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Infrared images often carry obvious streak noises due to the non-uniformity of the infrared detector and the readout circuit. These streak noises greatly affect the image quality, adding difficulty to subsequent image processing. Compared with current elimination algorithms for infrared stripe noises, our approach fully utilizes the difference between the stripe noise components and the actual information components, takes the gradient sparsity along the stripe direction and the global sparsity of the stripe noises as regular terms, and treats the sparsity of the components across the stripe direction as a fidelity term. On this basis, an adaptive edge-preserving operator (AEPO) based on edge contrast was proposed to protect the image edge and, thus, prevent the loss of edge details. The final solution was obtained by the alternating direction method of multipliers (ADMM). To verify the effectiveness of our approach, many real experiments were carried out to compare it with state-of-the-art methods in two aspects: subjective judgment and objective indices. Experimental results demonstrate the superiority of our approach.