High-resolution image reconstruction obtains one high-resolution image from multiple low-resolution, shifted, degraded samples of a true scene. This is a typical ill-posed problem and optimization models such as the $\ell^2$/TV model are previously studied for solving this problem. It is based on the assumption that during acquisition digital images are polluted by Gaussian noise. In this work, we propose a new optimization model arising from the statistical assumption for mixed Gaussian and impulse noises, which leads us to choose the Moreau envelop of the $\ell^1$-norm as the fidelity term. The developed env$_{\ell^1}$/TV model is effective to suppress mixed noises, combining the advantages of the $\ell^1$/TV and the $\ell^2$/TV models. Furthermore, a fixed-point proximity algorithm is developed for solving the proposed optimization model and convergence analysis is provided. An adaptive parameter choice strategy for the developed algorithm is also proposed for fast convergence. The experimental results confirm the superiority of the proposed model compared to the previous $\ell^2$/TV model besides the robustness and effectiveness of the derived algorithm.