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36
QuadCover – Surface Parameterization using Branched Coverings.
 COMPUT. GRAPH. FORUM
, 2007
"... We introduce an algorithm for automatic computation of global parameterizations on arbitrary simplicial 2manifolds whose parameter lines are guided by a given frame field, for example by principal curvature frames. The parameter lines are globally continuous, and allow a remeshing of the surface in ..."
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Cited by 62 (6 self)
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We introduce an algorithm for automatic computation of global parameterizations on arbitrary simplicial 2manifolds whose parameter lines are guided by a given frame field, for example by principal curvature frames. The parameter lines are globally continuous, and allow a remeshing of the surface into quadrilaterals. The algorithm converts a given frame field into a single vector field on a branched covering of the 2manifold, and generates an integrable vector field by a Hodge decomposition on the covering space. Except for an optional smoothing and alignment of the initial frame field, the algorithm is fully automatic and generates high quality quadrilateral meshes.
Discrete conformal mappings via circle patterns
 ACM Trans. Graph
, 2006
"... We introduce a novel method for the construction of discrete conformal mappings from surface meshes of arbitrary topology to the plane. Our approach is based on circle patterns, i.e., arrangements of circles—one for each face—with prescribed intersection angles. Given these angles the circle radii f ..."
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Cited by 53 (1 self)
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We introduce a novel method for the construction of discrete conformal mappings from surface meshes of arbitrary topology to the plane. Our approach is based on circle patterns, i.e., arrangements of circles—one for each face—with prescribed intersection angles. Given these angles the circle radii follow as the unique minimizer of a convex energy. The method supports very flexible boundary conditions ranging from free boundaries to control of the boundary shape via prescribed curvatures. Closed meshes of genus zero can be parameterized over the sphere. To parameterize higher genus meshes we introduce cone singularities at designated vertices. The parameter domain is then a piecewise Euclidean surface. Cone singularities can also help to reduce the often very large area distortion of global conformal maps to moderate levels. Our method involves two optimization problems: a quadratic program and the unconstrained minimization of the circle pattern energy. The latter is a convex function of logarithmic radius variables with simple explicit expressions for gradient and Hessian. We demonstrate the versatility and performance of our algorithm with a variety of examples.
Mesh parameterization methods and their applications
 FOUNDATIONS AND TRENDSÂŐ IN COMPUTER GRAPHICS AND VISION
, 2006
"... We present a survey of recent methods for creating piecewise linear mappings between triangulations in 3D and simpler domains such as planar regions, simplicial complexes, and spheres. We also discuss emerging tools such as global parameterization, intersurface mapping, and parameterization with co ..."
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Cited by 43 (0 self)
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We present a survey of recent methods for creating piecewise linear mappings between triangulations in 3D and simpler domains such as planar regions, simplicial complexes, and spheres. We also discuss emerging tools such as global parameterization, intersurface mapping, and parameterization with constraints. We start by describing the wide range of applications where parameterization tools have been used in recent years. We then briefly review the pertinent mathematical background and terminology, before proceeding to survey the existing parameterization techniques. Our survey summarizes the main ideas of each technique and discusses its main properties, comparing it to other methods available. Thus it aims to provide guidance to researchers and developers when assessing the suitability of different methods for various applications. This survey focuses on the practical aspects of the methods available, such as time complexity and robustness and shows multiple examples of parameterizations generated using different methods, allowing the reader to visually evaluate and compare the results.
Conformal Flattening by Curvature Prescription and Metric Scaling
, 2008
"... We present an efficient method to conformally parameterize 3D mesh data sets to the plane. The idea behind our method is to concentrate all the 3D curvature at a small number of select mesh vertices, called cone singularities, and then cut the mesh through those singular vertices to obtain disk topo ..."
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Cited by 35 (2 self)
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We present an efficient method to conformally parameterize 3D mesh data sets to the plane. The idea behind our method is to concentrate all the 3D curvature at a small number of select mesh vertices, called cone singularities, and then cut the mesh through those singular vertices to obtain disk topology. The singular vertices are chosen automatically. As opposed to most previous methods, our flattening process involves only the solution of linear systems of Poisson equations, thus is very efficient. Our method is shown to be faster than existing methods, yet generates parameterizations having comparable quasiconformal distortion.
