Linear Surface Reconstruction from Discrete Fundamental Forms on Triangle Meshes

We present a linear algorithm to reconstruct the vertex coordinates for a surface mesh given its edge lengths and dihedral angles, unique up to rotation and translation. A local integrability condition for the existence of an immersion of the mesh in 3D Euclidean space is provided, mirroring the fun...

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Veröffentlicht in:Computer graphics forum Jg. 31; H. 8; S. 2277 - 2287
Hauptverfasser: Wang, Y., Liu, B., Tong, Y.
Format: Journal Article
Sprache:Englisch
Veröffentlicht: Oxford, UK Blackwell Publishing Ltd 01.12.2012
Schlagworte:
Algorithms
Analysis
Computer graphics
differential coordinates
Euclidean space
first fundamental forms
Gauss equation
I.3.5 [Computer Graphics]: Computational Geometry and Object Modelling-Curve
Image processing systems
Immersion
Mathematical analysis
Reconstruction
second fundamental forms
solid and object representations
Studies
surface
surface deformation
Theorems
Three dimensional
Topological manifolds
ISSN:0167-7055, 1467-8659
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Zusammenfassung:We present a linear algorithm to reconstruct the vertex coordinates for a surface mesh given its edge lengths and dihedral angles, unique up to rotation and translation. A local integrability condition for the existence of an immersion of the mesh in 3D Euclidean space is provided, mirroring the fundamental theorem of surfaces in the continuous setting (i.e. Gauss's equation and the Mainardi–Codazzi equations) if we regard edge lengths as the discrete first fundamental form and dihedral angles as the discrete second fundamental form. The resulting sparse linear system to solve for the immersion is derived from the convex optimization of a quadratic energy based on a lift from the immersion in the 3D Euclidean space to the 6D rigid motion space. This discrete representation and linear reconstruction can benefit a wide range of geometry processing tasks such as surface deformation and shape analysis. A rotation‐invariant surface deformation through point and orientation constraints is demonstrated as well. We present a linear algorithm to reconstruct the vertex coordinates for a surface mesh given its edge lengths and dihedral angles, unique up to rotation and translation. A local integrability condition for the existence of an immersion of the mesh in 3D Euclidean space is provided, mirroring the fundamental theorem of surfaces in the continuous setting (i.e., Gauss's equation and the Mainardi‐Codazzi equations) if we regard edge lengths as the discrete first fundamental form and dihedral angles as the discrete second fundamental form.
Bibliographie:ark:/67375/WNG-4GH6C10G-0
ArticleID:CGF3153
istex:C97FF448ED021AE9B9ACB6A30AB1972B06BEFE33
SourceType-Scholarly Journals-1
ObjectType-Feature-1
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ISSN:0167-7055
1467-8659
DOI:10.1111/j.1467-8659.2012.03153.x