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Centroidal Voronoi Tesselation of Line Segments and Graphs
"... Figure 1: Starting from a mesh (A) and a template skeleton (B), our method fits the skeleton to the mesh (C) and outputs a segmentation (D). Our main contribution is an extension of Centroidal Voronoi Tesselation to line segments, using approximated Voronoi Diagrams of segments (E). Segment Voronoi ..."
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Figure 1: Starting from a mesh (A) and a template skeleton (B), our method fits the skeleton to the mesh (C) and outputs a segmentation (D). Our main contribution is an extension of Centroidal Voronoi Tesselation to line segments, using approximated Voronoi Diagrams of segments (E). Segment Voronoi cells (colors) are approximated by the union of sampled point’s Voronoi cells (thin lines, right half of D). Clipped 3D Voronoi cells are accurately computed, at a subfacet precision (F). Centroidal Voronoi Tesselation (CVT) of points has many applications in geometry processing, including remeshing and segmentation to name but a few. In this paper, we propose a new extension of CVT, generalized to graphs. Given a graph and a 3D polygonal surface, our method optimizes the placement of the vertices of the graph in such a way that the graph segments best approximate the shape of the surface. We formulate the computation of CVT for graphs as a continuous variational problem, and present a simple approximated method to solve this problem. Our method is robust in the sense that it is independent of degeneracies in the input mesh, such as skinny triangles, Tjunctions, small gaps or multiple connected components. We present some applications, to skeleton fitting and to shape segmentation.
Boundary Aligned Smooth 3D CrossFrame Field
"... Figure 1: Snapshots of the optimization procedure to construct a boundary aligned 3D crossframe field. The top row shows the internal streamlines. The next row contains another visualization with cubes spread by a parameterization along the current crossframe field and rotated by the current local ..."
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Cited by 11 (2 self)
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Figure 1: Snapshots of the optimization procedure to construct a boundary aligned 3D crossframe field. The top row shows the internal streamlines. The next row contains another visualization with cubes spread by a parameterization along the current crossframe field and rotated by the current local frame R(Φ). The corresponding number of iteration is shown at the bottom. In this paper, we present a method for constructing a 3D crossframe field, a 3D extension of the 2D crossframe field as applied to surfaces in applications such as quadrangulation and texture synthesis. In contrast to the surface crossframe field (equivalent to a 4Way RotationalSymmetry vector field), symmetry for 3D crossframe fields cannot be formulated by simple oneparameter 2D rotations in the tangent planes. To address this critical issue, we represent the 3D frames by spherical harmonics, in a manner invariant to combinations of rotations around any axis by multiples of π/2. With such a representation, we can formulate an efficient smoothness measure of the crossframe field. Through minimization of this measure under certain boundary conditions, we can construct a smooth 3D crossframe field that is aligned with the surface normal at the boundary. We visualize the resulting crossframe field through restrictions to the boundary surface, streamline tracing in the volume, and singularities. We also demonstrate the application of the 3D crossframe field to producing hexahedrondominant meshes for given volumes, and discuss its potential in highquality hexahedralization, much as its 2D counterpart has shown in quadrangulation.
DesignDriven Quadrangulation of Closed 3D Curves
"... (a) input curve network (c) pairing and iterative refinement (f) design rendering (b) initial segmentation (d) final quadrangulation and quadmesh (e) designdriven quadrangulation Figure 1: Steps to quadrangulating a design network of closed 3D curves (a) : Closed curves are independently segmented ..."
