Local Surface Analysis

This part of the course notes describes some of the fundamental techniques that are used to define more sophisticated processing algorithms for point-based surfaces. In particular, efficient methods for estimating local surface properties, such as normal vector and curvature, are described. These computations are based on local neighborhoods of point samples, which are defined as subsets of a given point cloud P that satisfy some local neighborhood relation. Also, the moving least squares (MLS) surface model is discussed, which defines a smooth manifold surface from the point cloud and allows the evaluation of the signed distance function. The concept of a 2-manifold surface embedded in 3-space is crucial for the algorithms described in these notes. The following definition of a manifold surface is taken from [dC76]: Definition: A subset S ⊂ IR is a 2-manifold surface, if for each x ∈ S, there exists a neighborhood V in IR and a map X : U → V ∩ S of an open set U ⊂ IR onto V ∩ S ⊂ IR such that