Triangular Berstein-Be´zier patches
Computer Aided Geometric Design
Scattered data interpolation and approximation with error bounds
Computer Aided Geometric Design
Marching cubes: A high resolution 3D surface construction algorithm
SIGGRAPH '87 Proceedings of the 14th annual conference on Computer graphics and interactive techniques
Fundamentals of computer aided geometric design
Fundamentals of computer aided geometric design
Least squares surface approximation using multiquadrics and parametric domain distortion
Computer Aided Geometric Design
Developments in bivariate spline interpolation
Journal of Computational and Applied Mathematics - Special issue on numerical analysis in the 20th century vol. 1: approximation theory
Proceedings of the sixth ACM symposium on Solid modeling and applications
Reconstruction and representation of 3D objects with radial basis functions
Proceedings of the 28th annual conference on Computer graphics and interactive techniques
Local Lagrange interpolation by bivariate C1 cubic splines
Mathematical Methods for Curves and Surfaces
Modelling with implicit surfaces that interpolate
ACM Transactions on Graphics (TOG)
Bezier and B-Spline Techniques
Bezier and B-Spline Techniques
Simplicial subdivisions and sampling artifacts
Proceedings of the conference on Visualization '01
Smooth approximation and rendering of large scattered data sets
Proceedings of the conference on Visualization '01
Remarks on Meshless Local Construction of Surfaces
Proceedings of the 9th IMA Conference on the Mathematics of Surfaces
Scattered Data Techniques for Surfaces
Dagstuhl '97, Scientific Visualization
Multi-level partition of unity implicits
ACM SIGGRAPH 2003 Papers
Dimension of C1-splines on type-6 tetrahedral partitions
Journal of Approximation Theory
Reconstruction of volume data with quadratic super splines
IEEE Transactions on Visualization and Computer Graphics
SARDF: signed approximate real distance functions in heterogeneous objects modeling
Heterogeneous objects modelling and applications
Distance to objects built with set operations in constructive solid modeling
Proceedings of the 13th International Conference on Humans and Computers
Mathematics and Computers in Simulation
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We present an efficient algorithm for approximating huge general volumetric data sets, i.e. the data is given over arbitrarily shaped volumes and consists of up to millions of samples. The method is based on cubic trivariate splines, i.e. piecewise polynomials of total degree three defined w.r.t, uniform type-6 tetrahedral partitions of the volumetric domain. Similar as in the recent bivariate approximation approaches (cf. [10, 15]), the splines in three variables are automatically determined from the discrete data as a result of a two-step method (see [40]), where local discrete least squares polynomial approximations of varying degrees are extended by using natural conditions, i.e. the continuity and smoothness properties which determine the underlying spline space. The main advantages of this approach with linear algorithmic complexity are as follows: no tetrahedral partition of the volume data is needed, only small linear systems have to be solved, the local variation and distribution of the data is automatically adapted, Bernstein-Bézier techniques well-known in Computer Aided Geometric Design (CAGD) can be fully exploited, noisy data are automatically smoothed. Our numerical examples with huge data sets for synthetic data as well as some real-world data confirm the efficiency of the methods, show the high quality of the spline approximation, and illustrate that the rendered iso-surfaces inherit a visual smooth appearance from the volume approximating splines.