  The minimal geodesic models based on the Eikonal equations are capable of
finding suitable solutions in various image segmentation scenarios. Existing
geodesic-based segmentation approaches usually exploit image features in
conjunction with geometric regularization terms, such as Euclidean curve length
or curvature-penalized length, for computing geodesic curves. In this paper, we
take into account a more complicated problem: finding curvature-penalized
geodesic paths with a convexity shape prior. We establish new geodesic models
relying on the strategy of orientation-lifting, by which a planar curve can be
mapped to an high-dimensional orientation-dependent space. The convexity shape
prior serves as a constraint for the construction of local geodesic metrics
encoding a particular curvature constraint. Then the geodesic distances and the
corresponding closed geodesic paths in the orientation-lifted space can be
efficiently computed through state-of-the-art Hamiltonian fast marching method.
In addition, we apply the proposed geodesic models to the active contours,
leading to efficient interactive image segmentation algorithms that preserve
the advantages of convexity shape prior and curvature penalization.
