"Active Contour Without Edges", on Parametric Manifolds

Region based active contour model has been widely used in image segmentation on planar images. However while a photo picture or a medical image is defined on 2D or 3D Euclidean spaces, in many cases the information is defined on the curved surfaces, or more general manifolds. In this work we extend the region based active contour method to work on the parametric manifolds. Essentially, it was noticed that in some region based active contour segmentation methods, it were only the signs of the level set function values, instead of the value themselves, which contribute to the cost functional. Thus a binary state function is enough to represent the two phase segmentation. This gives an alternative view of the level set based optimization and it is especially useful when the image domain is curved because the signed distance function and its derivative are relative difficult to be evaluated in a curved space. Based on this, the segmentation on the curved space is proceed by consecutively changing the binary state function, to optimize the cost functional. Finally, the converged binary function gives the segmentation on the manifold. The method is stable and fast. We demonstrate the applications of this method, with the cost functional defined using the Chan-Vese model, in neuroimaging, fluid mechanics and geographic fields where the information is naturally defined on curved surfaces.