Variational level set methods for image segmentation based on both L2L2 and Sobolev gradients

Variational level set methods for image segmentation involve minimizing an energy functional over a space of level set functions using a continuous gradient descent method. The functional includes the internal energy (curve length, usually) for regularization and the external energy that aligns the curves with object boundaries. Current practice is, in general, to minimize the energy functional by calculating the L2L2 gradient of the total energy. However, the Sobolev gradient is particularly effective for minimizing the curve length functional by the gradient descent method in that it produces the solution in a single iteration. In this paper, we thus propose to use the Sobolev gradient for the internal energy (curve length), while still using the L2L2 gradient for the external energy. The test results show that the “L2L2 plus Sobolev” gradient scheme is significantly more computationally efficient than the methods only based on the L2L2 gradient.