A new metric for statistical analysis of rigid transformations: Application to the rib cage

Abstract

Statistical analysis of an anatomical structure composed of multiple objects is useful for many computational anatomy tasks as registration or classification. As rigid transformations do not belong to an Euclidean space, conventional mean and covariance formulas could not be applied to study the movement of each object with respect to the others. Some tools from Riemannian geometry are used instead, requiring the definition of a metric. We show that common metrics are not intuitive in the case of an object with an elongated shape and we propose a new one based on displacements of all the points of the structure. We describe the method to study the pose variability of a multi-object structure with this new metric. It is then applied to the statistical analysis of the rib cage which is composed of 24 elongated bones.