Shape-Up: Shaping Discrete Geometry with Projections

We introduce a unified optimization framework for geometry processing based on shape constraints. These constraints preserve or prescribe the shape of subsets of the points of a geometric data set, such as polygons, one‐ring cells, volume elements, or feature curves. Our method is based on two key c...

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Bibliographic Details
Published in:Computer graphics forum Vol. 31; no. 5; pp. 1657 - 1667
Main Authors: Bouaziz, Sofien, Deuss, Mario, Schwartzburg, Yuliy, Weise, Thibaut, Pauly, Mark
Format: Journal Article
Language:English
Published: Oxford, UK Blackwell Publishing Ltd 01.08.2012
Subjects:
3-D graphics
and systems
Computer graphics
Finite element method
Geometry
I.3.5 [Computer Graphics]: Computational Geometry and Object Modeling-Geometric algorithms
I.3.5 [Computer Graphics]: Computational Geometry and Object Modeling—Geometric algorithms, languages, and systems
languages
Mathematical analysis
Mathematical models
Operators
Optimization
Preserves
Projection
Proximity
Studies
ISSN:0167-7055, 1467-8659
Online Access:Get full text
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Summary:We introduce a unified optimization framework for geometry processing based on shape constraints. These constraints preserve or prescribe the shape of subsets of the points of a geometric data set, such as polygons, one‐ring cells, volume elements, or feature curves. Our method is based on two key concepts: a shape proximity function and shape projection operators. The proximity function encodes the distance of a desired least‐squares fitted elementary target shape to the corresponding vertices of the 3D model. Projection operators are employed to minimize the proximity function by relocating vertices in a minimal way to match the imposed shape constraints. We demonstrate that this approach leads to a simple, robust, and efficient algorithm that allows implementing a variety of geometry processing applications, simply by combining suitable projection operators. We show examples for computing planar and circular meshes, shape space exploration, mesh quality improvement, shape‐preserving deformation, and conformal parametrization. Our optimization framework provides a systematic way of building new solvers for geometry processing and produces similar or better results than state‐of‐the‐art methods.
Bibliography:ark:/67375/WNG-R9XTRGR7-F
ArticleID:CGF3171
istex:A1CEE08C937676A8B7345B088ACDA88F68C95F4F
SourceType-Scholarly Journals-1
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ISSN:0167-7055
1467-8659
DOI:10.1111/j.1467-8659.2012.03171.x