Type-constrained robust fitting of quadrics with application to the 3D morphological characterization of saddle-shaped articular surfaces

Stéphane Allaire (1) , Jean-José Jacq (2, 3) , Valérie Burdin (2, 3) , Christian Roux (2, 3) , Christine Couture (4)

1 Radiation medicine program (Princess Margaret Hopital)
2 LaTIM - Laboratoire de Traitement de l'Information Medicale
3 ITI - Département Image et Traitement Information
4 PACEA - De la Préhistoire à l'Actuel : Culture, Environnement et Anthropologie

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 bias- corrected 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

Mots clés: QSM

Domaines: Traitement du signal et de l'image [eess.SP]

Stéphane Allaire, Jean-José Jacq, Valérie Burdin, Christian Roux, Christine Couture. Type-constrained robust fitting of quadrics with application to the 3D morphological characterization of saddle-shaped articular surfaces. MMBIA'07 : IEEE 8th Computer Society Workshop on Mathematical Methods in Biomedical Image Analysis 11th IEEE International Conference on Computer Vision (ICCV'07), Oct 2007, Rio De Janeiro, Brazil. pp.1 - 8, ⟨10.1109/ICCV.2007.4409163⟩. ⟨hal-02164443⟩