E. Puppo. Variable resolution terrain surfaces. In Proceedings of the 8th Canadian Conference on Computational Geometry, 1996.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....6 concludes the paper. 2. RELATED WORK There has been extensive work on TIN based triangle mesh simplification, refinement methods and multiresolution triangulation for terrains that goes beyond the scope of this paper. We refer the interested reader to the literature for more details (see [3, 5, 6, 9, 12, 13, 19]) Because such TIN based methods work on arbitrary irregular point input data sets they tend to have higher computa 1. hregular triangulation of elevation points in the 2D projection tional costs associated with simplification and refinement operations compared to regular hierarchical methods. ....

Endco Puppo. Variable resolution terrain surfaces. In Proceedings of the 8th Canadian Conference on Computational Geometry, pages 202-210, 1996.

....height field data. The Delaunay pyramid [DeF89, VBH94] is a multiresolution extension of Delaunay based TINs in which groups of connected triangles are hierarchically replaced by sets with fewer triangles, see also [deBD95] Other hierarchical triangulations as discussed in [DP95] DMP96] and [Pup96] are mainly based on recursive subdivision of triangles. In [Hop98b] a hierarchical mul tiresolution terrain triangulation method is presented that is based on progressive application of edge collapse and vertexsplit mesh simplification and refinement operations [Hop96] Similar to the ....

.... given in b) RQT constraints are enforced on the range query as shown in c) The restricted quadtree triangulation (RQT) approach [Paj98a] provides an efficient top down triangulation algorithm similar to the other methods that is output sensitive and theoretically optimal in the sense of [Pup96] Moreover, the introduced error saturation technique allows fast extraction of a matching triangulation at arbitrary LOD without the need for handling any forced split propagation. The bottom up RQT vertex selection algorithm as presented in [Paj98a] can efficiently be used to perform the ....

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E. Puppo. Variable resolution terrain surfaces. In Proceedings of the 8th Canadian Conference on Computational Geometry, pages 202--210, 1996.

....a of vertices from the approximation at each step. Because of this geometry based construction, the tolerance of each level is not explicitly controlled. Although variable resolution surfaces can be extracted at linear time complexity, the accuracy of the model may not be guaranteed. In [Puppo96] a theoretical framework is proposed in order to formalize hierarchical triangulations, and an abstract structure called MultiTriangulation (MT) is presented. MultiTtriangulations are axiomatically defined and do not exist by themselves. In his work Puppo shows some interesting properties of MTs ....

- Puppo, E., "Variable Resolution Terrain Surfaces", Genova University, tech.rep DISI-TR-96-6.

....quadtree was used to triangulate a parametric surface. However, no algorithm was provided to efficiently build the restricted one from a plain quadtree. In addition to the algorithms in [14] we will provide two slightly different construction methods, one theoretically optimal in the sense of [13] and 2 an implementation efficient one. Furthermore, we present a new triangle strip construction, progressive meshing and spatial access, all based on the restricted quadtree. Because of space limitations only preliminary details are presented in this introduction. The assignment of grid ....

....procedures set.insert( and set.contains( can be implemented by setting and testing a flag for every point. The complexity of Algorithm 1 and the respective triangulation is linear in the size of the extracted restricted quadtree. Moreover, the complexity is theoretically optimal in the sense of [13] as the RQT is a multi triangulation. n 2 k 1 = P i , i j , 0. n 1 = l 0 L l P i j h l 0 1 . 2 l , i ; j , 0 d l . h , l d l P i j L l 1 = d l n 1 ( 2 l = L l center L l L k 0 = l k = L Q ( L l level 2 level 0 ....

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E. Puppo. "Variable resolution terrain surfaces". In Proceedings of the 8th Canadian Conference on Computational Geometry, pages 202--210, 1996.

....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 level of the ....

