Fractional distance regularized level set evolution with its application to image segmentation

Abstract

To avoid the irregularities during the level set evolution, a fractional distance regularized variational model is proposed for image segmentation. We first define a fractional distance regularization term which punishes the deviation of the level set function (LSF) and the signed distance function. Since the fractional derivative of the constant value function outside the starting point is nonzero, the fractional gradient modular of the LSF does not approach infinity where the integer order gradient modular is close to 0. This prevents the sharp reverse diffusion of LSF in flat areas and ensures the smooth evolution of LSF. Then, we use the Grünwald-Letnikov (G-L) fractional derivative to derive the discrete forms of the conjugate of fractional derivatives and fractional divergence. To facilitate the calculation of fractional derivatives and their conjugates, we designed their covering templates. Finally, a numerical solution to the minimization of the energy functional is obtained from these discrete forms and covering templates. Numerical experiments of medical images with different modalities show that the model in this paper can well segment weak images and intensity inhomogeneity images.