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23
Detecting and Reconstructing Subdivision Connectivity
, 2001
"... In this paper we introduce fast and efficient inverse subdivision algorithms, with linear time and space complexity, to detect and reconstruct uniform Loop, CatmullClark, and DooSabin subdivision structure in irregular triangular, quadrilateral, and polygonal meshes. We consider two main applicati ..."
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In this paper we introduce fast and efficient inverse subdivision algorithms, with linear time and space complexity, to detect and reconstruct uniform Loop, CatmullClark, and DooSabin subdivision structure in irregular triangular, quadrilateral, and polygonal meshes. We consider two main applications for these algorithm. The first one is to enable interactive modeling systems that support uniform subdivision surfaces, to use popular interchange file formats which do not preserve the subdivision structure, such as VRML, without loss of information. The second application is to improve the compression efficiency of existing lossless connectivity compression schemes, by optimally compressing meshes with Loop subdivision connectivity. Our Loop inverse subdivision algorithm is based on global connectivity properties of the covering mesh, a concept motivated by the covering surface from Algebraic Topology. Although the same approach can be used for other subdivision schemes such as CatmullClark, we present a CatmullClark inverse subdivision algorithm based on a much simpler graph coloring algorithm, and a DooSabin inverse subdivision algorithm based on properties of the dual mesh. Straightforward extensions of these approaches to other popular uniform subdivision schemes are also discussed.
Teaching meshes, subdivision and multiresolution techniques
, 2004
"... In recent years, geometry processing algorithms that directly operate on polygonal meshes have become an indispensable tool in computer graphics, CAD/CAM applications, numerical simulations, and medical imaging. Because the demand for people that are specialized in these techniques increases steadil ..."
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In recent years, geometry processing algorithms that directly operate on polygonal meshes have become an indispensable tool in computer graphics, CAD/CAM applications, numerical simulations, and medical imaging. Because the demand for people that are specialized in these techniques increases steadily the topic is finding its way into the standard curricula of related lectures on computer graphics and geometric modeling and is often the subject of seminars and presentations. In this article we suggest a toolbox to educators who are planning to set up a lecture or talk about geometry processing for a specific audience. For this we propose a set of teaching blocks, each of which covers a specific subtopic. These teaching blocks can be assembled so as to fit different occasions like lectures, courses, seminars and talks and different audiences like students and industrial practitioners. We also provide examples that can be used to deepen the subject matter and give references to the most relevant work.
Efficient and robust algorithm for overlaying nonmatching surface meshes
 In 10th International Meshing Roundtable
, 2001
"... This paper describes an efficient and robust algorithm for computing a common refinement of two meshes modeling a common surface of arbitrary shape by overlaying them on top of each other. A common refinement is an important data structure for transferring data between meshes that have different top ..."
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This paper describes an efficient and robust algorithm for computing a common refinement of two meshes modeling a common surface of arbitrary shape by overlaying them on top of each other. A common refinement is an important data structure for transferring data between meshes that have different topological structures. Our algorithm is optimal in time and space, with linear complexity. Special treatments are introduced to handle discretization and rounding errors and to ensure robustness with imprecise computations. It also addresses the additional complexities caused by degeneracies, sharp edges, sharp corners, and nonmatching boundaries. The algorithm has been implemented and demonstrated to be robust for complex geometries from realworld applications.
Generic Programming Techniques that Make Planar Cell Complexes Easy to Use
 Proc. of Dagstuhl WS on Digital and Image Geometry 2000, Springer LNCS
, 2000
"... Cell complexes are potentially very useful in many fields, including image segmentation, numerical analysis, and computer graphics. However, in practice they are not used as widely as they could be. This is partly due to the difficulties in actually implementing algoritStms on top of cell complex ..."
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Cell complexes are potentially very useful in many fields, including image segmentation, numerical analysis, and computer graphics. However, in practice they are not used as widely as they could be. This is partly due to the difficulties in actually implementing algoritStms on top of cell complexes. We propose to use generic programming to design cell complex data stxuctures that are easy to use, efficient, and flexible. The implementation of the new design is demonstrated for a number of common cell complex types and an example algorithm.
A Control of Smooth Deformations With Topological Change on a Polyhedral Mesh Based on Curves and Loops
, 2002
"... We propose a method to model and control topological changes by a smooth deformation of a polyhedral mesh using curves and loops. As changing the genus of a surface is not a continuous transformation, the topological change is made when an intermediate shape between the two topologies has been obtai ..."
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We propose a method to model and control topological changes by a smooth deformation of a polyhedral mesh using curves and loops. As changing the genus of a surface is not a continuous transformation, the topological change is made when an intermediate shape between the two topologies has been obtained. The creation and the deletion of holes are studied. The deletion of a hole uses non nullhomotopic loops to designate the hole to be deleted. A method computing two independent loops associated to a hole is presented.
Three Dimensional Triangulations in CGAL
, 1999
"... This paper describes the design and the implementation of the threedimensional triangulation package 1 of the Computational Geometric Algorithms Library Cgal 2 . We focus on representation issues and especially insist on how the cases of degenerate dimensions are treated. The algorithmic issues ..."
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This paper describes the design and the implementation of the threedimensional triangulation package 1 of the Computational Geometric Algorithms Library Cgal 2 . We focus on representation issues and especially insist on how the cases of degenerate dimensions are treated. The algorithmic issues are not examined in this short paper. 1 Introduction A threedimensional triangulation is a threedimensional simplicial complex, pure connected and without singularities [BY98]. It is a set of cells (tetrahedra) such that two cells either do not intersect or share a common facet, edge or vertex. Generalizing the storage of 2D triangulations [tri99] to the 3D case, we choose to explicitly represent only cells and vertices, together with adjacency and incidence relations: a cell has pointers to its four vertices and to its four neighbors, a vertex has a pointer to one of the cells having this vertex. Design Overview We follow the design in three layers proposed for polyhedral surfaces by L...
