended version as UU-CS1995 -12, Utrecht University, Dept. of Computer Science, 1995. Available from

.... selective refinement has been done quite recently (see [20] for a survey) Most LOD data structures are built through local updates, which are popular methods for mesh simplification (see [12, 20] for surveys) Different data structures adopt different hierarchical organizations, such as pyramids [1, 6], trees of nested meshes [7, 10, 16] higher dimensional embeddings [5] directed acyclic graphs (DAGs) 3, 19, 11] linear sequences [13, 15] and vertex hierarchies [22, 14, 17, 18] Algorithms for selective refinement have been proposed for all data structures cited above. They are based on a ....

....16] higher dimensional embeddings [5] directed acyclic graphs (DAGs) 3, 19, 11] linear sequences [13, 15] and vertex hierarchies [22, 14, 17, 18] Algorithms for selective refinement have been proposed for all data structures cited above. They are based on a top down visit of the hierarchy [3, 1, 10, 14, 17, 18, 19, 22], on a bottom up traversal of the hierarchy [16] on a breadth first traversal of the surface [5, 7] or on a linear scan with backtracking of a sequence of modifications [13, 15] Algorithms proposed in [3, 1, 14, 19, 22] may achieve linear time in the size of their output: the results they ....

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M. de Berg, K.T.G. Dobrindt, On the Levels of Detail in Terrains, Proceedings 11th ACM Symposium on Computational Geometry, Vancouver, pp.c26-c27, 1995.

....a specific operator (see Figure 6) In [Hop96, XESV97, Hop97, MMS97, GTLH98, KCVS98] the refinement pattern is a vertex split, which expands a vertex p into an edge, two edges incident at p into two triangles, and warps the triangles surrounding p accordingly. In [DF89, CDFM 94, dBD95, BFM95, KS97, CPS97, DFMP98] the refinement pattern consists of inserting a new vertex p, and replacing a subset of triangles in the neighborhood of p with a set of triangles incident at p. Note (b) a) Fig. 5. A hierarchy of right triangles (a) with meshes shown level by level, and the ....

M. de Berg and K. Dobrindt. On levels of detail in terrains. In Proceedings 11th ACM Symposium on Computational Geometry, pages C26--C27, Vancouver (Canada), 1995. ACM Press.

.... start with a single triangle (or tetrahedron for general surfaces) and refine it locally until the resulting surface becomes an approximation, or they start with a fine triangulation and coarsen it locally (by removing a vertex and filling the hole) until one can no longer remove a vertex [12, 13, 14, 15, 17, 16, 27, 33, 34, 36, 38]. The former method is called refinement, and the latter is called decimation. Some other approaches that extend to arbitrary surfaces have also been proposed. They include an optimization method that formulates the problem as an energy optimization problem [28, 29] and an approach based on ....

M. de Berg and K. Dobrindt, On levels of detail in terrain, Proc. 11th Annual ACM Symp. on Comput. Geom. 1995, C26--C27.

....resolution is decreasing gradually with distance from the viewpoint. A multiresolution model is a model that can provide different representations, depending on the level of detail required. In order to be effective, such a model must fulfill requirements that have been outlined by several authors [4, 6, 7, 13, 16, 21], and can be summarized as follows: ffl continuity through domain: there cannot be cracks or aliasing due to abrupt transition between different resolutions; ffl continuity across resolution: abrupt changes should be avoided in changing a representation into another at a close LOD; ffl ....

....representations made with right triangles. Layered models, on the other hand, did not support the extraction of representations whose LOD is variable through space, also known as selective refinement. Selective refinement from non nested hierarchies was studied first, and independently, in [4, 6], and later in [13, 15, 21] The hierarchical representation proposed by de Berg and Dobrindt [6] is a sequence of Delaunay triangulations at increasingly coarser level of resolution. The model is built through a technique developed earlier by Kirkpatrick for arbitrary triangulations [14] each ....

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M. de Berg, M., K.T.G. Dobrindt, 1995, On the levels of detail in terrains, 11th ACM Symposium on Computational Geometry, Vancouver (Canada), June 5-7, pp.c26-c27. Also available in long version as Techical Report UU-CS-1995-12, Utrecht University, Dept. of Computer Science, April 1995.

....the given precision requirement at each point in the domain but cannot guarantee continuity between triangles. A further triangulation step is required to produce a complete surface, but this may produce a mesh which violates the previously satisfied precision requirement. De Berg and Dobrindt [dD95] present a system of linking a hierarchical sequence of Delaunay triangulations such that a selectively refined surface can be extracted which satisfies the Delaunay 1 A Delaunay triangulation T is such that for each triangle t in T , there is no vertex of T in the interior of t s circumcircle ....