Mesh Parameterization: Theory and Practice
 SIGGRAPH ASIA 2008 COURSE NOTES
, 2008
"... Mesh parameterization is a powerful geometry processing tool with numerous computer graphics applications, from texture mapping to animation transfer. This course outlines its mathematical foundations, describes recent methods for parameterizing meshes over various domains, discusses emerging tools ..."
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Cited by 33 (2 self)
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Mesh parameterization is a powerful geometry processing tool with numerous computer graphics applications, from texture mapping to animation transfer. This course outlines its mathematical foundations, describes recent methods for parameterizing meshes over various domains, discusses emerging tools like global parameterization and intersurface mapping, and demonstrates a variety of parameterization applications.
NSymmetry Direction Field Design
, 2008
"... Many algorithms in computer graphics and geometry processing use two orthogonal smooth direction fields (unit tangent vector fields) defined over a surface. For instance, these direction fields are used in texture synthesis, in geometry processing or in nonphotorealistic rendering to distribute and ..."
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Cited by 31 (0 self)
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Many algorithms in computer graphics and geometry processing use two orthogonal smooth direction fields (unit tangent vector fields) defined over a surface. For instance, these direction fields are used in texture synthesis, in geometry processing or in nonphotorealistic rendering to distribute and orient elements on the surface. Such direction fields can be designed in fundamentally different ways, according to the symmetry requested: inverting a direction or swapping two directions may be allowed or not. Despite the advances realized in the last few years in the domain of geometry processing, a unified formalism is still lacking for the mathematical object that characterizes these generalized direction fields. As a consequence, existing direction field design algorithms are limited to use nonoptimum local relaxation procedures. In this paper, we formalize Nsymmetry direction fields, a generalization of classical direction fields. We give a new definition of their singularities to explain how they relate with the topology of the surface. Namely, we provide an accessible demonstration of the PoincaréHopf theorem in the case of Nsymmetry direction fields on 2manifolds. Based on this theorem, we explain how to control the topology of Nsymmetry direction fields on meshes. We demonstrate the validity and robustness of this formalism by deriving a highly efficient algorithm to design a smooth eld interpolating user de ned singularities and directions.
Polycube splines
, 2008
"... This paper proposes a new concept of polycube splines and develops novel modeling techniques for using the polycube splines in solid modeling and shape computing. Polycube splines are essentially a novel variant of manifold splines which are built upon the polycube map, serving as its parametric dom ..."
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Cited by 13 (5 self)
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This paper proposes a new concept of polycube splines and develops novel modeling techniques for using the polycube splines in solid modeling and shape computing. Polycube splines are essentially a novel variant of manifold splines which are built upon the polycube map, serving as its parametric domain. Our rationale for defining spline surfaces over polycubes is that polycubes have rectangular structures everywhere over their domains, except a very small number of corner points. The boundary of polycubes can be naturally decomposed into a set of regular structures, which facilitate tensorproduct surface definition, GPUcentric geometric computing, and imagebased geometric processing. We develop algorithms to construct polycube maps, and show that the introduced polycube map naturally induces the affine structure with a finite number of extraordinary points. Besides its intrinsic rectangular structure, the polycube map may approximate any original scanned dataset with a very low geometric distortion, so our method for building polycube splines is both natural and necessary, as its parametric domain can mimic the geometry of modeled objects in a topologically correct and geometrically meaningful manner. We design a new data structure that facilitates the intuitive and rapid construction of polycube splines in this paper. We demonstrate the polycube splines with applications in surface reconstruction and shape computing.
Global Structure Optimization of Quadrilateral Meshes
"... We introduce a fully automatic algorithm which optimizes the highlevel structure of a given quadrilateral mesh to achieve a coarser quadrangular base complex. Such a topological optimization is highly desirable, since stateoftheart quadrangulation techniques lead to meshes which have an appropria ..."
