Efficient Numerical Solvers for Geometric Modelling Applications to Mesh Parameterization

Bruno Levy and Bruno Vallet and Alla Sheffer. ( 2004 )
in: 24th gOcad Meeting, ASGA

Abstract

These last few years, much time and effort have been devoted to designing parameterization algorithms for mesh models, minimizing area or angle deformation. Authalic parameterizations preserve areas. Conformal (i.e. angle preserving) parameterizations show suitable properties for remeshing algorithms and PDE solvers. This paper presents new parameterization methods that overcome several theoretical and practical limitations of previous work. Instead of directly optimizing u,v coordinates, our approach uses alternative variables that are naturally adapted to the problems. Our authalic parameterization DPBF directly optimizes the Jacobian of the parameterization in dot-product space, providing fine control on area deformation. Our conformal parameterization ABF++ is a highly efficient extension of the non-linear ABF method. In ABF++, algebraic transforms dramatically reduce the dimension of the linear systems solved by ABF at each iteration. ABF++ extends the feasibility of the original formulation to models with several hundred thousand faces. To reconstruct the u,v’s from those variables, we propose LSAM, a new anisotropic generalization of LSCM. In LSAM, the u,v’s are globally optimized avoiding the numerical instabilities caused by accumulated errors in greedy algorithms, such as the reconstruction phase used in ABF. The alternative formulation facilitates incorporating additional constraints. For instance, we present a simple mechanism to compute periodic, globally smooth maps. For genus other than one where a fully periodic mapping is not available, a similar but weaker condition is enforced.

Download / Links

    BibTeX Reference

    @inproceedings{LévyRM2004,
 abstract = { These last few years, much time and effort have been devoted to designing parameterization algorithms for mesh models, minimizing area or angle deformation. Authalic parameterizations preserve areas. Conformal (i.e. angle preserving) parameterizations show suitable properties for remeshing algorithms and PDE solvers. This paper presents new parameterization methods that overcome several theoretical and practical limitations of previous work. Instead of directly optimizing u,v coordinates, our approach uses alternative variables that are naturally adapted to the problems. Our authalic parameterization DPBF directly optimizes the Jacobian of the parameterization in dot-product space, providing fine control on area deformation. Our conformal parameterization ABF++ is a highly efficient extension of the non-linear ABF method. In ABF++, algebraic transforms dramatically reduce the dimension of the linear systems solved by ABF at each iteration. ABF++ extends the feasibility of the original formulation to models with several hundred thousand faces. To reconstruct the u,v’s from those variables, we propose LSAM, a new anisotropic generalization of LSCM. In LSAM, the u,v’s are globally optimized avoiding the numerical instabilities caused by accumulated errors in greedy algorithms, such as the reconstruction phase used in ABF. The alternative formulation facilitates incorporating additional constraints. For instance, we present a simple mechanism to compute periodic, globally smooth maps. For genus other than one where a fully periodic mapping is not available, a similar but weaker condition is enforced. },
 author = { Levy, Bruno AND Vallet, Bruno AND Sheffer, Alla },
 booktitle = { 24th gOcad Meeting },
 month = { "june" },
 publisher = { ASGA },
 title = { Efficient Numerical Solvers for Geometric Modelling Applications to Mesh Parameterization },
 year = { 2004 }
}