No abstract

The scope of this paper is the guaranteed fitting of specified types of quadratic surfaces to scattered 3D point clouds. Since we chose quadrics to account for articular surfaces of various shapes in medical images, the models thus estimated usefully extract global symmetry-related intrinsic features in human joints: centers, axes, extremal curvatures. The unified type-enforcing method is based on a constrained weighted least-squares minimization of algebraic residuals which uses a robust and biascorrected metric. Provided that at most one quadratic constraint is involved, every step produces closed-form eigenvector solutions. In this framework, guaranteeing the occurrence of 3D primitives of certain types among this eigendecomposition is not a straightforward transcription of the priorly handled 2D case. To explore possibilities, we re-exploit a mapping to a 2D space called the Quadric Shape Map (QSM) where the influence of any constraint on shape and type can in fact be studied visually. As a result, we provide a new enforceable quadratic constraint that practically ensures types such as hyperboloids, which helps characterize saddle-like articular surfaces. Application to a database shows how this guarantee is needed to coherently extract the center and axes of the ankle joint.

This paper addresses the ellipsoid-type-specified fitting of quadratic surfaces, in the scope of model-based global feature extraction within scattered 3D point clouds. At characterizing articular bone surfaces, the quadrics estimated indicate useful overall-symmetry-related intrinsic centers and axes in joints. A constrained weighted least-squares minimization of algebraic residuals is used, with a robust and bias-corrected metric. With only one quadratic constraint involved, every step produces closed-form eigenvector solutions. To guarantee that an ellipsoid is output, we originally exploit a 2D representation called the Quadric Shape Map (QSM) by carrying out a visual study of the influence of shape constraints. The identified ellipsoid guarantee is needed to extract the center and axes in a wrist joint data stemming from 3D medical images.