Extracting Manifold Information from Point Clouds
Patrick Guidotti
A kernel based method is proposed for the construction of signature
(defining) functions of subsets of $\mathbb{R}^d$. The subsets can range from
full dimensional manifolds (open subsets) to point clouds (a finite number of
points) and include bounded smooth manifolds of any codimension. The
interpolation and analysis of point clouds are the main application. Two
extreme cases in terms of regularity are considered, where the data set is
interpolated by an analytic surface, at the one extreme, and by a H\"older
continuous surface, at the other. The signature function can be computed as a
linear combination of translated kernels, the coefficients of which are the
solution of a finite dimensional linear problem. Once it is obtained, it can be
used to estimate the dimension as well as the normal and the curvatures of the
interpolated surface. The method is global and does not require explicit
knowledge of local neighborhoods or any other structure present in the data
set. It admits a variational formulation with a natural ``regularized''
counterpart, that proves to be useful in dealing with data sets corrupted by
numerical error or noise. The underlying analytical structure of the approach
is presented in general before it is applied to the case of point clouds.