Solving Sobolev active contour models via level set

摘要

All the closed smooth curves C's embedded in R~2 form a differential manifold when endowed with a differential structure. We can define inner products on the tangent space of this manifold and then introduce corresponding gradients for the functional defined on the manifold. All previous geometric active contour models use the same L~2-typed inner product to define the notions of gradient. Recent works had shown that this kind of inner products induce pathological Riemannian metrics on the manifold of smooth curves. Sundaramoorthi et al. reformulated the generic geometric active contour model as Sobolev active contour model via redefining the notions of gradient in accordance with Sobolev-typed inner products. But they had directly discretized the parametric representation of Sobolev-gradient when implementing their algorithms and resulting in very complex calculations, which can in fact be solved via Level Set Method if the concept of Sobolev-gradient from the functional theory on the Sobolev-space H~(1,2)(R~2) would be adopted. The calculation would be much easier because rich results in Level Set theory could be utilized. The purpose of this paper is to show how the work can be done.