Parameterization and reconstruction from 3D scattered points based on neural network and PDE techniques

Reverse engineering ordinarily uses laser scanners since they can sample 3D data quickly and accurately relative to other systems. These laser scanner systems, however, yield an enormous amount of irregular and scattered digitized point data that requires intensive reconstruction processing. Reconstruction of freeform objects consists of two main stages: 1) parameterization and 2) surface fitting. Selection of an appropriate parameterization is essential for topology reconstruction as well as surface fitness. Current parameterization methods have topological problems that lead to undesired surface fitting results, such as noisy selfintersecting surfaces. Such problems are particularly common with concave shapes whose parametric grid is self-intersecting, resulting in a fitted surface that considerably twists and changes its original shape. In such cases, other parameterization approaches should be used in order to guarantee non-self-intersecting behavior. The parameterization method described in this paper is based on two stages: 1) 2D initial parameterization and 2) 3D adaptive parameterization. Two methods were developed for the first stage: Partial Differential Equation (PDE) parameterization and neural network Self Organizing Maps (SOM) parameterization. PDE parameterization yields a parametric grid without self-intersections. Neural network SOM parameterization creates a grid where all the sampled points, not only the boundary points, affect the grid, leading to a uniform and smooth surface. In the second stage, a 3D base surface was created and then adaptively modified. To this end, the Gradient Descent Algorithm (GDA) and Random Surface Error Correction (RSEC), both of which are iterative surface fitting methods, were developed and implemented. The feasibility of the developed parameterization methods and fitting algorithms is demonstrated on several examples using sculptured free objects.