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Abstract

In this paper, we propose a hierarchical approach to 3D scattered
data interpolation with compactly supported basis functions.
Our numerical experiments suggest that the approach integrates
the best aspects of scattered data fitting with locally and globally
supported basis functions. Employing locally supported functions leads
to an efficient computational procedure, while a coarse-to-fine
hierarchy makes our method insensitive to the density of
scattered data and allows us to restore large parts of
missed data.

Given a point cloud distributed along a surface, we first use
spatial down sampling to construct a coarse-to-fine hierarchy
of point sets. Then we interpolate the sets starting from the
coarsest level. We interpolate a point set of the hierarchy,
as an offsetting of the interpolating function computed at
the previous level. Fig.\,\ref{risu_multi} shows an original
point set (the leftmost image) and its coarse-to-fine hierarchy
of interpolated sets.

According to our numerical experiments, the method
is essentially faster than the state-of-art scattered data
approximation with globally supported RBFs \cite{rbf}
and much simpler to implement.