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49
Geometry images
 IN PROC. 29TH SIGGRAPH
, 2002
"... Surface geometry is often modeled with irregular triangle meshes. The process of remeshing refers to approximating such geometry using a mesh with (semi)regular connectivity, which has advantages for many graphics applications. However, current techniques for remeshing arbitrary surfaces create onl ..."
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Cited by 295 (22 self)
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Surface geometry is often modeled with irregular triangle meshes. The process of remeshing refers to approximating such geometry using a mesh with (semi)regular connectivity, which has advantages for many graphics applications. However, current techniques for remeshing arbitrary surfaces create only semiregular meshes. The original mesh is typically decomposed into a set of disklike charts, onto which the geometry is parametrized and sampled. In this paper, we propose to remesh an arbitrary surface onto a completely regular structure we call a geometry image. It captures geometry as a simple 2D array of quantized points. Surface signals like normals and colors are stored in similar 2D arrays using the same implicit surface parametrization — texture coordinates are absent. To create a geometry image, we cut an arbitrary mesh along a network of edge paths, and parametrize the resulting single chart onto a square. Geometry images can be encoded using traditional image compression algorithms, such as waveletbased coders.
Global Conformal Surface Parameterization
, 2003
"... We solve the problem of computing global conformal parameterizations for surfaces with nontrivial topologies. The parameterization is global in the sense that it preserves the conformality everywhere except for a few points, and has no boundary of discontinuity. We analyze the structure of the space ..."
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Cited by 117 (22 self)
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We solve the problem of computing global conformal parameterizations for surfaces with nontrivial topologies. The parameterization is global in the sense that it preserves the conformality everywhere except for a few points, and has no boundary of discontinuity. We analyze the structure of the space of all global conformal parameterizations of a given surface and find all possible solutions by constructing a basis of the underlying linear solution space. This space has a natural structure solely determined by the surface geometry, so our computing result is independent of connectivity, insensitive to resolution, and independent of the algorithms to discover it. Our algorithm is based on the properties of gradient fields of conformal maps, which are closedness, harmonity, conjugacy, duality and symmetry. These properties can be formulated by sparse linear systems, so the method is easy to implement and the entire process is automatic. We also introduce a novel topological modification method to improve the uniformity of the parameterization. Based on the global conformal parameterization of a surface, we can construct a conformal atlas and use it to build conformal geometry images which have very accurate reconstructed normals.
Removing excess topology from isosurfaces
 ACM Trans. Graph
"... Many highresolution surfaces are created through isosurface extraction from volumetric representations, obtained by 3D photography, CT, or MRI. Noise inherent in the acquisition process can lead to geometrical and topological errors. Reducing geometrical errors during reconstruction is well studied ..."
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Cited by 76 (1 self)
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Many highresolution surfaces are created through isosurface extraction from volumetric representations, obtained by 3D photography, CT, or MRI. Noise inherent in the acquisition process can lead to geometrical and topological errors. Reducing geometrical errors during reconstruction is well studied. However, isosurfaces often contain many topological errors in the form of tiny handles. These nearly invisible artifacts hinder subsequent operations like mesh simplification, remeshing, and parametrization. In this article we present a practical method for removing handles in an isosurface. Our algorithm makes an axisaligned sweep through the volume to locate handles, compute their sizes, and selectively remove them. The algorithm is designed to facilitate outofcore execution. It finds the handles by incrementally constructing and analyzing a Reeb graph. The size of a handle is measured by a short nonseparating cycle. Handles are removed robustly by modifying the volume rather than attempting “mesh surgery. ” Finally, the volumetric modifications are spatially localized to preserve geometrical detail. We demonstrate topology simplification on several complex models, and show its benefits for subsequent surface processing.
Featurebased surface parameterization and texture mapping
 ACM Transactions on Graphics
, 2005
"... and precomputation of solid textures. The stretch caused by a given parameterization determines the sampling rate on the surface. In this article, we present an automatic parameterization method for segmenting a surface into patches that are then flattened with little stretch. Many objects consist o ..."
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Cited by 71 (5 self)
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and precomputation of solid textures. The stretch caused by a given parameterization determines the sampling rate on the surface. In this article, we present an automatic parameterization method for segmenting a surface into patches that are then flattened with little stretch. Many objects consist of regions of relatively simple shapes, each of which has a natural parameterization. Based on this observation, we describe a threestage featurebased patch creation method for manifold surfaces. The first two stages, genus reduction and feature identification, are performed with the help of distancebased surface functions. In the last stage, we create one or two patches for each feature region based on a covariance matrix of the feature’s surface points. To reduce stretch during patch unfolding, we notice that stretch is a 2 × 2 tensor, which in ideal situations is the identity. Therefore, we use the GreenLagrange tensor to measure and to guide the optimization process. Furthermore, we allow the boundary vertices of a patch to be optimized by adding scaffold triangles. We demonstrate our featurebased patch creation and patch unfolding methods for several textured models. Finally, to evaluate the quality of a given parameterization, we describe an imagebased error measure that takes into account stretch, seams, smoothness, packing efficiency, and surface visibility.
