Curve and Surface Reconstruction
Curve
and surface
reconstruction from a set of sample points is a
problem occurring in many domains, e.g., reverse engineering,
CAD, and data interpretation. Given a finite set S
of sample points from an unknown
curve or
surface γ, reconstruct the
curve or surface. In the
case of a curve this means to connect two sample points if and
only if they are adjacent on γ and in the case of a
surface this means to construct a triangulated surface with
vertices in S
homeomorphic to γ. The figures below
show sample sets and the reconstructed curve or
surface,
respectively.
The
problems are
well-studied since the early 80s and a number of effective
heuristics were developed in the 80s and 90s. In the past 10
years, the research shifted to algorithms which come with a
guarantee, i.e., which provably solve the problem for a
certain class of curves or surfaces and for samples satisfying
a certain sampling condition.
The
first
algorithms worked for smooth closed curves. This was then extended
to collections of open and closed smoothed curves, and finally
to non-smooth curves (the example on the left illustrates such
a reconstruction). In the case of surfaces, the case of smooth
surfaces is solved. Current work investigates generalizations
to non-smooth surfaces.