We propose a novel first-order non-convex model for the fusion of infrared and visible images. It maintains thermal radiation information by ensuring that the fused image has similar pixel intensities as the infrared image, and it preserves the appearance information, including the edges and texture of the source images, by enforcing similar gray gradients and pixel intensities as the visible image. Our model could effectively reduce the staircase effect and enhance the preservation of sharp edges. The maximum-minimum principle of the model with Neumann boundary condition is discussed and the existence of a minimizer of our model in $W^{1,2} (Ω)$ is also proved. We employ the augmented Lagrangian method (ALM) to design a fast algorithm to minimize the proposed model and establish the convergence analysis of the proposed algorithm. Numerical experiments are conducted to showcase the distinctive features of the model and to provide a comparison with other image fusion techniques.