Hierarchical Spline Approximations

We discuss spline refinement methods that approximate multi-valued data defined over one, two, and three dimensions. The input to our method is a coarse decomposition of the compact domain of the function to be approximated consisting of intervals (univariate case), triangles (bivariate case), and tetrahedra (trivariate case). We first describe a best linear spline approximation scheme, understood in a least squares sense, and refine on initial mesh using repeated bisection of simplices (intervals, triangles, or tetrahedra) of maximal error. We discuss three enhancements that improve the performance and quality of our basic bisection approach. The enhancements we discuss are: (i) using a finite element approach that only considers original data sites during subdivision, (ii) including first-derivative information in the error functional and spline-coefficient computations, and (iii) using quadratic (deformed, “curved”) rather than linear simplices to better approximate bivariate and trivariate data. We improve efficiency of our refinement algorithms by subdividing multiple simplices simultaneously and by using a sparsematrix representation and system solver.

[1]  Martin Reimers,et al.  Meshless parameterization and surface reconstruction , 2001, Comput. Aided Geom. Des..

[2]  Gregory M. Nielson,et al.  Scattered data modeling , 1993, IEEE Computer Graphics and Applications.

[3]  Bernd Hamann,et al.  A data reduction scheme for triangulated surfaces , 1994, Comput. Aided Geom. Des..

[4]  Bernd Hamann,et al.  Constructing Hierarchies for Triangle Meshes , 1998, IEEE Trans. Vis. Comput. Graph..

[5]  Markus Gross,et al.  Multiresolution compression and reconstruction , 1997 .

[6]  Gregory M. Nielson,et al.  BLaC-wavelets: a multiresolution analysis with non-nested spaces , 1996, Proceedings of Seventh Annual IEEE Visualization '96.

[7]  Paolo Cignoni,et al.  Multiresolution Representation and Visualization of Volume Data , 1997, IEEE Trans. Vis. Comput. Graph..

[8]  M. Gross,et al.  Fast multiresolution surface meshing , 1995, Proceedings Visualization '95.

[9]  N. Dyn,et al.  Adaptive thinning for bivariate scattered data , 2002 .

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

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

[12]  Thomas Ertl,et al.  The multilevel finite element method for adaptive mesh optimization and visualization of volume data , 1997 .

[13]  R. L. Hardy.  Multiquadric equations of topography and other irregular surfaces , 1971 .

[14]  D. A. Duce,et al.  Visualization in Scientific Computing , 1994, Focus on Computer Graphics.

[15]  Hugues Hoppe,et al.  View-dependent refinement of progressive meshes , 1997, SIGGRAPH.

[16]  S. Rippa.  Long and thin triangles can be good for linear interpolation , 1992 .

[17]  Wolfgang Böhm,et al.  Numerical methods , 1993 .

[18]  Hans Hagen,et al.  Scientific Visualization: Overviews, Methodologies, and Techniques , 1997 .

[19]  Arie E. Kaufman.  Volume visualization , 1996, CSUR.

[20]  R. Franke.  Scattered data interpolation: tests of some methods , 1982 .

[21]  I. Duff,et al.  Direct Methods for Sparse Matrices , 1987 .

[22]  Hans Hagen,et al.  Focus on Scientific Visualization , 1993, Computer Graphics: Systems and Applications.

[23]  Lawrence J. Rosenblum,et al.  Scientific visualization : advances and challenges , 1994 .

[24]  Leila De Floriani,et al.  Multiresolution modeling and visualization of volume data based on simplicial complexes , 1994, VVS '94.

[25]  Issac J. Trotts,et al.  Smooth hierarchical surface triangulations , 1997 .

[26]  Les A. Piegl,et al.  The NURBS Book , 1995, Monographs in Visual Communication.

[27]  Bernd Hamann,et al.  Errata: Corrections to "On a Construction of a Hierarchy of Best Linear Spline Approximations Using Repeated Bisection" , 1999, IEEE Trans. Vis. Comput. Graph..

[28]  Gerald Farin,et al.  Curves and surfaces for computer aided geometric design , 1990 .

[29]  Bruce Shriver,et al.  Visualization in Scientific Computing , 1990 .