....Theorem 4.4 If the input base triangulation, T 0 , and sequence of refinement operations, fR 1 , R m g satisfies the Delaunay criterion [PS85] then the output triangulation, T , also satisfies this criterion. Proof: In the similar way to the Delaunay satisfiability proof in [dD95] we note that the vertices in each refinement region R i ; 1 i m, cannot have an influence, in terms of the Delaunay criterion, outside that region. Hence when we add triangles to the current triangulation, T , these triangles will not affect the Delaunay nature of T . 5 Geomorphing ....

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Mark de Berg and Katrin T. G. Dobrindt. On levels of detail in terrains. In 11th ACM Symposium on Computational Geometry, pages 26--27, June 1995. Also published in longer version as Technical Report UU-CS-1995-12, Utrecht University, Dept. of Computer Science, April 1995.

....is maintained between successive layers of resolution. This is achieved by starting with the finest resolution mesh, and removing an independent set of nodes at each level. 3.1. Independent Set An independent set is a set of vertices among which no two vertices are adjacent to each other [12]. The open circles in Figure 1 depict an independent set of nodes. The removal of an independent set of nodes implies that the bounding polygons of the removed nodes can be preserved from Figure 1: Open dots indicate an independent set of nodes that can be removed. one layer to the next (shown ....

Berg M. de, Dobrindt K.T.G., "On Levels of Detail in Terrains", Technical Report, Dept. of Computer Science, Utrecht Univ., 1995.

.... as well as the ideas underying the second algorithm for variable resolution described in Section 7 of this paper (which were added during revision) were inspired by the work in [Cig95] Independently of the above work, other algorithms and data structures for the Delaunay pyramid were proposed in [deB95]. In such work, emphasis was put on the possibility of performing point location in logarithmic time, and of extracting a Delaunay triangulation that represents a surface at variable resolution in linear time. The pyramid is built bottom up through a sim41 plification technique essentially based ....

de Berg, M., Dobrindt, K.T.G., "On the levels of detail in terrains", 11th ACM Symposium on COmputational Geometry, Vancouver, BC (Canada), June 5-7, 1995.

....an optimal balance between the computing power of the server, and the rest of the architecture. 3. Related work Selective refinement of meshes has been addressed by several authors using hierarchical structures based on either Progressive Meshes [Tau98,Hop97,Xia97] or Delaunay triangulations [Bro96,Cig97,deB95]. The MT is indeed a unifying framework for all such models, which is independent on the construction technique, and offers simple and efficient accessing methods based on the manipulation of cuts in a DAG. For a thorough discussion of MTs, and their relations with other models see [Pup97] Static ....

de Berg, M., Dobrindt, K.T.G., 1995, On the levels of detail in terrains, Proceedings 11th ACM Symposium on Computational Geometry, Vancouver, BC (Canada), pp.c26--c27.

....it to construct approximations of the surface at various levels of detail [3, 16] When rendering the height field, we can choose an approximation with an appropriate level of detail and use it in place of the original. The various levels of detail can be combined into a hierarchical triangulation [6, 5]. In some applications, such as flight simulators, the speed of simplification is unimportant, because database preparation is done off line, once, while rendering of the simplified terrain is done thousands of times. In more general computer graphics and computer animation applications, the scene ....

Mark de Berg and Katrin Dobrindt. On levels of detail in terrains. In Proc. 11th Annual ACM Symp. on Computational Geometry, Vancouver, B.C., June 1995. Also available as Utrecht University tech report UU-CS-1995-12, URL=ftp://ftp.cs.ruu.nl/pub/RUU/CS/techreps/CS-1995/.

....complexity can extract data of different quality levels from this single bit stream. Hierarchical representation of 3D meshes has been addressed in computer graphics for adaptive level of detail (LOD) rendering of 3D objects [17] Geometric methods for fine to coarse 3D mesh simplification include [27], 16] 19] A wavelet based multi resolution mesh approximation was proposed in [18] In VRML, rendering at multiple resolutions is enabled by the LOD node which requires definition of L separate meshes which are stored or transmitted independently (simulcast) Compression of 3D triangular ....

