Morphological Analysis of Shapes

We rst describe a multiresolutional discretization scheme for the boundary of a planar connected shape, by means of the Haar wavelet transform. This scheme yields a hierarchy of polygons of increasing complexity that adaptively captures the shape's boundary at increasing resolutions. Next, we present a new morphological transform called the Chordal Axis Transform for planar shapes that leads to a constructive definition of the skeleton of a discretized shape. We describe an eecient, adaptive algorithm based on the Constrained Delaunay Triangulation of polygons for skeletonizing the shape. The resulting skeleton is connected and faithfully captures the structure of the shape. Finally, we obtain a simple quantitative descriptor of the morphology of a discretized shape based on its Delaunay triangulation, and introduce the concept of morphological congruence. This leads to a graph theoretic characterization of the nature of structural diierences between shapes that are morphologically congruent.