Infrared (IR) images are often contaminated by obvious intensity bias and stripes, which severely affect the visual quality and subsequent applications. It is challenging to eliminate simultaneously the mixed nonuniformity noise without blurring the fine-image details in low-textured IR images. In this article, we present a new model for simultaneous intensity bias correction and destriping through introducing two sparsity constraints. One is that model fit on the intensity bias should be as accurate as possible. A bivariate polynomial model is built to characterize the global smoothness of the intensity bias. The other constraint is that the unidirectional variational sparse model can concisely represent the direction characteristic of stripe noise. A computationally efficient numerical algorithm based on split Bregman iteration is used to solve the complex optimization problem. The proposed method is fundamentally different from the existing denoising techniques and simultaneously estimates the sharp image, intensity bias, and stripe components. Significant improvement on image quality is achieved on both simulated and real studies. Both qualitative and quantitative comparisons with the state-of-the-art correction methods demonstrate its superiority.