The Foundation of Computational Geometry: Theory and Application of the Point-Lattice-Concept within Modern Structure Analysis

Summary

The recent progress in various disciplines of quantitative structure analysis such as Stereology, Image Analysis and Processing, Pattern Recognition and Scene Analysis, opened the question for a common methodological background. This back — ground can be found in probabilistic and computational geometry and is subject to this paper. In using geometrical figures to convey information about structure, the concept of a lattice of points or domains (pixels) becomes decisive. Interaction of structure on one hand and a lattice on the other results in a point-texture which may or not preserve original properties such as the position-invariant, additive and continuous MINKOWSKI-measures known from Integral-Geometry. Some of those measures have a very obvious geometrical meaning such as volume, surface, norm and topological character of the structure but are only defined for certain classes of figures, the most fundamental one being the class of convex-bodies. Obviously, the condition of convexity is a restriction which can hardly ever meet the practical situation found in biological research. As a consequence of this, a new and very general class of figures is introduced for which the Minkowski-functionals can be defined and which can be described quantitatively by a “number of representation” regarding the lattice used to generate the corresponding point-texture. This opens a wide range of practical applications such as the determination of the largest possible grid-constant for Stereology and level of digitalization in Image Processing.