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Cited by 13 (4 self)
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We introduce a fully automatic algorithm which optimizes the highlevel structure of a given quadrilateral mesh to achieve a coarser quadrangular base complex. Such a topological optimization is highly desirable, since stateoftheart quadrangulation techniques lead to meshes which have an appropriate singularity distribution and an anisotropic element alignment, but usually they are still far away from the highlevel structure which is typical for carefully designed meshes manually created by specialists and used e.g. in animation or simulation. In this paper we show that the quality of the highlevel structure is negatively affected by helical configurations within the quadrilateral mesh. Consequently we present an algorithm which detects helices and is able to remove most of them by applying a novel grid preserving simplification operator (GPoperator) which is guaranteed to maintain an allquadrilateral mesh. Additionally it preserves the given singularity distribution and in particular does not introduce new singularities. For each helix we construct a directed graph in which cycles through the start vertex encode operations to remove the corresponding helix. Therefore a simple graph search algorithm can be performed iteratively to remove as many helices as possible and thus improve the highlevel structure in a greedy fashion. We demonstrate the usefulness of our automatic structure optimization technique by showing several examples with varying complexity. Categories and Subject Descriptors (according to ACM CCS): I.3.5 [Computer Graphics]: Computational Geometry and Object Modeling—Hierarchy and geometric transformations, Curve, surface, solid, and object representations
A Robust TwoStep Procedure for QuadDominant Remeshing
 Computer Graphics Forum
, 2006
"... We propose a new technique for quaddominant remeshing which separates the local regularity requirements from the global alignment requirements by working in two steps. In the first step, we apply a slight variant of variational shape approximation in order to segment the input mesh into patches w ..."
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Cited by 12 (2 self)
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We propose a new technique for quaddominant remeshing which separates the local regularity requirements from the global alignment requirements by working in two steps. In the first step, we apply a slight variant of variational shape approximation in order to segment the input mesh into patches which capture the global structure of the processed object. Then we compute an optimized quadmesh for every patch by generating a finite set of candidate curves and applying a combinatorial optimization procedure. Since the optimization is performed independently for each patch, we can afford more complex operations while keeping the overall computation times at a reasonable level. Our quadmeshing technique is robust even for noisy meshes and meshes with isotropic or flat regions since it does not rely on the generation of curves by integration along estimated principal curvature directions.
Representing higherorder singularities in vector fields on piecewise linear surfaces
 IEEE Transactions on Visualization and Computer Graphics
, 2006
"... Abstract—Accurately representing higherorder singularities of vector fields defined on piecewise linear surfaces is a nontrivial problem. In this work, we introduce a concise yet complete interpolation scheme of vector fields on arbitrary triangulated surfaces. The scheme enables arbitrary singula ..."
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Cited by 9 (4 self)
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Abstract—Accurately representing higherorder singularities of vector fields defined on piecewise linear surfaces is a nontrivial problem. In this work, we introduce a concise yet complete interpolation scheme of vector fields on arbitrary triangulated surfaces. The scheme enables arbitrary singularities to be represented at vertices. The representation can be considered as a facetbased “encoding ” of vector fields on piecewise linear surfaces. The vector field is described in polar coordinates over each facet, with a facet edge being chosen as the reference to define the angle. An integer called the period jump is associated to each edge of the triangulation to remove the ambiguity when interpolating the direction of the vector field between two facets that share an edge. To interpolate the vector field, we first linearly interpolate the angle of rotation of the vectors along the edges of the facet graph. Then, we use a variant of Nielson’s sidevertex scheme to interpolate the vector field over the entire surface. With our representation, we remove the bound imposed on the complexity of singularities that a vertex can represent by its connectivity. This bound is a limitation generally exists in vertexbased linear schemes. Furthermore, using our data structure, the index of a vertex of a vector field can be combinatorily determined. We show the simplicity of the interpolation scheme with a GPUaccelerated algorithm for a LICbased visualization of the sodefined vector fields, operating in image space. We demonstrate the algorithm applied to various vector fields on curved surfaces. Index Terms—vector field visualization, higherorder singularities, line integral convolution, GPU. 1