Harmonic functions for quadrilateral remeshing of arbitrary manifolds
 COMPUTERAIDED GEOMETRIC DESIGN
, 2005
"... In this paper, we propose a new quadrilateral remeshing method for manifolds of arbitrary genus that is at once general, flexible, and efficient. Our technique is based on the use of smooth harmonic scalar fields defined over the mesh. Given such a field, we compute its gradient field and a second v ..."
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Cited by 60 (1 self)
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In this paper, we propose a new quadrilateral remeshing method for manifolds of arbitrary genus that is at once general, flexible, and efficient. Our technique is based on the use of smooth harmonic scalar fields defined over the mesh. Given such a field, we compute its gradient field and a second vector field that is everywhere orthogonal to the gradient. We then trace integral lines through these vector fields to sample the mesh. The two nets of integral lines together are used to form the polygons of the output mesh. Curvaturesensitive spacing of the lines provides for anisotropic meshes that adapt to the local shape. Our scalar field construction allows users to exercise extensive control over the structure of the final mesh. The entire process is performed without computing an explicit parameterization of the surface, and is thus applicable to manifolds of any genus without the need for cutting the surface into patches.
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 58 (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.
Finding shortest nonseparating and noncontractible cycles for topologically embedded graphs
 Discrete Comput. Geom
, 2005
"... We present an algorithm for finding shortest surface nonseparating cycles in graphs embedded on surfaces in O(g 3/2 V 3/2 log V + g 5/2 V 1/2) time, where V is the number of vertices in the graph and g is the genus of the surface. If g = o(V 1/3−ε), this represents a considerable improvement over p ..."
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Cited by 40 (9 self)
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We present an algorithm for finding shortest surface nonseparating cycles in graphs embedded on surfaces in O(g 3/2 V 3/2 log V + g 5/2 V 1/2) time, where V is the number of vertices in the graph and g is the genus of the surface. If g = o(V 1/3−ε), this represents a considerable improvement over previous results by Thomassen, and Erickson and HarPeled. We also give algorithms to find a shortest noncontractible cycle in O(g O(g) V 3/2) time, which improves previous results for fixed genus. This result can be applied for computing the (nonseparating) facewidth of embedded graphs. Using similar ideas we provide the first nearlinear running time algorithm for computing the facewidth of a graph embedded on the projective plane, and an algorithm to find the facewidth of embedded toroidal graphs in O(V 5/4 log V) time. 1
Dynamic Generators of Topologically Embedded Graphs
, 2003
"... We provide a data structure for maintaining an embedding of a graph on a surface (represented combinatorially by a permutation of edges around each vertex) and computing generators of the fundamental group of the surface, in amortized time O(logn + logg(loglogg) 3) per update on a surface of genus g ..."
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Cited by 39 (1 self)
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We provide a data structure for maintaining an embedding of a graph on a surface (represented combinatorially by a permutation of edges around each vertex) and computing generators of the fundamental group of the surface, in amortized time O(logn + logg(loglogg) 3) per update on a surface of genus g; we can also test orientability of the surface in the same time, and maintain the minimum and maximum spanning tree of the graph in time O(log n + log 4 g) per update. Our data structure allows edge insertion and deletion as well as the dual operations; these operations may implicitly change the genus of the embedding surface. We apply similar ideas to improve the constant factor in a separator theorem for lowgenus graphs, and to find in linear time a treedecomposition of lowgenus lowdiameter graphs.
Isotropic remeshing of surfaces: A local parameterization approach
 In Proceedings of 12th International Meshing Roundtable
, 2003
"... We present a method for isotropic remeshing of arbitrary genus surfaces. The method is based on a mesh adaptation process, namely, a sequence of local modifications performed on a copy of the original mesh, while referring to the original mesh geometry. The algorithm has three stages. In the first s ..."
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Cited by 38 (3 self)
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We present a method for isotropic remeshing of arbitrary genus surfaces. The method is based on a mesh adaptation process, namely, a sequence of local modifications performed on a copy of the original mesh, while referring to the original mesh geometry. The algorithm has three stages. In the first stage the required number or vertices are generated by iterative simplification or refinement. The second stage performs an initial vertex partition using an areabased relaxation method. The third stage achieves precise isotropic vertex sampling prescribed by a given density function on the mesh. We use a modification of Lloyd’s relaxation method to construct a weighted centroidal Voronoi tessellation of the mesh. We apply these iterations locally on small patches of the mesh that are parameterized into the 2D plane. This allows us to handle arbitrary complex meshes with any genus and any number of boundaries. The efficiency and the accuracy of the remeshing process is achieved using a patchwise parameterization technique.