Area minimizing flows

Several active contour models have been proposed to unify the curve evolution framework with classical energy minimization techniques for segmentation, such as snakes. The essential idea is to evolve a curve (in 2D) or a surface (in 3D) under constraints from image forces so that it clings to features of interest in an intensity image. The evolution equation has been derived from first principles as the gradient flow that minimizes a modified length functional, tailored to features such as edges. However, because the flow may be slow to converge in practice, a constant (hyperbolic) term is added to keep the curve/surface moving in the desired direction. We provide a justification, for this term based on the gradient flow derived from a weighted area functional, with an image dependent weighting factor. When combined with the earlier modified length gradient flow we obtain a partial differential equation (PDE) which offers a number of advantages, as illustrated by several examples of shape segmentation on medical images. In many cases the weighted area flow may be used on its own, with significant computational savings