Simultaneous cartoon and texture reconstruction for image restoration by bivariate function

We study the simultaneous cartoon and texture reconstruction problem. We propose a new model to approximate the cartoon and texture part by a sparse linear combination of some bases. A bivariate function is employed as the cost function. One of the variables is the decomposition image and the other is the sparse representation of the decomposition image. An alternating minimization algorithm is used to solve the minimization problem. We prove that the algorithm converges for both the l1-norm and the l0-norm. Numerical simulations are given to illustrate the efficiency of our method.