The paper presents a novel approach to image restoration, specifically focusing on deblurring and super-resolution. The authors propose an Adaptive Sparse Domain Selection (ASDS) scheme and Adaptive Regularization (AReg) to improve the quality of image restoration. The key contributions are: 1. **ASDS**: This scheme learns a set of compact sub-dictionaries from a pre-collected dataset of image patches. For each given patch, the best sub-dictionary is adaptively selected to characterize the local sparse domain, enhancing the representation accuracy compared to using a universal dictionary. 2. **AReg**: Two types of regularization terms are introduced: - **Autoregressive (AR) Models**: A set of AR models are learned from the training dataset to regularize the local image structures. - **Non-Local Self-Similarity (NLS)**: This term leverages the non-local redundancy in images to preserve edge sharpness and suppress noise. 3. **Iterative Shrinkage Algorithm**: An efficient iterative shrinkage algorithm is used to solve the $l_1$-minimization problem, which is the core of the sparse representation framework. 4. **Adaptive Sparsity Regularization**: The sparsity regularization parameter is adaptively estimated to further improve the restoration performance. The proposed method is evaluated on both deblurring and super-resolution tasks, showing superior results in terms of both PSNR and visual perception compared to state-of-the-art algorithms. The experiments demonstrate the effectiveness of the proposed approach in handling various image contents and achieving high-quality image restoration.The paper presents a novel approach to image restoration, specifically focusing on deblurring and super-resolution. The authors propose an Adaptive Sparse Domain Selection (ASDS) scheme and Adaptive Regularization (AReg) to improve the quality of image restoration. The key contributions are: 1. **ASDS**: This scheme learns a set of compact sub-dictionaries from a pre-collected dataset of image patches. For each given patch, the best sub-dictionary is adaptively selected to characterize the local sparse domain, enhancing the representation accuracy compared to using a universal dictionary. 2. **AReg**: Two types of regularization terms are introduced: - **Autoregressive (AR) Models**: A set of AR models are learned from the training dataset to regularize the local image structures. - **Non-Local Self-Similarity (NLS)**: This term leverages the non-local redundancy in images to preserve edge sharpness and suppress noise. 3. **Iterative Shrinkage Algorithm**: An efficient iterative shrinkage algorithm is used to solve the $l_1$-minimization problem, which is the core of the sparse representation framework. 4. **Adaptive Sparsity Regularization**: The sparsity regularization parameter is adaptively estimated to further improve the restoration performance. The proposed method is evaluated on both deblurring and super-resolution tasks, showing superior results in terms of both PSNR and visual perception compared to state-of-the-art algorithms. The experiments demonstrate the effectiveness of the proposed approach in handling various image contents and achieving high-quality image restoration.