Shape Descriptions and Classes of Shapes. A Proximal Physical Geometry Approach

Abstract

This chapter introduces the notion of classes of shapes that have descriptive proximity to each other in planar digital 2D image object shape detection. A finite planar shape is planar region with a boundary (shape contour) and a non-empty interior (shape surface). The focus here is on the triangulation of image object shapes, resulting in maximal nerve complexes (MNCs) from which shape contours and shape interiors can be detected and described. An MNC is collection of filled triangles (called 2-simplexes) that have a vertex in common. Interesting MNCs are those collections of 2-simplexes that have a shape vertex in common. The basic approach is to decompose an planar region containing an image object shape into 2-simplexes in such a way that the filled triangles cover either part or all of a shape. After that, an unknown shape can be compared with a known shape by comparing the measurable areas of a collection of 2-simplexes covering both known and unknown shapes. Each known shape with a known triangulation belongs to a class of shapes that is used to classify unknown triangulated shapes. Unlike the conventional Delaunay triangulation of spatial regions, the proposed triangulation results in simplexes that are filled triangles, derived by the intersection of half spaces, where the edge of each half space contains a line segment connected between vertices called sites (generating points). A straightforward result of this approach to image geometry is a rich source of simple descriptions of plane shapes of image objects based on the detection of nerve complexes that are maximal nerve complexes or MNCs. The end result of this work is a proximal physical geometric approach to detecting and classifying image object shapes.