Geometric computation theory for morphological filtering on freeform surfaces

Surfaces govern functional behaviours of geometrical products, especially high-precision and high-added-value products. Compared with the mean line-based filters, morphological filters, evolved from the traditional E-system, are relevant to functional performance of surfaces. The conventional implementation of morphological filters based on image-processing does not work for state-of-the-art surfaces, for example, freeform surfaces. A set of novel geometric computation theory is developed by applying the alpha shape to the computation. Divide and conquer optimization is employed to speed up the computational performance of the alpha-shape method and reduce memory usage. To release the dependence of the alpha-shape method on the Delaunay triangulation, a set of definitions and propositions for the search of contact points is presented and mathematically proved based on alpha shape theory, which are applicable to both circular and horizontal flat structuring elements. The developed methods are verified through experimentation.