Smooth signed distance surface reconstruction and applications

Abstract

We describe a new and simple variational formulation to reconstruct the surface geometry, topology, and color map of a 3D scene from a finite set of colored oriented points. Point clouds are nowadays obtained using a variety of techniques, including structured lighting systems, pasive multi-view stereo algorithms, and 3D laser scanning. In our formulation the implicit function is forced to be a smooth approximation of the signed distance function to the surface. The formulation allows for a number of different efficient discretizations, reduces to a finite dimensional least squares problem for all linearly parameterized families of functions, does not require the specification of boundary conditions, and it is particularly good at extrapolating missing and/or irregularly sampled data. The resulting algorithms are significantly simpler and easier to implement than alternative methods. In particular, our implementation based on a primal-graph octree-based hybrid finite element-finite difference discretization, and the Dual Marching Cubes isosurface extraction algorithm is very efficient, and produces high quality crack-free adaptive manifold polygon meshes. After the geometry and topology are reconstructed, the color information from the points is smoothly extrapolated to the surface by solving a second variational problem which also reduces to a finite dimensional least squares problem. The resulting method produces high quality polygon meshes with smooth color maps, which accurately approximate the source colored oriented points. An open source implementation of this method is available for download. We describe applications to digital archaeology, 3D forensics, and 3D broadcasting. © 2012 Springer-Verlag.