....Figure 8. 2) Furthermore, although the hierarchy has depth O(log n) there are constant factors hidden in the big O notation (Dobrindt and De Berg report hierarchies of with depths varying between 1:7 log 2 n and 2:4 log 2 n depending on the method used to select vertices for deletion) Puppo [58] takes the DAG based approach one step further by describing a general DAG based framework called the MultiTriangulation. In this scheme, a group of triangles can be replaced by another group of triangles as long as the boundaries of CHAPTER 8. SURVEY OF EXISTING WORK 56 Figure 8.3: Some ....

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E. Puppo. Variable resolution terrain surfaces. In Proceedings of the 8th Canadian Conference on Computational Geometry, 1996.

....where the next set of refinings had to be determined. The rendering time of each frame is still significantly less than the average time to render a scene of the animation from the highest resolution triangulation (0. 57s) 7 Related Work Independently, and in parallel with this work, Puppo [Pup96] has developed a theoretical method for meshing a set of single resolution TINs into a static variable resolution representation of a surface. His algorithm differs from our selective refinement process in that it combines the region expansion and surface extraction steps. This prevents any ....

Enrico Puppo. Variable resolution terrain surfaces. In 8th Canadian Conference on Computational Geometry, August 1996.

....Bertocci Computer and Information Science Department, University of Genova Via Dodecaneso, 35, 16146 Genova (Italy) Tel. 39 10 3536704 Fax: 39 10 3536699 Email: fmagillo disi, bertocci educ.disig . unige.it 1 Introduction A Multi Triangulation (MT) is a multiresolution model proposed in [Pup96a, Pup96b] for the description of bivariate surfaces (such as terrains) The intuitive idea behind an MT is that of having a collection of local modifications progressively refining an initial triangulation of a plane domain, which are arranged according to a partial order. By selecting consistent subsets ....

....on a previous modification M 1 if and only if it removes some triangles introduced by M 1 . Dependency means that M 2 cannot be performed unless M 1 has been performed. Under suitable and reasonable constraints on the modifications of the sequence, the relation of dependency is a partial order [Pup96a, Pup96b]. The modifications of the given sequence can be applied to the initial triangulation in a different order, provided that such order is compatible with the relation of dependency. In any such permuted order, the final result is the same but the set of intermediate meshes is different (see Figure ....

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E. Puppo. Variable resolution terrain surfaces. In Proceedings Eight Canadian Conference on Computational Geometry, pages 202--210, Ottawa, Canada, August 12-15 1996. Extended version to appear under title "Variable resolution triangulations " in Computational Geometry Theory and Applications.

....the approximation tolerance, no algorithm was provided in [20] to efficiently build the restricted one from a plain quadtree. In addition to the algorithms presented in [16] we will provide two different algorithms to build a restricted quadtree, one theoretically optimal in the sense of [14] and an implementation efficient one. When using the restricted quadtree for terrain triangulation, the quadtree decomposition can directly be applied to the height field data in the object space, instead of using an intermediate parameter space. In [11] a screen space error function was used as ....

....in ResolveDependencies( visits all vertices of Q at most four times as no more than four arcs of the dependency graph end in one vertex, see also Figure 6. calls to set.insert( and set.contains( by ResolveDependencies( Moreover, the complexity is theoretically optimal in the sens of [14]. The RQT is a multitriangulation and it can be constructed in linear time of the output size, the number of triangles issued. Contrarily to the top down method in [16] we do not predict and interpolate next level vertices to guide recursive tree traversal. We can use the exact approximation ....

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E. Puppo. "Variable resolution terrain surfaces". In Proceedings of the 8th Canadian Conference on Computational Geometry, pages 202--210, 1996.

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Puppo, E., 1996, Variable resolution terrain surfaces, Proceedings Eight Canadian Conference on Computational Geometry, Ottawa (Canada), 12-15 August, pp.202-210. Also appeared in extended version as: Variable resolution triangulations, Technical Report 16/96, IMA-CNR (submitted for publication).

