Modeling and analysis of 3-D elongated shapes with applications to long bone morphometry

This paper presents a geometric model to be used as a framework for the description and analysis of three-dimensional (3-D) elongated shapes, Elongated shapes can be decomposed into two different parts: a 3-D curve (the central axis) and a 3-D surface (the straight surface), The central axis is described in terms of curvature and torsion. A novel concept of torsion image is introduced which allows the user to study the torsion of some relevant 3-D structures such as the medulla of long bones, without computing the third derivative, The description of the straight surface is based on an ordered set of Fourier Descriptors (FD's), each set representing a 2-D slice of the structure, These descriptors possess completeness, continuity, and stability properties, and some geometrical invariancies. A polar diagram is built which contains the anatomical information of the straight surface and can be used as a tool for the analysis and discrimination of 3-D structures. A technique for the reconstruction of the 3-D surface from the model's two components is presented, Various applications to the analysis of long bone structures, such as the ulna and radius, are derived from the model, namely, data compression, comparison of 3-D shapes, segmentation into 3-D primitives, and torsion and curvature analysis, The relevance of the method to morphometry and to clinical applications is discussed.