Image Segmentation Based on Modified Fractional Allen–Cahn Equation

Abstract

We present the image segmentation model using the modified Allen–Cahn equation with a fractional Laplacian. The motion of the interface for the classical Allen–Cahn equation is known as the mean curvature flows, whereas its dynamics is changed to the macroscopic limit of Lévy process by replacing the Laplacian operator with the fractional one. To numerical implementation, we prove the unconditionally unique solvability and energy stability of the numerical scheme for the proposed model. The effect of a fractional Laplacian operator in our own and in the Allen–Cahn equation is checked by numerical simulations. Finally, we give some image segmentation results with different fractional order, including the standard Laplacian operator.