....interactively and dynamically obtained from such model. Such approach has been extensively applied for view dependent rendering of triangle meshes [13, 19, 22, 27, 37] In our previous work, we have developed a completely general multi resolution model for triangle meshes with a manifold domain [31], called a Multi Tessellation (MT) We have defined and implemented instances of such model which are dependent on the specific simplification operator applied in its construction (see De Floriani and Magillo [5] for a recent survey) Our approach is not restricted to rendering, since we want to ....

....models based on manifold triangle meshes. There has been a lot of work on regular meshes based on nested recursive subdivisions for terrain modeling [11, 14, 25] and for subdivision surfaces [23, 35, 39] Multi resolution models for irregular meshes are structured as directed acyclic graphs [6, 7, 19, 31], or similar structures [37] or as forests of binary trees of vertices [22, 27] sometimes enriched with vertex labeling [13] The multi resolution structures proposed in [13, 27] for view dependent rendering support non manifold meshes (also non regular meshes in [27] but they extract just ....

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E. Puppo. Variable resolution terrain surfaces. In Proceedings Eight Canadian Conference on Computational Geometry, pages 202--210, Ottawa, Canada, August 12-15 1996.

....different Levels Of Detail (LODs) can be extracted, based on efficient graph traversal strategies. Here, we present a volume visualization system, called TAn2 (Tetrahedra Analyzer 2) which is based on a multiresolution model which specializes for the three dimensional case the model proposed in [4, 9], called the Multi Tesselation (MT) CNR Research Area S. Cataldo 56100 Pisa, ITALY Email: cignoniiei. pi. cnr. it iVia Dodecaneso, 35, 16146 Genova, ITALY Emaih de1o ,magillo ,puppo disi. unige. it CNR Research Area S. Cataldo 56100 Pisa, ITALY Emaih roberto. scopignocnuce. cnr. ....

E. Puppo. Variable resolution terrain surfaces. In Pro- ceedings Eight Canadian Conference on Computational Geometry, pages 202 210, Ottawa, Canada, August 12-15 1996.

....mesh. Set S, however, must be consistent with the partial order de ned in the DAG: for every node u 2 S, all the parents of u must be also in S. A consistent set S may contain more updates in some areas, and fewer updates elsewhere, hence de ning a mesh S whose resolution is variable in space. In [22] it has been shown that all possible meshes formed by triangles described by an MT are in one to one correspondence with its consistent subsets of updates. As de ned in [6] the fundamental query on a multiresolution model, called selective re nement, consists of extracting a mesh such that the ....

E. Puppo. Variable resolution terrain surfaces. In Proceedings Eight Canadian Conference on Computational Geometry, pages 202-210, Ottawa, Canada, August 12-15 1996.

.... set of meshes at di#er9 t Levels Of Detail (LODs) can beextr4q65H based on e#cient grB4 tr ver9O strq65O2B5 Her we pr5L5 t a volume visualization system, called TAn2 (Tetrahedra An 2) which is based on a multir65525Oqr model which specializesfor the thr4 dimensional case the modelpr5 osed in [4, 9], called the Multi Tesselation (MT) # CNR ResearchA rea S. Cataldo 56100 Pisa, ITA LY Email: cignoni iei.pi.cnr.it ia Dodecaneso, 35, 16146 Genova, ITA LY Email: deflo,magillo,puppo disi.unige.it # CNR ResearchA rea S. Cataldo 56100 Pisa, ITA LY Email: ....

E. Puppo. Variable resolution terrain surfaces. In Proceedings Eight Canadian Conference on Computational Geometry, pages 202--210, Ottawa, CanadT August 12-15 1996.

....mature stage, which allows developers to use it through a common interface, without the need to care about many details. In our previous work, we have developed a general model, called a Simplicial Multi Complex (SMC) that can capture all LOD models based on simplicial complexes as special cases [13, 5, 14]. Based on such model, we have built systems for managing the level of detail in terrains [2] and in free form surfaces [3] and we are currently developing an application in volume visualization. In this paper, we consider general operations that can be performed on LOD models and propose an ....

