  This paper proposes to solve the Total Variation regularized models by
finding the residual between the input and the unknown optimal solution. After
analyzing a previous method, we developed a new iterative algorithm, named as
Residual Solver, which implicitly solves the model in gradient domain. We
theoretically prove the uniqueness of the gradient field in our algorithm. We
further numerically confirm that the residual solver can reach the same global
optimal solutions as the classical method on 500 natural images. Moreover, we
unfold our iterative algorithm into a convolution neural network (named as
Residual Solver Network). This network is unsupervised and can be considered as
an "enhanced version" of our iterative algorithm. Finally, both the proposed
algorithm and neural network are successfully applied on several problems to
demonstrate their effectiveness and efficiency, including image smoothing,
denoising, and biomedical image reconstruction. The proposed network is general
and can be applied to solve other total variation regularized models.
