Hamiltonian triangulations for fast rendering

Arkin, Esther M.; Held, Martin; Mitchell, Joseph S. B.; Skiena, Steven

doi:10.1007/bfb0049395

Cited by 46 publications

(50 citation statements)

References 9 publications

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“…In graph theory, the problem of Hamiltonian cycle is to find a cycle passing through every vertex exactly once on a given graph. For triangular mesh, Hamiltonian cycle is a closed curve visiting vertices of the mesh once and only once and has been widely used in triangular mesh processing, such as model rendering [3], mesh compression [34], shape sculpture [1]. For generating Hamiltonian cycle on 3D surface, a greedy RING-EXPANDER algorithm [9] is adopted here, which is simple and linear in time and space.…”

Section: Hamiltonian Cycle On 3d Surfacementioning

confidence: 99%

Efficient EMD and Hilbert spectra computation for 3D geometry processing and analysis via space-filling curve

Wang

Zhang

et al. 2015

Vis Comput

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Empirical Mode Decomposition (EMD) has proved to be an effective and powerful analytical tool for non-stationary time series and starts to exhibit its modeling potential for 3D geometry analysis. Yet, existing EMD-based geometry processing algorithms only concentrate on multiscale data decomposition by way of computing intrinsic mode functions. More in-depth analytical properties, such as Hilbert spectra, are hard to study for 3D surface signals due to the lack of theoretical and algorithmic tools. This has hindered much more broader penetration of EMD-centric algorithms into various new applications on 3D surface. To tackle this challenge, in this paper we propose a novel and efficient EMD and Hilbert spectra computational scheme for 3D geometry processing and analysis. At the core of our scheme is the strategy of dimensionality reduction via spacefilling curve. This strategy transforms the problem of 3D geometry analysis to 1D time series processing, leading to two major advantages. First, the envelope computation is carried out for 1D signal by cubic spline interpolation, which is much faster than existing envelope computation directly over B Jianping Hu 3D surface. Second, it enables us to calculate Hilbert spectra directly on 3D surface. We could take advantages of Hilbert spectra that contain a wealth of unexploited properties and utilize them as a viable indicator to guide our EMD-based 3D surface processing. Furthermore, to preserve sharp features, we develop a divide-and-conquer scheme of EMD by explicitly separating the feature signals from non-feature signals. Extensive experiments have been carried out to demonstrate that our new EMD and Hilbert spectra based method is both fast and powerful for 3D surface processing and analysis.

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Section: Hamiltonian Cycle On 3d Surfacementioning

confidence: 99%

Efficient EMD and Hilbert spectra computation for 3D geometry processing and analysis via space-filling curve

Wang

Zhang

et al. 2015

Vis Comput

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“…PROOF. Any set S of n points has a path triangulation (a triangulation whose dual graph has a Hamiltonian path), which can be constructed in O(n log n) time [2], [8] (Figure 8 illustrates such a triangulation of a point set). Denote by t the number of triangles in any triangulation of n points with h extreme points (t = 2n − 2 − h).…”

Section: Upper Boundmentioning

confidence: 99%

Small Strictly Convex Quadrilateral Meshes of Point Sets

et al. 2003

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In this paper we give upper and lower bounds on the number of Steiner points required to construct a strictly convex quadrilateral mesh for a planar point set. In particular, we show that 3 n/2 internal Steiner points are always sufficient for a convex quadrilateral mesh of n points in the plane. Furthermore, for any given n ≥ 4, there are point sets for which (n − 3)/2 − 1 Steiner points are necessary for a convex quadrilateral mesh. Introduction.Discrete approximations of a surface or volume are necessary in numerous applications. Some examples are models of human organs in medical imaging, terrain models in GIS, or models of parts in a CAD/CAM system. These applications typically assume that the geometric domain under consideration is divided into small, simple pieces called finite elements. The collection of finite elements is referred to as a mesh. For several applications, quadrilateral/hexahedral mesh elements are preferred over triangles/tetrahedra owing to their numerous benefits, both geometric and numerical; for example, quadrilateral meshes give lower approximation errors in finite element methods for elasticity analysis [1], [3] or metal forming processes [15]. However, much less is known about quadrilateralizations and hexahedralizations and, in general, highquality quadrilateral/hexahedral (quad/hex) meshes are harder to generate than good triangular/tetrahedral (tri/tet) ones. Indeed, there are several important open questions, both combinatorial as well as algorithmic, about quad/hex meshes for sets of objects such as polygons, points, etc., even in two dimensions. Whereas triangulations of polygons and two-dimensional point sets and tetrahedralizations of three-dimensional point sets and convex polyhedra always exist (not so for non-convex polyhedra [29]), quadrilateralizations of two-dimensional point sets do not. Hence it becomes necessary to add extra points, called Steiner points, to the geometric domain. This raises the issue of bound-

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“…To achieve this, quantization techniques [50] are used, applied either globally, on the entire mesh [8,14,24,57], or partially, per region [12]. A vertex position is predicted from one or several of its neighbors by using delta coding [38] or linear prediction [4,23] along the vertex ordering imposed by the coding of the connectivity. Being imposed by the connectivity, this mesh traversal is still not optimal for geometry coding, other techniques [16,39] proposing to use the geometry as a driver of mesh traversal.…”

Section: Static Mesh Compressionmentioning

confidence: 99%