....between the levels, by performing a new Delaunay triangulation using an arbitrarily selected subset of the initial node points in a fine to coarse scheme. Here, we use an approach that represents the middle ground between these two options, by preserving part of the topology from level to level [27], 28] A simplification step starts September 14, 1998 DRAFT IEEE TRANSACTIONS ON CSVT, VOL. XX, NO. Y, MONTH 1998 6 by removing an independent set of nodes from M l as illustrated in Figure 2. An independent set is a set of nodes among which no two nodes are connected to each other by an edge. ....

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M. de Berg and K.T.G. Dobrindt, "On levels of detail in terrains," Tech. Rep. UU-CS-1995-12, Dept. of Computer Science, Utrecht University, the Netherlands, Apr. 1995.

.... irregular networks (TINs) A number of different approaches have been developed to create TINs from height fields using Delaunay and other triangulations [9, 10, 19] and hierarchical triangulation representations have been proposed that lend themselves to usage in level of detail algorithms [3, 4, 18]. TINs allow variable spacing between vertices of the triangular mesh, approximating a surface at any desired level of accuracy with fewer polygons than other representations. However, the algorithms required to create TIN models are generally computationally expensive, prohibiting use of ....

DE BERG, M. and DOBRINDT, K. T. G. On Levels of Detail in Terrains. In 11th ACM Symposium on Computational Geometry, June 1995.

....over the domain. In particular, an algorithm was presented, which extracts a representation whose accuracy is monotonically decreasing proportionally to the distance from a given point. Such a representation is particularly suitable for visualization in the context of flight simulators. In (de Berg and Dobrindt 1995), a pyramidal model at implicit multiresolution was proposed, which is also based on Delaunay triangulations. In such work, emphasis was put on the efficiency of operations, rather than on the accuracy of representation. The model (called here the ) is built bottom up through a simplification ....

....storage. Efficient data structures for the main memory have been proposed for most of the remaining HTMs (De Floriani et al. 1984; De Floriani and Puppo 1995) as well as for the Delaunay pyramid (De Floriani 1989, Bertolotto et al. 1994, Cignoni et al. 1995) and for the implicit Delaunay pyramid (de Berg and Dobrindt 1995). To our knowledge, construction algorithms have been implemented for all models described in the paper, although they were not all available to us. The worst case time complexity of such algorithms is seldom significative, since it is usually quite pessimistic with respect to the practical ....

De Berg M, Dobrindt KTG (1995) On the levels of details in terrains. In: Proceedings 11th ACM Symposium on Computational Geometry, Vancouver BC.

....if the accuracy of each triangle is smaller than the minimum of the function on the triangle itself. Most multiresolution models support only constant thresholds. A few models supporting more general functions, called variable resolution models, were proposed recently by de Berg and Dobrindt [4], Cignoni et al. 3] and De Floriani and Puppo [5] The hierarchical representation proposed in [4] is defined as a pyramid of triangulations, whose structure is essentially based on an earlier hierarchical triangulation scheme proposed by Kirkpatrick [9] The pyramid is built bottom up: each ....

....Most multiresolution models support only constant thresholds. A few models supporting more general functions, called variable resolution models, were proposed recently by de Berg and Dobrindt [4] Cignoni et al. 3] and De Floriani and Puppo [5] The hierarchical representation proposed in [4] is defined as a pyramid of triangulations, whose structure is essentially based on an earlier hierarchical triangulation scheme proposed by Kirkpatrick [9] The pyramid is built bottom up: each layer is obtained by removing a constant fraction of the vertices from the previous layer. A traversal ....

M. de Berg, M., K.T.G. Dobrindt, 1995, On the levels of detail in terrains, 11th ACM Symposium on Computational Geometry, Vancouver, BC (Canada), June 5-7, 1995, pp.c26-c27. Also published in longer version as Techical Report UU-CS-1995-12, Utrecht University, Dept. of Computer Science, April 1995.

.... point is equivalent to computing the upper envelope of a set of O(n) segments [Ata83] The horizon has complexity O(n ff(n) 1 and can be computed in optimal time O(n log n) Her89] Alternatively, a hierarchical model can be used to represent the surface at various resolutions [DP88, SP90, dBD95] In this model, the horizon can be computed for a single point at different resolutions in time O(n log n) DM95] 1 ff(n) is the extremely slowly growing inverse of Ackermann s function, and can be considered constant for any practical n. Technical Report 349, Department of Computer ....

Mark de Berg and Katrin Dobrindt. On levels of detail in terrains. In Proc. 11th Annu. ACM Sympos. Comput. Geom., pages C26--C27, 1995.

