Resumen
In order to better decompose the images and protect their edges, in this paper, we proposed a coupled variational system consisting of two steps. The first step, an improved weighted variational model is introduced to obtain the cartoon and texture. Using the obtained cartoon image, in the second step, a new vector function is obtained for describing the pseudo edge of the considered image by one Tikhonov regularization variational model. Because Tikhonov regularization model is equivalent to carrying out a Gaussian linear filtering, the obtained vector function is smoother. To solve the coupled system, we give the alternating direction method, primal-dual method and Gauss-Seidel iteration. Using the coupled system, we can not only separate out the cartoon and texture parts, but also extract the edge. Extensive numerical experiments are given to show the effectiveness of the proposed method compared with other variational methods.