On the computation of zone and double zone diagrams

Classical objects in computational geometry are defined by explicit relations. Several years ago the pioneering works of T. Asano, J. Matousek and T. Tokuyama introduced "implicit computational geometry", in which the geometric objects are defined by implicit relations involving sets. An important member in this family is called "a zone diagram". The implicit nature of zone diagrams implies, as already observed in the original works, that their computation is a challenging task. In a continuous setting this task has been addressed (briefly) only by these authors in the Euclidean plane with point sites. We discuss the possibility to compute zone diagrams in a wide class of spaces and also shed new light on their computation in the original setting. The class of spaces, which is introduced here, includes, in particular, Euclidean spheres and finite dimensional strictly convex normed spaces. Sites of a general form are allowed and it is shown that a generalization of the iterative method suggested by Asano, Matousek and Tokuyama converges to a double zone diagram, another implicit geometric object whose existence is known in general. Occasionally a zone diagram can be obtained from this procedure. The actual (approximate) computation of the iterations is based on a simple algorithm which enables the approximate computation of Voronoi diagrams in a general setting. Our analysis also yields a few byproducts of independent interest, such as certain topological properties of Voronoi cells (e.g., that in the considered setting their boundaries cannot be "fat").Comment: Very slight improvements (mainly correction of a few typos); add DOI; Ref [51] points to a freely available computer application which implements the algorithms; to appear in Discrete & Computational Geometry (available online

Voronoi image segmentation and its applications to geoinformatics

As various geospatial images are available for analysis, there is a strong need for an intelligent geospatial image processing method. Segmenting and districting digital images is a core process and is of great importance in many geo-related applications. We propose a flexible image segmentation framework based on generalized Voronoi diagrams through Euclidean distance transforms. We introduce a three-scan algorithm that segments images in O(N) time when N is the number of pixels. The algorithm is capable of handling generators of complex types (point, line and area), Minkowski metrics and different weights. This paper also provides applications of the proposed method in various geoinformation datasets. Illustrated examples demonstrate the usefulness and robustness of our proposed method

Exact Generalized Voronoi Diagram Computation using a Sweepline Algorithm

Voronoi Diagrams can provide useful spatial information. Little work has been done on computing exact Voronoi Diagrams when the sites are more complex than a point. We introduce a technique that measures the exact Generalized Voronoi Diagram from points, line segments and, connected lines including lines that connect to form simple polygons. Our technique is an extension of Fortune’s method. Our approach treats connected lines (or polygons) as a single site