....currently specified LOD. 19 FIGURE 12. LOD distribution Avoiding discontinuity, or cracks, between regions of different terrain complexity or LOD, is another requirement of aspect two of continuous LOD rendering. This problem is difficult to solve for multiresolution triangulations, see also [1,4]. However, it is efficiently solved for grid based models by the restricted quadtree triangulation (RQT) presented in Section 2. Nevertheless, neatly stitching together independent terrain patches of different resolutions is yet another problem. In [9] discontinuities were allowed between quadtree ....

....introduction Section 1. For other multiresolution models the selection of points for the different LODs is closely coupled with the triangulation. In general, a multiresolution triangulation model replaces a number of triangles by more and smaller triangles to refine the approximation, see also [1,14]. The construction of such a triangulation model is quite complex but can be done beforehand. However, the extraction and especially the incremental refinement of a specific triangulation is very slow for on demand requirements. The topology representation is very complicated in comparison to the ....

M. de Berg and K. Dobrindt. "On levels of detail in terrains". In 11th ACM Symposium on Computational Geometry, pages C26--C27. ACM, 1995.

....hierarchical structures on top of TINs [7, 21] and on techniques to improve the quality of TIN meshes [22] Scarlatos dissertation [19] is a good survey of terrain modeling and representation. A very recent approach to building hierarchical models of terrains is given by de Berg and Dobrindt [6], who apply a hierarchical refinement of the Delaunay triangulation to represent terrain TINs at many levels of detail. See also [13, 14] for an approach called the drop heuristic and its comparison with other methods. Common to all these previous methods is the necessity to have a complete ....

....features or of edge features, requiring that the surface approximation include these points and segments in the output TIN. In top down algorithms, such requirements can be incorporated using constraints; for example, line segments can be preserved using constrained Delaunay triangulation (e.g. [6]) In our bottom up algorithm, we can incorporate such constraints directly, at low cost, within the test for triangle feasibility: A triangle T 0 is not feasible if its projection, T , contains a point feature on its interior or boundary, except at a vertex, or intersects an edge feature, ....

M. de Berg and K. Dobrindt. On levels of detail in terrains. In Proc. 11th Annu. ACM Sympos. Comput. Geom., pages C26--C27, v 1995.

....for the finest level of approximation. The price of such small storage is the rigidity and uniformity of the sampling pattern. A TIN hierarchy allows varying levels of detail within an approximation at the expense of more storage. There are several hierarchies based on irregular triangulations [2, 4]. One may roughly divide them into those that preserve the boundaries of coarse triangulations in going to finer triangulations, and those that do not. In the first case, the hierarchy is organized in a tree structure; each node represents a region, and the children of a node represent the ....

....the degree of the vertices of the atomic polygon within the partition, in cyclic order around the polygon. In this case, the right angled vertex has degree 4, and the two other vertices both have degree 8. Other Laves nets which may be used as the basis for single shape hierarchies are the square [4 4 ], the equilateral triangle [6 3 ] and the 30 60 right triangle [4:6:12] See figure 2. These nets all share the crucial property that they are infinitely refinable using similar polygons: a given polygon in any net can itself be partitioned using pieces of the same shape. Figure 2: Pieces of ....

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M. de Berg and K. Dobrindt. On levels of detail in terrains. In Proc. 11th Annual ACM Symp. on Computational Geometry, June 1995. Also available as Utrecht University tech report UU-CS-1995-12, http://www.cs.ruu.nl/docs/research/publication/TechRep.html.

....18 we decide that the light grey triangle should be refined, but that the dark grey triangle is detailed enough (because it is slightly further from the view point) Then we have a problem, because the children of the light grey triangle overlap with the dark grey triangle. De Berg and Dobrindt [3] have shown that if the hierarchy is constructed in a slightly different manner, then it is still possible to extract a variable resolution representation. Their idea is to construct the hierarchy as follows. They start with the most detailed level, remove a subset of the vertices and ....

M. de Berg and K.T. Dobrindt. On levels of detail in terrains. In Proc. 11th Annu. ACM Sympos. Comput. Geom., pages C26--C27, 1995.

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# 1994. #25# M. de Berg# M. Overmars# and O. Schwarzkopf. Computing and verifying depth orders. SIAM J. Comput.# 23#437#446# 1994. #26# M. de Berg and K. Dobrindt. On levels of detail in terrains. In Proc.

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M. de Berg and K. Dobrindt. On levels of detail in terrains. In 11th ACM Symposium on Computational Geometry, pages C26--C27. ACM, 1995.

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