On a fractional reaction–diffusion system applied to image decomposition and restoration

In this paper, we propose a new model using a fractional reaction–diffusion system for image restoration and image decomposition which is motivated by work of S. Osher, A. Solé and L. Vese's. Our model decomposes the degraded image into cartoon component belonging to a fractional Sobolev space and textured component belonging to a negative Hilbert space. We are interested in the existence and uniqueness of weak solutions of the system obtained from the model. To overcome the difficulties of the singularity in the system, we use a regularization method. We also give a numerical algorithm for solving the system, and present numerical experimental results to show the efficiency of our model.