Geodesic Models with Convexity Shape Prior

Da Chen (1) , Jean-Marie Mirebeau (2) , Minglei Shu (1) , Xuecheng Tai (3) , Laurent D. Cohen (4)

1 Shandong Artificial Intelligence Institute
2 CNRS - Centre National de la Recherche Scientifique
3 HKBU - Department of Mathematics [Hong Kong Baptist University]
4 CEREMADE - CEntre de REcherches en MAthématiques de la DEcision

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 curves which are imposed 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 curves 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.

Mots clés: Geodesic curve; convexity shape prior; curvature penalization; Eikonal equation; fast marching; image segmentation

Domaines: Analyse numérique [math.NA]; Equations aux dérivées partielles [math.AP]

Da Chen, Jean-Marie Mirebeau, Minglei Shu, Xuecheng Tai, Laurent D. Cohen. Geodesic Models with Convexity Shape Prior. IEEE Transactions on Pattern Analysis and Machine Intelligence, In press. ⟨hal-03359125v1⟩