....of spatial relations. In Section 6, we present some concluding remarks. 2 Simplicial Multi Complexes In this section, we briefly review the main concepts about the Simplicial MultiComplex, a dimension independent multiresolution simplicial model which extends the Multi Triangulation presented in [13, 5, 14]. For the sake of brevity, this subject is treated informally here. For a formal treatment and details see [11] In the remainder of the paper, we denote with k and d two integer numbers such that 0 k d. A k dimensional simplex oe is the locus of points that can be expressed as the convex ....

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E. Puppo. Variable resolution terrain surfaces. In Proceedings Eight Canadian Conference on Computational Geometry, pages 202--210, Ottawa, Canada, August 12-15 1996.

....structures and optimization techniques they adopt, as well as custom tailored to perform speci c operations. On the other hand, a wide range of potential applications for LOD would bene t from having a common, general and customizable set of operations which could be considered as a standard. In [11] we have rst introduced a general model that can capture all LOD models based on triangle meshes as special cases. Construction algorithms, and algorithms for extracting variable resolution meshes from it are presented in [3, 5] Here, we call Multi Tesselation (MT) the extension of such model to ....

E. Puppo. Variable resolution terrain surfaces. In Proceedings Eight Canadian Conference on Computational Geometry, pages 202-210, 1996.

....server answers to queries by traversing the multiresolution model and transmitting updates to the client, which uses them to progressively modify a current mesh. We study this problem in the context of a vertex based multiresolution model, which is a special instance of the Multi Triangulation [Pup96, DFPM97], based on vertex insertion and removal. We de ne a compact data structure for such a model which exploits the speci c update rule. We propose a dynamic algorithm for selective re nement and we discuss in detail its implementation as a client server application. In order to reduce memory ....

....structure for a multiresolution model to handle huge meshes on the server, and an ecient selective re nement algorithm capable of dynamically modifying previously extracted meshes and operating in a client server environment. We develop here an instance of the Multi Triangulation as de ned in [Pup96, DFPM97], that we call a Vertex based Multi Triangulation (MT) since it is built based on a vertex decimation strategy. Vertex decimation is a very popular technique used in simpli cation which allows preserving the topology of the mesh and the shape of the triangles (which is important when simplifying, ....

[Article contains additional citation context not shown here]

E. Puppo. Variable resolution terrain surfaces. In Proceedings Eight Canadian Conference on Computational Geometry, pages 202-210, Ottawa, Canada, August 12-15 1996.

....server answers to queries by traversing the multiresolution model and transmitting updates to the client, which uses them to progressively modify a current mesh. We study this problem in the context of a vertex based multiresolution model, which is a special instance of the Multi Triangulation [Pup96, DFPM97], based on vertex insertion and removal. We define a compact data structure for such a model which exploits the specific update rule. We propose a dynamic algorithm for selective refinement and we discuss in detail its implementation as a client server application. In order to reduce memory ....

....structure for a multiresolution model to handle huge meshes on the server, and an efficient selective refinement algorithm capable of dynamically modifying previously extracted meshes and operating in a clientserver environment. We develop here an instance of the Multi Triangulation as defined in [Pup96, DFPM97], that we call a Vertex based Multi Triangulation (MT) since it is built based on a vertex decimation strategy. Vertex decimation is a very popular technique used in simplification which allows preserving the topology of the mesh and the shape of the triangles (which is important when ....

[Article contains additional citation context not shown here]

E. Puppo. Variable resolution terrain surfaces. In Proceedings Eight Canadian Conference on Computational Geometry, pages 202--210, Ottawa, Canada, August 12-15 1996.

....stage, which allows developers to use it through a common interface, without the need to care about too many details. In our previous work, we have developed a general model, called a Simplicial Multi Complex (SMC) that can capture all LOD models based on siplicial complexes as special cases [Pup96, DFPM97a] Based on such model, we have built systems for managing the level of detail in terrains [DFMP97b] and in free form surfaces [DFMP98] and we are currently developing an application in volume visualization. On the basis of our experience, we have elaborated a general study of ....

