Medial manifolds and hierarchical description of 2D and 3D objects with applications to MRI data of the human brain

Abstract

A mathematical description and representation of 2D and 3D shape, capable of hierarchically decomposing complex objects, is at the focus of this paper. The development is based on a hierarchic extension to the Medial Axis Transformation (MAT). Our implementation of the hierarchic MAT (HMAT) combines full Voronoi tessellation generated by the set of border points with regularization procedures to obtain a hierarchy of geometrically and topologically correct medial manifolds. This hierarchy defines a skeleton pyramid allowing shape-driven decomposition in 2D and 3D. The proposed methodology is illustrated in 2D on a planar section through a 3D MRI data set of the human brain. The hierarchical decomposition indicates the process history of brain development. For further analysis, it is converted into a boundary representation. The 3D extension of the HMAT concept is tested and illustrated on synthetic objects. It is applied to the full 3D MRI data set to obtain a description of the sulci ..