Parameterization of closed surfaces for parametric surface description

A procedure for the parameterization of surface meshes of objects with spherical topology is presented. The generation of such a parameterisation has been formulated and solved as a large constrained optimization problem by C. Brechbuhler (1995), but the convergence of this algorithm becomes unstable for object meshes consisting of several thousand vertices. We propose a new more stable algorithm to overcome this problem using multi-resolution meshes. A triangular mesh is mapped to a sphere by harmonic mapping. Next, a mesh hierarchy is constructed. The coarsest level is then optimized using a modification of the original procedure to map object surfaces to the unit sphere. The result is used as a starting point for the mapping of the next finer mesh, a process which is repeated until the final result is obtained. The new approach is compared to the original one and some parameterized object surfaces are presented.

[1]  Paul S. Heckbert,et al.  Survey of Polygonal Surface Simplification Algorithms , 1997 .

[2]  Christian Michael Brechbühler.  Description and analysis of 3-D shapes by parametrization of closed surfaces , 1995 .

[3]  James S. Duncan,et al.  Deformable Fourier models for surface finding in 3-D images , 1992, Other Conferences.

[4]  Pascal Fua,et al.  Imposing Hard Constraints on Deformable Models through Optimization in Orthogonal Subspaces , 1997, Comput. Vis. Image Underst..

[5]  David P. Dobkin,et al.  MAPS: multiresolution adaptive parameterization of surfaces , 1998, SIGGRAPH.

[6]  Baba C. Vemuri,et al.  Multiresolution stochastic hybrid shape models with fractal priors , 1994, TOGS.

[7]  William H. Press,et al.  Numerical recipes in C , 2002 .

[8]  Dimitris N. Metaxas,et al.  Dynamic 3D models with local and global deformations: deformable superquadrics , 1990, [1990] Proceedings Third International Conference on Computer Vision.

[9]  Laurent D. Cohen,et al.  Fitting 3-D data using superquadrics and free-form deformations , 1994, Proceedings of 12th International Conference on Pattern Recognition.

[10]  Stéphane Mallat,et al.  A Theory for Multiresolution Signal Decomposition: The Wavelet Representation , 1989, IEEE Trans. Pattern Anal. Mach. Intell..

[11]  Guido Gerig,et al.  Elastic model-based segmentation of 3-D neuroradiological data sets , 1999, IEEE Transactions on Medical Imaging.

[12]  Guido Gerig,et al.  Parametrization of Closed Surfaces for 3-D Shape Description , 1995, Comput. Vis. Image Underst..

[13]  Matthias Eck,et al.  Automatic reconstruction of B-spline surfaces of arbitrary topological type , 1996, SIGGRAPH.

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

[15]  Barr,et al.  Superquadrics and Angle-Preserving Transformations , 1981, IEEE Computer Graphics and Applications.

[16]  James S. Duncan,et al.  Model-based deformable surface finding for medical images , 1996, IEEE Trans. Medical Imaging.

[17]  Dana H. Ballard,et al.  Computer Vision , 1982 .

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