....relations. Finally, we present some concluding remarks in Section 6. 2 Simplicial Multi Complexes In this section we briefly review main concepts about the Simplicial Multi Complex, a dimension independent multiresolution simplicial model which extends the Multi Triangulation presented in [Pup96, DFPM97a] For the sake of brevity, this subject is treated informally here (for a formal treatment, and details see [Mag99] A k dimensional simplex oe is the locus of points that can be expressed as the convex combination of k 1 affinely independent points in IR d , called the vertices of ....

[Article contains additional citation context not shown here]

E. Puppo. Variable resolution terrain surfaces. In Proceedings Eight Canadian Conference on Computational Geometry, pages 202--210, Ottawa, Canada, August 12-15 1996. Extended version to appear under title "Variable resolution triangulations" in Computational Geometry Theory and Applications.

....ordered set, where the order relation depends on spatial interferences of components, and on the possibility to combine them to obtain simplicial complexes describing the given object at different resolutions, and possibly at different levels of resolution through the domain. In a companion paper [15], more properties of MSMs are investigated in the two dimensional case, and their application to the representation of terrain surfaces is presented. Multiresolution models proposed in the literature are usually classified into and models on the basis of the characteristics of their domain ....

....navigation of the model within a given resolution, and the solution of spatial queries at an assigned level of resolution. In the final part of the paper, we analyze within the above framework different data structures that have been proposed in the literature for multiresolution models [1, 5, 8, 14, 15, 16, 18]. The paper is organized as follows: in Section 2 we introduce the multiresolution simplicial model, related definitions and properties, and show how models proposed in the literature can be interpreted as MSMs. In Section 3 we define three basic operations on MSMs, namely the extraction of a 0 ....

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E. Puppo, 1996, Variable resolution terrain surfaces, , Ottawa (Canada), 12-15 August, 1996, to appear. Also published in longer version as N.6/96, Istituto per la Matematica Applicata, C.N.R., Genova, Italy, 1996. Available from

....ffl provide always conforming meshes, i.e. avoid cracks due to abrupt transition between different levels of detail within a mesh; ffl have a size not much higher than the size of the model at full resolution. In the following, we describe a general framework, called a Multiresolution Mesh (MM) [DeF96, Pup96]. Multiresolution models proposed in the literature will be reviewed next as instances of such a framework. Existing multiresolution models based on domain decomposition are all obtained from an initial model, which is progressively modified by a generalization algorithm (either by refinement or ....

....set of components, which can be represented by a DAG (see Figure 3) The collection of such components, plus the partial order, form a Multiresolution Mesh (MM) which is at the heart of any multiresolution model. An MM has several interesting properties that have been investigated in detail in [Pup96]. In particular, any consistent (i.e. dependency closed) subset of components defines a mesh representing the terrain at a given level of detail: such a mesh is obtained by performing the corresponding local modifications in any temporal sequence consistent with the dependency relation. Moreover, ....

[Article contains additional citation context not shown here]

Puppo, E., Variable resolution terrain surfaces, Proceedings Canadian Conference on Computational Geometry, Ottawa (Canada) 12-15 August, 1996; appeared in longer version as Variable resolution triangulations, Technical Report 12/96, Istituto per la Matematica Applicata, C.N.R., Genova, Italy, 1996.

....space, ensuring controlled image quality and no popping effects. The same strategy can also be used in our approach: in variable resolution extraction, error can be evaluated by taking into account the screen space magnitude (computed on the fly) of the error associated to each facet. Finally, (Puppo, 1996) gives a comprehensive multiresolution model, which includes all models mentioned above, and the one described in this paper, as special cases: an algorithm has been also proposed, which extracts from such a model a representation of minimal size for a given threshold, variable over the domain, in ....

Puppo, E. (1996). Variable resolution terrain surfaces. In Proceedings Eight Canadian Conference on Computational Geometry, Ottawa, Canada, pages 202--210.

