With the fast development of multi-sensor technologies, image fusion has played an essential role in modern military and civilian. To better integrate thermal radiation information in infrared images and detailed appearance information in visible images, we investigate a novel norm formulation via joint contrast and gradient preservation, for infrared-visible image fusion. Specifically, we employ a structure tensor measurement to characterize the similarity between the fused image and the infrared image in terms of thermal radiation information, to better integrate visible appearance details. Since natural image gradients follow the hyper-Laplacian distribution, we employ $ \ell _{ p\in [ 0.5,0.8]}$ norm instead of $ \ell _0$ or $ \ell _1$ norm to measure the gradient term to further extract texture information from visible images. To cope with the computational problem introduced by the coupled structure tensor and non-convex $ \ell _{p\in [ 0.5,0.8]}$ norm, we propose a computational efficient solver based on half-quadratic splitting scheme. Experiments on public datasets validate the competitiveness of FCGP from both the subjective and objective evaluation perspective.