COLUMN TESSELLATIONS

A new class of non facet-to-facet random tessellations in three-dimensional space is introduced -- the so-called column tessellations. The spatial construction is based on a stationary planar tessellation; each cell of the spatial tessellation is a prism whose base facet is congruent to a cell of the planar tessellation. Thus intensities, topological and metric mean values of the spatial tessellation can be calculated from suitably chosen parameters of the planar tessellation.

Data Visualization Using Weighted Voronoi Diagrams

A Voronoi diagram is a standard spatial tessellation that partitions a domain into sub-regions based on proximity to a fixed set of landmark points. In order to maintain control over the size and shape of these sub-regions, a weighting scheme is often used, in which each landmark has a scalar value associated with it. This suggests a natural “inverse” problem: given a fixed set of landmark points in a given planar region and a set of “desired” areas, is it possible to calculate a set of weights so that each sub-region has a particular area? In this chapter, the authors give a fast scheme for determining these weights based on theory from convex optimization, which is then applied to a variety of problems in data visualization.

EVALUATING THE EULER-POINCARÉ CHARACTERISTIC OF A SET USING A SPATIAL TESSELLATION

A new formula is established to evaluate the Euler-Poincaré characteristic of a polyconvex subset X in Rd starting only from measurements of X in the cells of a tessellation. Simplifications occur when X is a union of cells of the tessellation, leading to another formula that unifies and extends several classical digitization results.