Abstract:
Ka-me’: A Voronoi Image Analyzer (VIA) is designed for image analysis of biological, medical, and environmental images. Our contention is that every image is replete with quantitative information that can be used to test hypotheses. One of the simplest hypotheses is that spatial arrangements arise by two or more agents spatially maintaining all of the region around them that is closer to them than that of any neighbor; agents may be fish defending a region around a nest, trees competing for sunlight with neighbors or avoiding contact and mechanical intertwining or leaving a gap to avoid herbivory by insects, packing of cells in tissues, or adjacent atomic shells in biopolymers repulsing the electronegativity of a neighboring atom’s shell or the packing of amino acid side chain polyhedra in protein folding.5 In an abstract sense, each of these inferences is stating that the causal interactions that define an observable pattern are due to local constraints that have a physical interaction between nearest neighbors. Herein we introduce our software’s features: (1) computational geometric tools such as Voronoi tessellations, convex hulls, and greatest open circles; (2) graph theoretic tools such as Delaunay triangulaion, Ulam trees, minimal spanning trees, Gabriel graphs, and nearest neighbor graphs; (3) spatial statistical analyses of dispersion, aggregation, clustering, random, and uniform distribution via such tools as the Clark-Evans Nearest Neighborhood, Variance to Mean Ratio, Index of Quantitative Variation, and Index of Relative Uncertainty as well as a Chi-squared test for bivariate data; (4) visual representation of quantitative histographic summaries of image features such as the number of edges per Voronoi polygon, the area of each cell (face in graph theoretic terminology), and an angliogram of the angles of the Voronoi cellular edges; (5) an acknowledgement of the quad-edge data structure which accounts for the speed of the software; (6) a graph theoretic feature called the Pitteway triangulation which can be used to evaluate shape features of polygonal tessellations; and, (6) several features of the human interface that have been implemented to focus on the biological image or the mathematical analysis of the image. Recent hypotheses in cellular and developmental biology related to proliferating epithelia have been examined with our software and suggestions for analyzing hypotheses about metastases, angiogenesis, morphogenesis, pattern formation, and spatial ecology will be briefly introduced.