A data reduction scheme for triangulated surfaces
Computer Aided Geometric Design
Data point selection for piecewise linear curve approximation
Computer Aided Geometric Design
Multiresolution modeling and visualization of volume data based on simplicial complexes
VVS '94 Proceedings of the 1994 symposium on Volume visualization
Multiresolution analysis of arbitrary meshes
SIGGRAPH '95 Proceedings of the 22nd annual conference on Computer graphics and interactive techniques
SIGGRAPH '96 Proceedings of the 23rd annual conference on Computer graphics and interactive techniques
BLaC-Wavelets: a multiresolution analysis with non-nested spaces
Proceedings of the 7th conference on Visualization '96
Dynamic view-dependent simplification for polygonal models
Proceedings of the 7th conference on Visualization '96
VIS '97 Proceedings of the 8th conference on Visualization '97
Smooth hierarchical surface triangulations
VIS '97 Proceedings of the 8th conference on Visualization '97
The multilevel finite element method for adaptive mesh optimization and visualization of volume data
VIS '97 Proceedings of the 8th conference on Visualization '97
Triangulations from repeated bisection
Proceedings of the international conference on Mathematical methods for curves and surfaces II Lillehammer, 1997
Simplification of tetrahedral meshes
Proceedings of the conference on Visualization '98
An efficient algorithm for terrain simplification
SODA '97 Proceedings of the eighth annual ACM-SIAM symposium on Discrete algorithms
Visualization in Scientific Computing
Visualization in Scientific Computing
Constructing Hierarchies for Triangle Meshes
IEEE Transactions on Visualization and Computer Graphics
Reducing the bandwidth of sparse symmetric matrices
ACM '69 Proceedings of the 1969 24th national conference
ACM '68 Proceedings of the 1968 23rd ACM national conference
Fast Multiresolution Surface Meshing
VIS '95 Proceedings of the 6th conference on Visualization '95
Best quadratic spline approximation for hierarchical visualization
VISSYM '02 Proceedings of the symposium on Data Visualisation 2002
On Simulated Annealing and the Construction of Linear Spline Approximations for Scattered Data
IEEE Transactions on Visualization and Computer Graphics
IEEE Transactions on Visualization and Computer Graphics
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We present a method for the construction of hierarchies of single-valued functions in one, two, and three variables. The input to our method is a coarse decomposition of the compact domain of a function in the form of an interval (univariate case), triangles (bivariate case), or tetrahedra (trivariate case). We compute best linear spline approximations, understood in an integral least squares sense, for functions defined over such triangulations and refine triangulations using repeated bisection. This requires the identification of the interval (triangle, tetrahedron) with largest error and splitting it into two intervals (triangles, tetrahedra). Each bisection step requires the recomputation of all spline coefficients due to the global nature of the best approximation problem. Nevertheless, this can be done efficiently by bisecting multiple intervals (triangles, tetrahedra) in one step and by reducing the bandwidths of the matrices resulting from the normal equations.