Non-rigid shape comparison of implicitly-defined curves

• Authors: Sheshadri R. Thiruvenkadam;David Groisser;Yunmei Chen
• Venue: VLSM'05 Proceedings of the Third international conference on Variational, Geometric, and Level Set Methods in Computer Vision
• Year: 2005

We present a novel variational model to find shape-based correspondences between two sets of level curves. While the usual correspondence techniques work with parametrized curves, we use a level-set formulation that enables us to handle curves with arbitrary topology. Given the functions $\Phi_{1}: (\Omega_{1} \subseteq IR^{2}) \longrightarrow IR$ and $\Phi_{2}: (\Omega_{2} \subseteq IR^{2}) \longrightarrow IR$ whose 0-level curves we want to match, we search for a diffeomorphism that minimizes the rate of change of the difference in tangential orientation of the zero-level sets. To make the formulation symmetric and to simplify computations, we map the domains of the level-set functions Φi to a common domain Ω by initial diffeomorphisms that are chosen arbitrarily. We then search for diffeomorphisms from Ω to itself, generating them by flows of certain vector fields on Ω. The resulting correspondences are scale- and rotation-invariant with respect to the curves. We show how this model can be used as a basis to compare curves of different topology. The model was tested on synthetic and MRI cardiac data,with good results.