Space: The Re-Visioning Frontier of Biological Image Analysis with Graph Theory, Computational Geometry, and Spatial Statistics

Abstract

Every biological image contains quantitative data that can be used to test hypotheses about how patterns were formed, what entities are associated with one another, and whether standard mathematical methods inform our understanding of biological phenomena. In particular, spatial point distributions and polygonal tessellations are particularly amendable to analysis with a variety of graph theoretic, computational geometric, and spatial statistical tools such as: Voronoi polygons; Delaunay triangulations; perpendicular bisectors; circumcenters; convex hulls; minimal spanning trees; Ulam trees; Pitteway violations; circularity; Clark-Evans spatial statistics; variance to mean ratios; Gabriel graphs; and, minimal spanning trees. Furthermore, biologists have developed a number of empirically related correlations for polygonal tessellations such as: Lewis’s law (the number of edges of convex polygons are positively correlated with the areas of these polygons): Desch’s Law (the number of edges of convex polygons are positively correlated with the perimeters of these polygons); and Errara’s Law (daughter cell areas should be roughly half that of their parent cells’ areas). We introduce a new Pitteway Law that the number of sides of the convex polygons in a Voronoi tessellation of biological epithelia is proportional to the minimal interior angle of the convex polygons as angles less than 90 degrees result in Pitteway violations of the Delaunay dual of the Voronoi tessellation.

Reticulate leaf venation inPilea peperomioidesis a Voronoi diagram

“…For example, when partitioning a town (space) into school districts (polygons), a Voronoi partitioning guarantees that all students living within a district are closer to that district’s school (center) than any other school, thus minimizing distances traveled. While many biological structures found in nature also seem to resemble Voronoi diagrams [31], in most of these examples (e.g., giraffe skin patterns [32] or secondary veins of dragonflies [33]) only the polygon boundaries but not the centers are present (Supplementary Text Section 1). Thus, rigorously testing whether these examples are naturally occurring Voronoi diagrams is not straight-forward.…”