Smoothing and Matching of 3-D Space Curves

Abstract

We present a new approach to the problem of matching 3D curves. The approach has an algorithmic complexity sublinear with the number of models, and can operate in the presence of noise and partial occlusions. Our method builds upon the seminal work of [9], where curves are first smoothed using B-splines, with matching based on hashing using curvature and torsion measures. However, we introduce two enhancements: We make use of non-uniform B-spline approximations, which permits us to better retain information at high curvature locations. The spline approximations are controlled (i.e., regularized) by making use of normal vectors to the surface in 3-D on which the curves lie, and by an explicit minimization of a bending energy. These measures allow a more accurate estimation of position, curvature, torsion and Frénet frames along the curve; The computational complexity of the recognition process is considerably decreased with explicit use of the Frénet frame for hypotheses generation. As opposed to previous approaches, the method better copes with partial occlusion. Moreover, following a statistical study of the curvature and torsion covariances, we optimize the hash table discretization and discover improved invariants for recognition, different than the torsion measure. Finally, knowledge of invariant uncertainties is used to compute an optimal global transformation using an extended Kalman filter. We present experimental results using synthetic data and also using characteristic curves extracted from 3D medical images.