Conformal Flattening by Curvature Prescription and Metric Scaling

Ben‐Chen, Mirela; Gotsman, Craig; Bunin, Guy

doi:10.1111/j.1467-8659.2008.01142.x

Cited by 144 publications

(127 citation statements)

References 22 publications

(28 reference statements)

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“…In addition, the problem of global parametrization was solved in ] using discrete Ricci flow and in [Kharevych et al 2006] by arranging circles on the surface, one for each face with prescribed intersection angles. Parametrization methods base on relation between curvature and metric was concerned in [Ben-Chen et al 2008] and [Sheffer and Hart 2002] and a parametrization with focus on minimization of signal approximation error was used in [Sander et al 2002].…”

Section: Related Workmentioning

confidence: 99%

Skeleton-based 3D Surface Parameterization Applied on Texture Mapping

Madaras¹,

Ďurikovič²

2012

Journal of Applied Mathematics, Statistics and Informatics

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Assume a 2D manifold surface topologically equivalent to a sphere with handles we propose a novel 3D surface parametrization along the surface skeleton. First, we use a global mapping of the surface vertices onto a computed skeleton. Second, we use local mapping of the surrounding area of each skeleton segment into a small rectangle whose size is derived based on the surface properties around the segment. Each rectangle can be textured by assigning the local u, v texture coordinates. Furthermore, these rectangles are packed into a large squared texture called skeleton texture map (STM) by approximately solving a palette loading problem.Our technique enables the mapping of a texture onto the surface without necessity to store texture coordinates together with the model data. In other words it is enough to store the geometry data with STM and the coordinates are calculated on the fly.

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Section: Related Workmentioning

confidence: 99%

Skeleton-based 3D Surface Parameterization Applied on Texture Mapping

Madaras¹,

Ďurikovič²

2012

Journal of Applied Mathematics, Statistics and Informatics

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“…Moreover, we can set several interior vertices as singular ones in order to minimize the area distortions. Several effective methods have been introduced to select singular vertices, including manual selection [30], vector field analysis [37], and spectral analysis [32].…”

Section: Selecting Singular Vertex Setmentioning

confidence: 99%

“…It can further reduce the distortion by incorporating manually selected cone singularities. Ben-Chen et al [32] introduced a conformal parameterization which automatically determines the locations and target curvatures of the cone singularities.…”

Section: Introductionmentioning

confidence: 99%

Optimal Surface Parameterization Using Inverse Curvature Map

Yang

Kim

Luo

et al. 2008

IEEE Trans. Visual. Comput. Graphics

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Abstract-Mesh parameterization is a fundamental technique in computer graphics. The major goals during mesh parameterization are to minimize both the angle distortion and the area distortion. Angle distortion can be eliminated by the use of conformal mapping, in principle. Our paper focuses on solving the problem of finding the best discrete conformal mapping that also minimizes area distortion. First, we deduce an exact analytical differential formula to represent area distortion by curvature change in the discrete conformal mapping, giving a dynamic Poisson equation. On a mesh, the vertex curvature is related to edge lengths by the curvature map. Our result shows the map is invertible, i.e., the edge lengths can be computed from the curvature (by integration). Furthermore, we give the explicit Jacobi matrix of the inverse curvature map. Second, we formulate the task of computing conformal parameterizations with least area distortions as a constrained nonlinear optimization problem in curvature space. We deduce explicit conditions for the optima. Third, we give an energy form to measure the area distortions, and show that it has a unique global minimum. We use this to design an efficient algorithm, called free boundary curvature diffusion, which is guaranteed to converge to the global minimum; it has a natural physical interpretation. This result proves the common belief that optimal parameterization with least area distortion has a unique solution and can be achieved by free boundary conformal mapping. Major theoretical results and practical algorithms are presented for optimal parameterization based on the inverse curvature map. Comparisons are conducted with existing methods and using different energies. Novel parameterization applications are also introduced. The theoretical framework of the inverse curvature map can be applied to further study discrete conformal mappings.

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“…However, the distortion is often suboptimal for the number of cones, and difficult to control. A number of methods [Gu and Yau 2003;Dong et al 2006;Tong et al 2006;Ben-Chen et al 2008;Springborn et al 2008] use global harmonic or conformal parametrizations with cones; among these, [Dong et al 2006] and [Ben-Chen et al 2008] and [Springborn et al 2008] present automatic cone placement. The Morse complex constructed in [Dong et al 2006] is not directly tied to parametrization distortion.…”

Section: Introductionmentioning

confidence: 99%

“…Our work builds on ideas from four sources: (1) featurealigned parametrization and quadrangulation, based on fitting parametrization gradients to a field, primarily [Bommes et al 2009], but also [Ray et al 2006;Kälberer et al 2007]; (2) asrigid-as-possible parametrization [Liu et al 2008]; (3) conformal parametrization/metric computation [Jin et al 2004;Ben-Chen et al 2008;Springborn et al 2008]; and (4) cross-field and Nsymmetry/rotational symmetry field and connection constructions [Hertzmann and Zorin 2000;Palacios and Zhang 2007;Ray et al 2008;Ray et al 2009;Lai et al 2009;Crane et al 2010]. We discuss the relationship in Sections 5 and 6 in more detail.…”

Section: Introductionmentioning

confidence: 99%

Global parametrization by incremental flattening

Myles

Zorin

2012

ACM Trans. Graph.

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Global parametrization of surfaces requires singularities (cones) to keep distortion minimal. We describe a method for finding cone locations and angles and an algorithm for global parametrization which aim to produce seamless parametrizations with low metric distortion. The idea of the method is to evolve the metric of the surface, starting with the original metric so that a growing fraction of the area of the surface is constrained to have zero Gaussian curvature; the curvature becomes gradually concentrated at a small set of vertices which become cones. We demonstrate that the resulting parametrizations have significantly lower metric distortion compared to previously proposed methods.

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Conformal Flattening by Curvature Prescription and Metric Scaling

Cited by 144 publications

References 22 publications

Skeleton-based 3D Surface Parameterization Applied on Texture Mapping

Skeleton-based 3D Surface Parameterization Applied on Texture Mapping

Optimal Surface Parameterization Using Inverse Curvature Map

Global parametrization by incremental flattening

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