Image fusion is an imaging technique to visualize information from multiple imaging sources in one single image, which is widely used in remote sensing, medical imaging etc. In this work, we study two variational approaches to image fusion which are closely related to the standard TV-$L_2$ and TV-$L_1$ image approximation methods. We investigate their convex optimization formulations, under the perspective of primal and dual, and propose their associated new image decomposition models. In addition, we consider the TV-$L_1$ based image fusion approach and study the specified problem of fusing two discrete-constrained images $f_1(x) ∈ \mathcal{L}_1$ and $f_2(x) ∈ \mathcal{L}_2$, where $\mathcal{L}_1$ and $\mathcal{L}_2$ are the sets of linearly-ordered discrete values. We prove that the TV-$L_1$ based image fusion actually gives rise to the exact convex relaxation to the corresponding nonconvex image fusion constrained by the discrete-valued set $u(x) ∈ \mathcal{L}_1 ∪ \mathcal{L}_2$. This extends the results for the global optimization of the discrete-constrained TV-$L_1$ image approximation [8, 36] to the case of image fusion. As a big numerical advantage of the two proposed dual models, we show both of them directly lead to new fast and reliable algorithms, based on modern convex optimization techniques. Experiments with medical images, remote sensing images and multi-focus images visibly show the qualitative differences between the two studied variational models of image fusion. We also apply the new variational approaches to fusing 3D medical images.