....of transformation operator. Most proposed models are a direct abstraction of the data structure used to implement them. Here, we formally define a general multiresolution model of a shape based on cell complexes, called the Multi Complex (MC) which extends the MultiTriangulation, proposed in [Pup96, DFPM97] The MC describes a partially ordered set of modification patterns specifying how an initial coarse cell complex can be iteratively refined in order to obtain more and more accurate representations. The partial order captures dependency relations between modifications that affect the ....

....sub MC is linear in the size of its front complex. In particular, the size of the MC itself is linear in the size of its front cell complex. This means that the multiresolution structure introduces only a linear storage overhead with respect to a simple cell complex at the maximum resolution. In [Pup96, Mag98] we have shown that a linear growth is fundamental to achieve optimal time complexity (i.e. linear in the output size) for algorithms which extract a cell complex at a given resolution from an MC: since these algorithms perform a DAG traversal to find the appropriate sub MC, linear ....

E. Puppo. Variable resolution terrain surfaces. In Proceedings Eight Canadian Conference on Computational Geometry, pages 202--210, Ottawa, Canada, August 12-15 1996. Extended version to appear under title "Variable resolution triangulations" in Computational Geometry Theory and Applications.

....a boundary, and using blending on such cells. A more rigourous solution of such a problem should be given at the level of the multiresolution extractor module, by providing a mechanism for extracting a mesh whose accuracy varies smoothly and consistently through domain. In recent works [9] [32] we proposed alternative multiresolution data structures that provide efficient solutions to this problem, and that produce effective results in the two dimensional case, e.g. for visualizing terrain models in the context of flight simulators. However, such structures may require a relevant ....

E. Puppo. Variable resolution terrain surfaces. In Proceedings Eight Canadian Conference on Computational Geometry, Ottawa, Canada, pages 202--210, August 12-15 1996.

....called a covering of Omega Gamma Given a generic t set T 0 having Omega as domain, any covering of Omega formed of triangles of T 0 is called a triangulation generated by T 0 . 4 Multi triangulations All results in this section are stated without any proof. Complete proofs are given in [10]. Given two triangulations T i and T j , we define the interference Omega , the subtraction Psi, and the pasting Phi, respectively, as follows: T i Omega T j = ft 2 T i j 9t 0 2 T j ; i(t) t 0 6= g (1) T i Psi T j = T i n (T i Omega T j ) 2) T i Phi T j = T i Psi T j ) T ....

....over its sub closure is a MT, called the reverse. Such a structure can be encoded together with the primary one, and, having the same associated t set, it generates the same set of triangulations. In case of monotone MTs, the reverse has a monotonicity opposed to the primary structure (see [10] for details) Definition 4.8 An increasing multi triangulation T has linear growth if and only if for each lower set T 0 T the size of T 0 is linear in the size of its upward pasting. A decreasing multi triangulation has linear growth if and only if its reverse has linear growth. In ....

E. Puppo, 1996, Variable resolution terrain surfaces, Techincal Report N.6/96, Istituto per la Matematica Applicata, C.N.R., Genova, Italy, April 1996.

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E. Puppo. Variable resolution terrain surfaces. In Proceedings of the 8th Canadian Conference on Computational Geometry, 1996.

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E. Puppo. Variable resolution terrain surfaces. In Proceedings of the 8th Canadian Conference on Computational Geometry, 1996. 3

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E. Puppo. Variable resolution terrain surfaces. In Proceedings Eight Canadian Conference on Computational Geometry, Ottawa, Canada, pages 202--210, August 12-15 1996. 2

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E. Puppo, Variable resolution terrain surfaces, Proceedings Eight Canadian Conference on Computational Geometry, Ottawa, Canada, August 12-15 1996, pp. 202--210.

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E. Puppo. Variable Resolution Terrain Surfaces, Proc. Eight Canadian Conference on Computational Geometry, 202--210, 1996.

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