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Abstract:
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, the splines in three variables
are automatically determined from the discrete data as a result of a
two-step method, 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-B{\'e}zier 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.
