Image inpainting is a crucial task in computer vision that aims to restore missing and occluded parts of damaged images. Deep-learning-based image inpainting methods have gained popularity in recent research. One such method is the deep image prior, which is unsupervised and does not require a large number of training samples. However, the deep image prior method often encounters overfitting problems, resulting in blurred image edges. In contrast, the second-order total generalized variation can effectively protect the image edge information. In this paper, we propose a novel image restoration model that combines the strengths of both the deep image prior and the second-order total generalized variation. Our model aims to better preserve the edges of the image structure. To effectively solve the optimization problem, we employ the augmented Lagrangian method and the alternating direction method of the multiplier. Numerical experiments show that the proposed method can repair images more effectively, retain more image details, and achieve higher performance than some recent methods in terms of peak signal-to-noise ratio and structural similarity.

In order to better decompose the images and protect their edges, in this paper, we proposed a coupled variational system consisting of two steps. The first step, an improved weighted variational model is introduced to obtain the cartoon and texture. Using the obtained cartoon image, in the second step, a new vector function is obtained for describing the pseudo edge of the considered image by one Tikhonov regularization variational model. Because Tikhonov regularization model is equivalent to carrying out a Gaussian linear filtering, the obtained vector function is smoother. To solve the coupled system, we give the alternating direction method, primal-dual method and Gauss-Seidel iteration. Using the coupled system, we can not only separate out the cartoon and texture parts, but also extract the edge. Extensive numerical experiments are given to show the effectiveness of the proposed method compared with other variational methods.