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Blind inpainting algorithms based on deep learning architectures have shown a remarkable performance in recent years, typically outperforming model-based methods both in terms of image quality and run time. However, neural network strategies typically lack a theoretical explanation, which contrasts with the well-understood theory underlying model-based methods. In this work, we leverage the advantages of both approaches by integrating theoretically founded concepts from transform domain methods and sparse approximations into a CNN-based approach for blind image inpainting. To this end, we present a novel strategy to learn convolutional kernels that applies a specifically designed filter dictionary whose elements are linearly combined with trainable weights. Numerical experiments demonstrate the competitiveness of this approach. Our results show not only an improved inpainting quality compared to conventional CNNs but also significantly faster network convergence within a lightweight network design. Our code is available at https://github.com/cv-stuttgart/SDPF_Blind-Inpainting.