We analyze a moving least squares algorithm for reconstructing a surface from point cloud data. Our algorithm defines an implicit function I whose zero set U is the reconstructed surface. We prove that I is a good approximation to the signed distance function of the sampled surface F and that U is geometrically close to and homeomorphic to F . Our proof requires sampling conditions similar to -sampling, used in Delaunay reconstruction algorithms.
We analyze a
moving least squares
(MLS) interpolation scheme for reconstructing a surface from point cloud data. The input is a sufficiently dense set of sample points that lie near a closed surface
F
with approximate surface normals. The output is a reconstructed surface passing near the sample points. For each sample point
s
in the input, we define a linear
point function
that represents the local shape of the surface near
s
. These point functions are combined by a weighted average, yielding a three-dimensional function
I
. The reconstructed surface is implicitly defined as the zero set of
I
.
We prove that the function
I
is a good approximation to the signed distance function of the sampled surface
F
and that the reconstructed surface is geometrically close to and isotopic to
F
. Our sampling requirements are derived from the
local feature size
function used in Delaunay-based surface reconstruction algorithms. Our analysis can handle noisy data provided the amount of noise in the input dataset is small compared to the feature size of
F
.
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