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Cited by 11 (4 self)
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(a) input curve network (c) pairing and iterative refinement (f) design rendering (b) initial segmentation (d) final quadrangulation and quadmesh (e) designdriven quadrangulation Figure 1: Steps to quadrangulating a design network of closed 3D curves (a) : Closed curves are independently segmented (b) and iteratively paired and refined to capture dominant flowlines as well as overall flowline quality (c); final quadrangulation in green and dense quadmesh (d); quadrangulations are aligned across adjacent cycles to generate a single densely sampled mesh (e), suitable for design rendering and downstream applications (f). We propose a novel, designdriven, approach to quadrangulation of closed 3D curves created by sketchbased or other curve modeling systems. Unlike the multitude of approaches for quadremeshing of existing surfaces, we rely solely on the input curves to both conceive and construct the quadmesh of an artist imagined surface bounded by them. We observe that viewers complete the intended shape by envisioning a dense network of smooth, gradually changing, flowlines that interpolates the input curves. Components of the network bridge pairs of input curve segments with similar orientation and shape. Our algorithm mimics this behavior. It first segments the input closed curves into pairs of matching segments, defining dominant flow line sequences across the surface. It then interpolates the input curves by a network of quadrilateral cycles whose isolines define the desired flow line network. We proceed to interpolate these networks with allquad meshes that convey designer intent. We evaluate our results by showing convincing quadrangulations of complex and diverse curve networks with concave, nonplanar cycles, and validate our approach by comparing our results to artist generated interpolating meshes. 1
QuadMesh Generation and Processing: a survey
"... Triangle meshes have been nearly ubiquitous in computer graphics, and a large body of data structures and geometry processing algorithms based on them has been developed in the literature. At the same time, quadrilateral meshes, especially semiregular ones, have advantages for many applications, an ..."
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Cited by 10 (3 self)
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Triangle meshes have been nearly ubiquitous in computer graphics, and a large body of data structures and geometry processing algorithms based on them has been developed in the literature. At the same time, quadrilateral meshes, especially semiregular ones, have advantages for many applications, and significant progress was made in quadrilateral mesh generation and processing during the last several years. In this survey we discuss the advantages and problems of techniques operating on quadrilateral meshes, including surface analysis and mesh quality, simplification, adaptive refinement, alignment with features, parametrization, and remeshing.
Variational Anisotropic Surface Meshing with Voronoi Parallel Linear Enumeration
"... This paper introduces a new method for anisotropic surface meshing. From an input polygonal mesh and a specified number of vertices, the method generates a curvatureadapted mesh. The main idea consists in transforming the 3d anisotropic space into a higher dimensional isotropic space (typically 6d ..."
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Cited by 6 (2 self)
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This paper introduces a new method for anisotropic surface meshing. From an input polygonal mesh and a specified number of vertices, the method generates a curvatureadapted mesh. The main idea consists in transforming the 3d anisotropic space into a higher dimensional isotropic space (typically 6d or larger). In this high dimensional space, the mesh is optimized by computing a Centroidal Voronoi Tessellation (CVT), i.e. the minimizer of a C 2 objective function that depends on the coordinates at the vertices (quantization noise power). Optimizing this objective function requires to compute the intersection between the (higher dimensional) Voronoi cells and the surface (Restricted Voronoi Diagram). The method overcomes the dfactorial cost of computing a Voronoi diagram of dimension d by directly computing the restricted Voronoi cells with a new algorithm that can be easily parallelized (Vorpaline: Voronoi Parallel Linear Enumeration). The method is demonstrated with several examples comprising CAD and scanned meshes.
Fitting polynomial surfaces to triangular meshes with Voronoi squared distance minimization
 In 20th International Meshing Roundtable–IMR2011. 601–618
, 2012
"... This paper introduces Voronoi Squared Distance Minimization (VSDM), an algorithm that fits a surface to an input mesh. VSDM minimizes an objective function that corresponds to a Voronoibased approximation of the overall squared distance function between the surface and the input mesh (SDM). This ob ..."
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This paper introduces Voronoi Squared Distance Minimization (VSDM), an algorithm that fits a surface to an input mesh. VSDM minimizes an objective function that corresponds to a Voronoibased approximation of the overall squared distance function between the surface and the input mesh (SDM). This objective function is a generalization of Centroidal Voronoi Tesselation (CVT), and can be minimized by a quasiNewton solver. VSDM naturally adapts the orientation of the mesh to best approximate the input, without estimating any differential quantities. Therefore it can be applied to triangle soups or surfaces with degenerate triangles, topological noise and sharp features. Applications of fitting quad meshes and polynomial surfaces to input triangular meshes are demonstrated. 1
State of the Art in Quad Meshing
"... Triangle meshes have been nearly ubiquitous in computer graphics, and a large body of data structures and geometry processing algorithms based on them has been developed in the literature. At the same time, quadrilateral meshes, especially semiregular ones, have advantages for many applications, an ..."