Interactive Volume Visualization of General Polyhedral Grids
"... Fig. 1. Interactive raycasting of the temperature distribution in an exhaust manifold that was simulated using a stateoftheart CFD solver on a complex grid composed of general polyhedral cells. Red color indicates warm, green cool regions. The complex structure of the underlying mesh is illustra ..."
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Fig. 1. Interactive raycasting of the temperature distribution in an exhaust manifold that was simulated using a stateoftheart CFD solver on a complex grid composed of general polyhedral cells. Red color indicates warm, green cool regions. The complex structure of the underlying mesh is illustrated through cell faces. The cells in this mesh are not only tetrahedra or other predefined cell types, but also are very general, often nonconvex, polyhedra with nonplanar faces, which our approach can nevertheless visualize directly. Abstract—This paper presents a novel framework for visualizing volumetric data specified on complex polyhedral grids, without the need to perform any kind of a priori tetrahedralization. These grids are composed of polyhedra that often are nonconvex and have an arbitrary number of faces, where the faces can be nonplanar with an arbitrary number of vertices. The importance of such grids in stateoftheart simulation packages is increasing rapidly. We propose a very compact, facebased data structure for representing such meshes for visualization, called twosided face sequence lists (TSFSL), as well as an algorithm for direct GPUbased raycasting using this representation. The TSFSL data structure is able to represent the entire mesh topology in a 1D TSFSL data array of face records, which facilitates the use of efficient 1D texture accesses for visualization. In order to scale to large data sizes, we employ a mesh decomposition into bricks that can be handled independently, where each brick is then composed of its own TSFSL array. This bricking enables memory savings and performance improvements for large meshes. We illustrate the feasibility of our approach with realworld application results, by visualizing highly complex polyhedral data from commercial stateoftheart simulation packages. Index Terms—Volume rendering, unstructured grids, polyhedral grids, GPUbased visualization. 1
C++ Classes for 2D Unstructured Mesh Programming
 Domaine de Voluceau, Rocquencourt, BP 105, 78153, Le Chesnay
, 1999
"... In this report, a set of C++ classes is presented for representing unstructured triangular meshes of intrinsic dimension two; i.e. oriented 2manifolds. Simple classes for the basic mesh objects, i.e. vertices, triangles, and line segments, are described. They define abstractions based on their i ..."
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In this report, a set of C++ classes is presented for representing unstructured triangular meshes of intrinsic dimension two; i.e. oriented 2manifolds. Simple classes for the basic mesh objects, i.e. vertices, triangles, and line segments, are described. They define abstractions based on their incidence relations and a few geometric primitives for a mesh class, which is an intelligent container class of three lists of these simple mesh objects. The classes are intended to be components in an object oriented approach to software for meshing applications described in the report. This context differentiates the roles of the mesh class and the simple mesh object classes; these latter can be extended as the carriers of the applications data. The capability of the classes of this report to simultaneously simplify the coding of mesh methods and facilitate generalization of the code is discussed with examples. The report provides an overview of the class design and use, tutorial e...
A Geometric Framework for Computer Graphics Addressing Modeling, Visibility, and Shadows
, 1999
"... The main question this dissertation addresses is the following: Is it possible to design a computer graphics API such that modeling primitives, computing visibility, and generating shadows from point, linear, and area light sources can be conveniently and concisely expressed? The thesis answers this ..."
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The main question this dissertation addresses is the following: Is it possible to design a computer graphics API such that modeling primitives, computing visibility, and generating shadows from point, linear, and area light sources can be conveniently and concisely expressed? The thesis answers this question in the affirmative by describing a framework for geometric computing in computer graphics. The classes in the layered system constituting the framework are described using the UML notation and each algorithm presented is encapsulated in a member method of a class in the hierarchy. We identify a number of abstractions for object–space graphics such as transparent visibility and opaque visibility. These abstractions are somewhat harder to implement than standard rasterized abstractions as they rely on graphs and planar maps. Nevertheless, these notions prove to be fundamental in this work on object–space graphics and also appear to be fundamental for computer graphics in general. We propose that clients of a graphics API such as the one presented here should be relieved from the onus of computing shadows and we show that the computation of shadows can be automated and encapsulated in the framework. We address illumination under a point, a linear,
Geometric Representations and Transformations
"... This chapter provides important background material that will be needed for Part II. Formulating and solving motion planning problems require defining and manipulating complicated geometric models of a system of bodies in space. Section 3.1 introduces geometric modeling, which focuses mainly on semi ..."
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This chapter provides important background material that will be needed for Part II. Formulating and solving motion planning problems require defining and manipulating complicated geometric models of a system of bodies in space. Section 3.1 introduces geometric modeling, which focuses mainly on semialgebraic modeling because it is an important part of Chapter 6. If your interest is mainly in Chapter 5, then understanding semialgebraic models is not critical. Sections 3.2 and 3.3 describe how to transform a single body and a chain of bodies, respectively. This will enable the robot to “move. ” These sections are essential for understanding all of Part II and many sections beyond. It is expected that many readers will already have some or all of this background (especially Section 3.2, but it is included for completeness). Section 3.4 extends the framework for transforming chains of bodies to transforming trees of bodies, which allows modeling of complicated systems, such as humanoid robots and flexible organic molecules. Finally, Section 3.5 briefly covers transformations that do not assume each body is rigid.