Geometric fairing of irregular meshes for free-form surface design

In this paper we present a new algorithm for smoothing arbitrary triangle meshes while satisfying G^1 boundary conditions. The algorithm is based on solving a nonlinear fourth order partial differential equation (PDE) that only depends on intrinsic surface properties instead of being derived from a particular surface parameterization. This continuous PDE has a (representation-independent) well-defined solution which we approximate by our triangle mesh. Hence, changing the mesh complexity (refinement) or the mesh connectivity (remeshing) leads to just another discretization of the same smooth surface and doesn't affect the resulting geometric shape beyond this. This is typically not true for filter-based mesh smoothing algorithms. To simplify the computation we factorize the fourth order PDE into a set of two nested second order problems thus avoiding the estimation of higher order derivatives. Further acceleration is achieved by applying multigrid techniques on a fine-to-coarse hierarchical mesh representation.

[1]  W. H. Frey,et al.  1. Approximation with Aesthetic Constraints , 1994, Designing Fair Curves and Surfaces.

[2]  L. Kobbelt,et al.  Discrete Fairing of Curves and Surfaces based on Linear Curvature Distribution , 2000 .

[3]  Hugues Hoppe,et al.  Progressive meshes , 1996, SIGGRAPH.

[4]  L. Kobbelt,et al.  Generating Fair Meshes with $G^1$ Boundary Conditions , 2000 .

[5]  Malcolm I. G. Bloor,et al.  Using partial differential equations to generate free-form surfaces , 1990, Comput. Aided Des..

[6]  Michael S. Floater,et al.  Parametrization and smooth approximation of surface triangulations , 1997, Comput. Aided Geom. Des..

[7]  Günther Greiner,et al.  Variational Design and Fairing of Spline Surfaces , 1994, Comput. Graph. Forum.

[8]  Peter Schröder,et al.  Multiresolution signal processing for meshes , 1999, SIGGRAPH.

[9]  Gene H. Golub,et al.  Matrix computations , 1983 .

[10]  Mark Meyer,et al.  Implicit fairing of irregular meshes using diffusion and curvature flow , 1999, SIGGRAPH.

[11]  Andrew P. Witkin,et al.  Free-form shape design using triangulated surfaces , 1994, SIGGRAPH.

[12]  Hans-Peter Seidel,et al.  Interactive multi-resolution modeling on arbitrary meshes , 1998, SIGGRAPH.

[13]  J. Sethian,et al.  Motion by intrinsic Laplacian of curvature , 1999 .

[14]  Hans-Peter Seidel,et al.  A General Framework for Mesh Decimation , 1998, Graphics Interface.

[15]  Gabriel Taubin,et al.  A signal processing approach to fair surface design , 1995, SIGGRAPH.

[16]  M. Docarmo.  Differential geometry of curves and surfaces , 1976 .

[17]  Yutaka Ohtake,et al.  Polyhedral surface smoothing with simultaneous mesh regularization , 2000, Proceedings Geometric Modeling and Processing 2000. Theory and Applications.

[18]  W. Stuetzle,et al.  HIERARCHICAL COMPUTATION OF PL HARMONIC EMBEDDINGS , 1997 .

[19]  John Sullivan,et al.  Minimizing the Squared Mean Curvature Integral for Surfaces in Space Forms , 1992, Exp. Math..

[20]  Leif Kobbelt,et al.  Generating fair meshes with G/sup 1/ boundary conditions , 2000, Proceedings Geometric Modeling and Processing 2000. Theory and Applications.

[21]  Kenneth A. Brakke,et al.  The Surface Evolver , 1992, Exp. Math..

[22]  Barbara Putz,et al.  Approximation of spirals by piecewise curves of fewest circular arc segments , 1984 .

[23]  Manfredo P. do Carmo,et al.  Differential geometry of curves and surfaces , 1976 .

[24]  H. Piaggio.  Differential Geometry of Curves and Surfaces , 1952, Nature.

[25]  A. Barr,et al.  Discrete Differential-Geometry Operators in n D , 2000 .

[26]  Joe D. Warren,et al.  Subdivision Schemes for Thin Plate Splines , 1998, Comput. Graph. Forum.

[27]  Gabriel Taubin,et al.  Estimating the tensor of curvature of a surface from a polyhedral approximation , 1995, Proceedings of IEEE International Conference on Computer Vision.

[28]  Ulrich Pinkall,et al.  Computing Discrete Minimal Surfaces and Their Conjugates , 1993, Exp. Math..

[29]  Carlo H. Séquin,et al.  Functional optimization for fair surface design , 1992, SIGGRAPH.

[30]  Dereck S. Meek,et al.  On surface normal and Gaussian curvature approximations given data sampled from a smooth surface , 2000, Comput. Aided Geom. Des..