Fast image inpainting and colorization by Chambolle's dual method

In this paper, we propose to use Chambolle's dual methods to solve Total Variation (TV) inpainting model and (weighted) TV colorization model. The fidelity coefficients in these two models are functions which taking zero in the inpainting region and a positive constant in the other region. Then Chambolle's dual method can not be directly used to solve these models since the fidelity coefficient will be denominator in the algorithm. In order to overcome this drawback, we propose to approximate these models by adding new variables. Then the approximated problems can be solved by alternating minimization method with Chambolle's dual method and closed form solutions which is fast and easy to implement. Mathematical results of existence of minimizers are proved for both the original and the approximated problems. Numerical results and comparison with other closely related methods demonstrate that our algorithms are quite efficient.