Topology preserving deformable image matching using constrained hierarchical parametric models

We present a topology preserving image matching method, based on a hierarchical parametric model of the deformation map. The transformation is parameterized at different scales, using a decomposition of the deformation vector field over a sequence of nested subspaces, generated from a single compactly supported scaling function. To preserve topology, the positivity of the Jacobian of the continuous deformation is enforced. We establish that, for the proposed hierarchical parametric model, the Jacobian J may be easily constrained over the whole deformation field, by only constraining its values at a limited number of control points, yielding a fast topology preserving matching algorithm. Experimental results are presented both on simulated and real-world medical MR images