Type-Constrained Robust Fitting of Quadrics with Application to the 3D Morphological Characterization of Saddle-Shaped Articular Surfaces

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.

Type-Constrained Direct Fitting of Quadric Surfaces

“…Others have proposed exploring the space of solutions returned by the fitting method: While the best fit may not have the desired type, algebraic fitting uses a generalized eigenvalue method that returns a basis of solutions, and one of the other eigenvectors might have the desired type [1]. However, these eigenvectors are not optimal in terms of fitting error -in fact, even the second-best eigenvector tends to correspond to a very poor fit, as we illustrate in Figs.…”

“…Previous direct fitting methods for ellipsoids and hyperboloids have used algebraic fitting with a custom normalization function ( ) q c to ensure that at least one of the resulting eigenvectors has the desired quadric type [1,14]. These methods guarantee a hyperboloid or ellipsoid, but the result may not be a good fit: The type-constraining normalizations introduce more bias than Taubin's method, leading to poorer fitting results in the presence of noise, as shown in Fig.…”

“…1, 6, 8), type-specific fitting is useful in any quadric fitting application where the correct or desired quadric type is known a-priori. There are many such applications: Allaire et al give the example of fitting a bone joint that is known to be a hyperboloid [1]; many CAD parts are known to be cylinders or cones by construction; hyperbolic paraboloids show up in architecture, art, and Pringles potato chips. In some cases it is important that the quadric surface be bounded -for example, if the whole quadric surface is expected to correspond to some real surface -and in this case, the type must be an ellipsoid and not a hyperboloid.…”

“…Methods for type-specific quadric fitting are scattered throughout the literature: Some papers handle spheres, circular cones and cylinders [15]; a few others handle ellipsoids [14] or hyperboloids [1]. Non-circular cones and general rotationally symmetric quadrics are not typically discussed.…”

Performing Accurate Joint Kinematics From 3-D In Vivo Image Sequences Through Consensus-Driven Simultaneous Registration

“…Since the IMCP algorithm simultaneously performs an implicit synthesis of the actual rigid evolving shape, one useful application could be to link the algorithm to a post-processing step aiming at explicit reconstruction of the MCS in super-resolution. Applying accurate morphological analysis techniques [2,45] would then Fig. 13.…”