Harmonic Active Contours

We propose a segmentation method based on the geometric representation of images as two-dimensional manifolds embedded in a higher dimensional space. The segmentation is formulated as a minimization problem, where the contours are described by a level set function and the objective functional corresponds to the surface of the image manifold. In this geometric framework, both data-fidelity and regularity terms of the segmentation are represented by a single functional that intrinsically aligns the gradients of the level set function with the gradients of the image and exploits this directional information to overcome image inhomogeneities and fragmented contours. The proposed formulation combines this robust alignment of gradients with attractive properties of previous methods developed in the same geometric framework: the natural coupling of image channels proposed for anisotropic diffusion and the ability of the subjective surfaces of Sarti and Sethian to detect weak edges and close fragmented boundaries. The potential of such a geometric approach lies in the general definition of Riemannian manifolds, which naturally generalizes existing segmentation methods (the geodesic active contours of Caselles et al., the active contours without edges of Chan and Vese and the robust edge integrator of Kimmel and Bruckstein ) to higher dimensional spaces, non-flat images and feature spaces. Our experiments show that the proposed technique improves the segmentation of multichannel images, images subject to inhomogeneities, and images characterized by geometric structures like ridges or valleys.