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Cited by 5 (3 self)
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Triangle meshes have been nearly ubiquitous in computer graphics, and a large body of data structures and geometry processing algorithms based on them has been developed in the literature. At the same time, quadrilateral meshes, especially semiregular ones, have advantages for many applications, and significant progress was made in quadrilateral mesh generation and processing during the last several years. In this State of the Art Report, we discuss the advantages and problems of techniques operating on quadrilateral meshes, including surface analysis and mesh quality, simplification, adaptive refinement, alignment with features, parametrization, and remeshing.
Computing 2D periodic centroidal Voronoi tessellation
 In Proceedings of the 2011 International Symposium on Voronoi Diagrams in Science and Engineering
, 2011
"... Abstract—In this paper, we propose an efficient algorithm to compute the centroidal Voronoi tessellation in 2D periodic space. We first present a simple algorithm for constructing the periodic Voronoi diagram (PVD) from a Euclidean Voronoi diagram. The presented PVD algorithm considers only a small ..."
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Abstract—In this paper, we propose an efficient algorithm to compute the centroidal Voronoi tessellation in 2D periodic space. We first present a simple algorithm for constructing the periodic Voronoi diagram (PVD) from a Euclidean Voronoi diagram. The presented PVD algorithm considers only a small set of periodic copies of the input sites, which is more efficient than previous approaches requiring full copies of the sites (9 in 2D and 27 in 3D). The presented PVD algorithm is applied in a fast Newtonbased framework for computing the centroidal Voronoi tessellation (CVT). We observe that fullhexagonal patterns can be obtained via periodic CVT optimization attributed to the convergence of the Newtonbased CVT computation. KeywordsPeriodic Voronoi diagram, Delaunay triangulation, centroidal Voronoi tessellation, hexagonal pattern
Embedded Thin Shell for Wrinkle Simulation
"... Embedded thin shells is a new technique for simulating high resolution surface wrinkling deformations of composite objects consisting of a soft interior and a harder skin. It combines high resolution thin shells with coarse finite element lattices and defines frequency based constraints that allow t ..."
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Cited by 3 (1 self)
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Embedded thin shells is a new technique for simulating high resolution surface wrinkling deformations of composite objects consisting of a soft interior and a harder skin. It combines high resolution thin shells with coarse finite element lattices and defines frequency based constraints that allow the formation of wrinkles with properties matching those predicted by the physical Coarse Lattice When building the coarse model, we use a spatial hash to determine the element to which a vertex belongs; however, care must be taken not to lump separate parts of the mesh into the same element. We use superimposed elements with node duplication to accommodate these situations. parameters of the composite object. Interiror deformation and lattice
ParticleBased Anisotropic Surface Meshing
"... Figure 1: Anisotropic meshing results generated by our particlebased method. This paper introduces a particlebased approach for anisotropic surface meshing. Given an input polygonal mesh endowed with a Riemannian metric and a specified number of vertices, the method generates a metricadapted mesh ..."
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Cited by 3 (2 self)
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Figure 1: Anisotropic meshing results generated by our particlebased method. This paper introduces a particlebased approach for anisotropic surface meshing. Given an input polygonal mesh endowed with a Riemannian metric and a specified number of vertices, the method generates a metricadapted mesh. The main idea consists of mapping the anisotropic space into a higher dimensional isotropic one, called “embedding space”. The vertices of the mesh are generated by uniformly sampling the surface in this higher dimensional embedding space, and the sampling is further regularized by optimizing an energy function with a quasiNewton algorithm. All the computations can be reexpressed in terms of the dot product in the embedding space, and the Jacobian matrices of the mappings that connect different spaces. This transform makes it unnecessary to explicitly represent the coordinates in the embedding space, and also provides all necessary expressions of energy and forces for efficient computations. Through energy optimization, it naturally leads to the desired anisotropic particle distributions in the original space. The triangles are then generated by computing the Restricted Anisotropic Voronoi Diagram and its dual Delaunay triangulation. We compare our results qualitatively and quantitatively with the stateoftheart in anisotropic surface meshing on several examples, using the standard measurement criteria.