Aiming to the problem that the corrupted pixels and strips in hyperspectral image (HSI) limit the subsequent processing and applications, the sparse representation theory is applied to model HSI inpainting as an image reconstruction problem from incomplete observations, and an HSI inpainting algorithm is proposed based on adaptive sparse coding. First, an HSI incomplete observation model under the additive noise assumption is studied. Then, by introducing an online learning optimization method based on stochastic approximation, an algorithm for constructing a dictionary directly from the hyperspectral data is proposed to obtain a spectral dictionary. After that, the corrupted HSI is sparsely encoded by applying the sparse regression by variable splitting and augmented Lagrangian. Finally, the inpainted HSI is obtained by sparse reconstruction. Experiments illustrate that, compared with the state-of-the-art algorithms, the proposed algorithm can yield better inpainted results under different noise conditions, with shorter time consumption than other dictionary learning based inpainting algorithms.