  Surface reconstruction from a set of scattered points, or a point cloud, has
many applications ranging from computer graphics to remote sensing. We present
a new method for this task that produces an implicit surface (zero-level set)
approximation for an oriented point cloud using only information about
(approximate) normals to the surface. The technique exploits the fundamental
result from vector calculus that the normals to an implicit surface are
curl-free. By using a curl-free radial basis function (RBF) interpolation of
the normals, we can extract a potential for the vector field whose zero-level
surface approximates the point cloud. We use curl-free RBFs based on
polyharmonic splines for this task, since they are free of any shape or support
parameters. Furthermore, to make this technique efficient and able to better
represent local sharp features, we combine it with a partition of unity (PU)
method. The result is the curl-free partition of unity (CFPU) method. We show
how CFPU can be adapted to enforce exact interpolation of a point cloud and can
be regularized to handle noise in both the normal vectors and the point
positions. Numerical results are presented that demonstrate how the method
converges for a known surface as the sampling density increases, how
regularization handles noisy data, and how the method performs on various
problems found in the literature.
