Parametrization and smooth approximation of surface triangulations

A method based on graph theory is investigated for creating global parametrizations for surface triangulations for the purpose of smooth surface fitting. The parametrizations, which are planar triangulations, are the solutions of linear systems based on convex combinations. A particular parametrization, called shape-preserving, is found to lead to visually smooth surface approximations.

[1]  Larry L. Schumaker,et al.  Topics in Multivariate Approximation , 1987 .

[2]  F. Frances Yao,et al.  Computational Geometry , 1991, Handbook of Theoretical Computer Science, Volume A: Algorithms and Complexity.

[3]  Larry L. Schumaker,et al.  A Bibliography of multivariate Approximation , 1987, Topics in Multivariate Approximation.

[4]  C. Micchelli,et al.  Computation of Curves and Surfaces , 1990 .

[5]  Larry L. Schumaker,et al.  Triangulations in CAGD , 1993, IEEE Computer Graphics and Applications.

[6]  Tom Lyche,et al.  Mathematical methods in computer aided geometric design II , 1992 .

[7]  Robert J. Renka,et al.  A Storage-efficient Method for Construction of a Thiessen Triangulation , 1982 .

[8]  Kendall E. Atkinson.  An introduction to numerical analysis , 1978 .

[9]  Tony DeRose,et al.  Parametric surface interpolation , 1992, IEEE Computer Graphics and Applications.

[10]  Colin Bradley,et al.  G1 continuity of B-spline surface patches in reverse engineering , 1995, Comput. Aided Des..

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

[12]  W. T. Tutte.  Convex Representations of Graphs , 1960 .

[13]  Tony DeRose,et al.  Surface reconstruction from unorganized points , 1992, SIGGRAPH.

[14]  S. Rippa,et al.  Data Dependent Triangulations for Piecewise Linear Interpolation , 1990 .

[15]  Gregory M. Nielson,et al.  Knot selection for parametric spline interpolation , 1989 .

[16]  Takao Nishizeki.  Planar Graph Problems , 1990 .

[17]  T. B. Boffey,et al.  Applied Graph Theory , 1973 .

[18]  Henk A. van der Vorst,et al.  Bi-CGSTAB: A Fast and Smoothly Converging Variant of Bi-CG for the Solution of Nonsymmetric Linear Systems , 1992, SIAM J. Sci. Comput..

[19]  Carl de Boor,et al.  A Practical Guide to Splines , 1978, Applied Mathematical Sciences.

[20]  Larry L. Schumaker,et al.  Reconstructing 3D Objects from Cross-Sections , 1990 .

[21]  W. T. Tutte.  How to Draw a Graph , 1963 .

[22]  Fujio Yamaguchi,et al.  Curves and Surfaces in Computer Aided Geometric Design , 1988, Springer Berlin Heidelberg.

[23]  Tom Lyche,et al.  Mathematical methods in computer aided geometric design , 1989 .

[24]  Gerald Farin,et al.  Triangular Bernstein-Bézier patches , 1986, Comput. Aided Geom. Des..

[25]  Z.C Li,et al.  Harmonic models of shape transformations in digital images and patterns , 1992, CVGIP Graph. Model. Image Process..

[26]  Anne Verroust-Blondet,et al.  Interactive texture mapping , 1993, SIGGRAPH.

[27]  Larry L. Schumaker,et al.  Triangulation Methods , 1987, Topics in Multivariate Approximation.