Summary: We define a strictly convex smooth potential function and use it to measure the data fidelity as well as the regularity for image denoising and cartoon-texture decomposition. The new model has several advantages over the well-known ROF or TV-\(L^2\) and the TV-\(L^1\) model. First, due to the two-modality property of the new potential function, the new regularity has strong regularizing properties in all directions and thus encourages removing noise in smooth areas, while, near edges, it smoothes the edge mainly along the tangent direction and thus can well preserve the edges. Second, the new potential function is very close to the \(L^1\) norm; thus using it to measure the data fidelity makes the new model perform very well in removing impulse noise and preserving the contrast. Lastly, the proposed fidelity and regularization term is strictly convex and smooth and thus allows a unique global minimizer and it can be solved by using the steepest descent method. Numerical experiments show that the proposed model outperforms TV-\(L^2\) and TV-\(L^1\) in removing impulse noise and mixed noise. It also outperforms some state-of-the-art methods specially designed for impulse noise. Tests on cartoon-texture decomposition show that our method is effective and performs better than TV-\(L^1\).