二阶总广义变分小波修复模型

目的 针对全变分小波修复模型易导致阶梯效应的缺陷,提出一种加权的二阶总广义变分小波修复模型。方法 不同于全变分小波修复模型,假设的新模型引入二阶导数项且能够自动地调解一阶和二阶导数项。另外,为有效地利用图像的局部结构信息,新模型引入了权函数,它既能保护图像的边缘又增强光滑区域的去噪能力。 为有效地计算新模型,利用交替方向法将该模型变为两个子模型, 然后对两个子模型分别给出相应的理论和算法推导。结果 相比最近基于全变分正则小波修复模型(平均信噪比,平均绝对误差及平均结构相似性指标分别为21.884 4,6.857 8,0.827 2),新模型得到更好的修复效果(平均信噪比,平均绝对误差及平均结构相似性指标分别为22.313 8,6.626 1,0.831 8)。结论 与全变分正则相比,二阶总广义变分正则更好地减轻阶梯效应。目前, 国内外学者对该问题的研究取得一些结果。由于原始-对偶算法需要较小的参数,所以运算的速度较慢,因此更快速的算法理论有待进一步研究。另外,该正则能应用于图像去噪、分割、放大等方面。

Second order total generalized variational model for wavelet inpainting

Objective 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. Method 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. Result 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. Conclusion 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.