A basic projection strategy for surface approximation is presented
and discussed. The projection is based upon the 'Moving-Least-Squares'
(MLS) approach, and the resulting surface is C^∞ smooth. The
projection involves a first stage of defining a local reference
domain and a second stage of constructing an MLS approximation
with respect to the reference domain. The general approach is
presented for the problem of approximating a (d - 1)-dimensional
manifold in R^d, d ≥ 2. The application for surface approximation
in R^3 yields a C^∞ surface, interpolating or smoothing the
data, which is mesh-independent. For example, the resulting surface
is independent upon the triangulation chosen for its parametrization.
In a recent work with Yaron Lipman and Danny Cohen-Or we have
consider the problem of approximating surfaces with sharp features.
The new method builds on the MLS projection methodology, but
introduces a fundamental modification: While the classical MLS
uses a fixed approximation space, i.e., polynomials of a certain
degree, the new method is data-dependent. For each projected
point, it finds a proper local approximation space of piecewise
polynomials (splines). The locally constructed spline encapsulates
the local singularities which may exist in the data. The optional
singularity for this local approximation space is modeled via
a Singularity Indicator Field (SIF) which is computed over the
input data points. The effectiveness of the method is demonstrated
by reconstructing surfaces from real scanned 3D data, while being
faithful to their most delicate features.