A Universal Variational Framework for Sparsity-Based Image Inpainting

In this paper, we extend an existing universal variational framework for image inpainting with new numerical algorithms. Given certain regularization operator
\(\Phi \)
and denoting
\(u\)
the latent image, the basic model is to minimize the
\(\ell _{p},(p=0,1)\)
norm of
\(\Phi u\)
preserving the pixel values outside the inpainting region. Utilizing the operator splitting technique, the original problem can be approximated by a new problem with extra variable. With the alternating minimization method, the new problem can be decomposed as two subproblems with exact solutions. There are many choices for
\(\Phi \)
in our approach such as gradient operator, wavelet transform, framelet transform, or other tight frames. Moreover, with slight modification, we can decouple our framework into two relatively independent parts: 1) denoising and 2) linear combination. Therefore, we can take any denoising method, including BM3D filter in the denoising step. The numerical experiments on various image inpainting tasks, such as scratch and text removal, randomly missing pixel filling, and block completion, clearly demonstrate the super performance of the proposed methods. Furthermore, the theoretical convergence of the proposed algorithms is proved.