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342
Surface Parameterization: a Tutorial and Survey
 In Advances in Multiresolution for Geometric Modelling, Mathematics and Visualization
, 2005
"... Summary. This paper provides a tutorial and survey of methods for parameterizing surfaces with a view to applications in geometric modelling and computer graphics. We gather various concepts from differential geometry which are relevant to surface mapping and use them to understand the strengths and ..."
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Cited by 239 (7 self)
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Summary. This paper provides a tutorial and survey of methods for parameterizing surfaces with a view to applications in geometric modelling and computer graphics. We gather various concepts from differential geometry which are relevant to surface mapping and use them to understand the strengths and weaknesses of the many methods for parameterizing piecewise linear surfaces and their relationship to one another. 1
Geometry clipmaps: terrain rendering using nested regular grids
 In SIGGRAPH ’04: ACM SIGGRAPH 2004 Papers
, 2004
"... Illustration using a coarse geometry clipmap (size n=31) View of the 216,000×93,600 U.S. dataset near Grand Canyon (n=255) Figure 1:Terrains rendered using geometry clipmaps, showing clipmap levels (size n×n) and transition regions (in blue on right). Rendering throughput has reached a level that en ..."
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Cited by 146 (2 self)
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Illustration using a coarse geometry clipmap (size n=31) View of the 216,000×93,600 U.S. dataset near Grand Canyon (n=255) Figure 1:Terrains rendered using geometry clipmaps, showing clipmap levels (size n×n) and transition regions (in blue on right). Rendering throughput has reached a level that enables a novel approach to levelofdetail (LOD) control in terrain rendering. We introduce the geometry clipmap, which caches the terrain in a set of nested regular grids centered about the viewer. The grids are stored as vertex buffers in fast video memory, and are incrementally refilled as the viewpoint moves. This simple framework provides visual continuity, uniform frame rate, complexity throttling, and graceful degradation. Moreover it allows two new exciting realtime functionalities: decompression and synthesis. Our main dataset is a 40GB height map of the United States. A compressed image pyramid reduces the size by a remarkable factor of 100, so that it fits entirely in memory. This compressed data also contributes normal maps for shading. As the viewer approaches the surface, we synthesize grid levels finer than the stored terrain using fractal noise displacement. Decompression, synthesis, and normalmap computations are incremental, thereby allowing interactive flight at 60 frames/sec.
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 139 (26 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.
Fundamentals of Spherical Parameterization for 3D Meshes
 PROCEEDINGS OF THE 2006 SYMPOSIUM ON INTERACTIVE 3D GRAPHICS AND GAMES, MARCH 1417, 2006
, 2003
"... Parametrization of 3D mesh data is important for many graphics applications, in particular for texture mapping, remeshing and morphing. Closed manifold genus0 meshes are topologically equivalent to a sphere, hence this is the natural parameter domain for them. Parametrizing a triangle mesh onto the ..."
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Cited by 124 (24 self)
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Parametrization of 3D mesh data is important for many graphics applications, in particular for texture mapping, remeshing and morphing. Closed manifold genus0 meshes are topologically equivalent to a sphere, hence this is the natural parameter domain for them. Parametrizing a triangle mesh onto the sphere means assigning a 3D position on the unit sphere to each of the mesh vertices, such that the spherical triangles induced by the mesh connectivity do not overlap. Satisfying the nonoverlapping requirement is the most difficult and critical component of this process. We present a generalization of the method of barycentric coordinates for planar parametrization which solves the spherical parametrization problem, prove its correctness by establishing a connection to spectral graph theory and describe efficient numerical methods for computing these parametrizations.
MultiChart Geometry Images
, 2003
"... We introduce multichart geometry images, a new representation for arbitrary surfaces. It is created by resampling a surface onto a regular 2D grid. Whereas the original scheme of Gu et al. maps the entire surface onto a single square, we use an atlas construction to map the surface piecewise onto c ..."
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Cited by 117 (4 self)
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We introduce multichart geometry images, a new representation for arbitrary surfaces. It is created by resampling a surface onto a regular 2D grid. Whereas the original scheme of Gu et al. maps the entire surface onto a single square, we use an atlas construction to map the surface piecewise onto charts of arbitrary shape. We demonstrate that this added flexibility reduces parametrization distortion and thus provides greater geometric fidelity, particularly for shapes with long extremities, high genus, or disconnected components. Traditional atlas constructions suffer from discontinuous reconstruction across chart boundaries, which in our context create unacceptable surface cracks. Our solution is a novel zippering algorithm that creates a watertight surface. In addition, we present a new atlas chartification scheme based on clustering optimization.
Globally Smooth Parameterizations with Low Distortion
, 2003
"... Good parameterizations are of central importance in many digital geometry processing tasks. Typically the behavior of such processing algorithms is related to the smoothness of the parameterization and how much distortion it contains, i.e., how rapidly the derivatives of the parameterization change. ..."
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Cited by 100 (2 self)
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Good parameterizations are of central importance in many digital geometry processing tasks. Typically the behavior of such processing algorithms is related to the smoothness of the parameterization and how much distortion it contains, i.e., how rapidly the derivatives of the parameterization change. Since a parameterization maps a bounded region of the plane to the surface, a parameterization for a surface which is not homeomorphic to a disc must be made up of multiple pieces. We present a novel parameterization algorithm for arbitrary topology surface meshes which computes a globally smooth parameterization with low distortion. We optimize the patch layout subject to criteria such as shape quality and parametric distortion, which are used to steer a mesh simplification approach for base complex construction. Global smoothness is achieved through simultaneous relaxation over all patches, with suitable transition functions between patches incorporated into the relaxation procedure. We demonstrate the quality of our parameterizations through numerical evaluation of distortion measures; the rate distortion behavior of semiregular remeshes produced with these parameterizations; and a comparison with globally smooth subdivision methods. The numerical algorithms required to compute the parameterizations are robust and run on the order of minutes even for large meshes.
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 90 (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.
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 84 (2 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.