Abstract: In view of the disadvantage of traditional Total Generalized Variational（TGV） wavelet inpainting model, which adopts a single wavelet transforms and has a good effect on the image with less textures details and simple structure merely, a new model named total generalized variational image inpainting model based on compact frame domain is proposed. In contrast to traditional wavelet inpainting model, compact frame system has many excellent characteristics, such as redundancy, time shift invariant and linear phase, which is important in image processing. New model can make constraint on image by introducing lower order and higher order derivative of multi-level tight frame decomposition coefficient, and get the characteristic information of many different directions in the multiscale. The numerical implementation of model adopts an optimization algorithm which connects splitting technique and primal-dual algorithm（PDSBA）, then solves two simple sub problems by alternate iteration, and improves the computational efficiency of image restoration. Compared to the related model, new model can not only reduce staircase effect and produce image edge, but also can restore images that have much detail and texture information. The experiment results show that three restoration performance indexes Peak Signal to Noise Ratio（PSNR）, Mean Square Error（MAE） and Structural Similarity?Index Measure （SSIM） are all significantly improved.

Key words: Total Generalized Variational（TGV） wavelet inpainting, tight frame system, multi-level compact frame decomposition, lower order and higher order derivative, splitting technique, primal-dual algorithm

摘要： 针对传统总广义变分（TGV）小波修复模型采用单一小波基变换，仅对纹理细节信息较少且结构简单的图像有很好修复能力的缺点，提出一种紧框架域下的总广义变分正则化修复模型。不同于经典小波变换，紧框架系统具有冗余、时移不变和线性相位等图像处理过程中较为重要的特性。新模型通过引入多层紧框架分解系数的低阶与高阶导数项建立正则化项，获取图像不同尺度多方向上的特征信息来对图像进行约束。模型的数值实现采用分裂技术与原始-对偶方法相结合的优化算法（PDSBA），交替迭代求解两个易于处理的子问题，提高了图像修复过程的处理效率。相比于传统模型，所提模型不仅具有保边性能，而且对含有较多细节或纹理信息的图像也有较好的修复效果。实验结果显示，三个修复性能指标：峰值信噪比（PSNR）、平均绝对误差（MAE）和结构相似测度（SSIM）均获得显著提升。

关键词: 总广义变分小波修复, 紧框架系统, 多层紧框架分解, 低阶与高阶导数项, 分裂技术, 原始-对偶算法