To address the drawback of the staircase effect of the total variational method

we proposed a weighted second-order total generalized variational model for wavelet inpainting. Unlike the total variational method

the proposed model contains a second-order derivative term and a first-order derivative term

which the model can automatically balance with two regularization parameters. To utilize the local structure of image information

we introduced an edge indicator function in the proposed model. The edge indicator function was 0 when the pixels belonged to the edge domain of the image and 1 when the pixels belonged to the smooth domain of the image. Thus

the proposed edge indicator function can preserve the edges and fine parts of restored images while improving noise removal in the smooth domain. To compute the new model effectively

we introduced a new variable and then used the alternative direction method to convert the original model into two submodels. For the first submodel

we used variational theory to solve the energy function and obtain the corresponding closed solution. For the second submodel

its non-convex characteristic made the derivation of solution considerably difficult. We sought to overcome such disadvantage by introducing the iteratively reweighted method. This method was employed to convert the original non-convex problem into several convex ones

that is

we improved the edge indicator function using the last restoration image and turned the non-convex submodel into a convex one. Subsequently

we introduced dual variables and transformed the second submodel into a minmax problem

which we then solved using a primal-dual algorithm. The comprehensive experimental results show that the new model obtains better results than recent total variation regularization wavelet inpainting methods.The average values of PSNR

MAE

and SSIM obtained with the total variation method are 21.884 4

6.857 8

and 0.827 2

respectively; whereas

the values obtained with the proposed method are 22.313 8

6.626 1

and 0.831 8

respectively. Total variation regularization is widely used in image processing. However

this technique produces the staircase effect. To overcome this drawback

we proposed a new weighted variational model for wavelet inpainting. This new model contains a total generalized regularization term that includes first-order derivative and second-order derivative terms. This feature allows the new model to automatically adjust the terms through two regularization parameters. To solve the new model

we first used the alternative direction method to transform the original problem into two subproblems. For the first subproblem

we obtained the necessary closed solution. For the second subproblem

we used the iterative reweighted method to reduce this subproblem into a convex problem

which we then solved using a primal-dual algorithm. For fair comparison

we used Daubechies 7-9 bi-orthogonal wavelets with symmetric extensions at the boundaries. The level of wavelet decomposition was 3

and the low frequency wavelet coefficients were lost randomly. We adopted a signal-to-noise ratio in decibels

mean absolute deviation error

and structural similarity to evaluate the quality of the restoration image. Experimental results show that compared with existing state-of-the-art algorithms

the new model is more effective in wavelet domain inpainting and is capable of obtaining better inpainting results for certain smooth images.