Construction of Point Set Surfaces through Quadric Polynomials

This paper constructs PSSs (Point Set Surfaces) by combining locally fitted quadric polynomials. First, an energy function is defined as the weighted sum of distances from a point to these quadric polynomials. Then, a vector field is constructed by the weighted sum of normal vectors at input points. Finally, points on a PSS are obtained by finding local minima of the energy function along the vector field. Experiments demonstrate that high quality PSSs can be obtained from the method presented for input point clouds sampled from various shapes.

Topologically Enhanced Slicing of MLS Surfaces

Growing use of massive scan data in various engineering applications has necessitated research on point-set surfaces. A point-set surface is a continuous surface, defined directly with a set of discrete points. This paper presents a new approach that extends our earlier work on slicing point-set surfaces into planar contours for rapid prototyping usage. This extended approach can decompose a point-set surface into slices with guaranteed topology. Such topological guarantee stems from the use of Morse theory based topological analysis of the slicing operation. The Morse function for slicing is a height function restricted to the point-set surface, an implicitly defined moving least-squares (MLS) surface. We introduce a Lagrangian multiplier formulation for critical point identification from the restricted surface. Integral lines are constructed to form Morse-Smale complex and the enhanced Reeb graph. This graph is then used to provide seed points for forming slicing contours, with the guarantee that the sliced model has the same topology as the input point-set surface. The extension of this approach to degenerate functions on point-set surface is also discussed.