基于分数阶全变差图像修补模型的快速算法

A Fast Algorithm Based on the Fractional Order Total Variation Image Inpainting Model

摘要: 针对分数阶全变差图像修补模型的梯度下降算法计算速度缓慢的问题，本文受到增广拉格朗日方法在图像去噪领域成功应用的启发，对基于分数阶全变分的图像修补模型也采用增广拉格朗日方法来进行求解。首先将原问题转化为等价的对应于增广拉格朗日泛函的鞍点问题，然后使用交替方向法将鞍点问题分解成u子问题和p子问题的序列来进行求解。接着对u子问题应用快速傅里叶变换(FFT)来求解，对p子问题应用Shrinkage算子来求解。参数的选取对数值仿真有很大的影响，本文对参数的选择进行了大量的实验并选取了较好的参数，实验表明数值算法是收敛的，并对含有文字遮挡及有人工划痕的实验图像有较好的修补效果。

Abstract: In view of the slow calculation speed of gradient descent algorithm of fractional order total variation image inpainting model, inspired by the successful application of augmented Lagrangian method in the field of image denoising, this paper also uses the augmented Lagrangian method to solve the image inpainting model based on the fractional order total variation. Firstly, the original problem is transformed into an equivalent saddle point problem corresponding to the augmented Lagrangian functional, and then the saddle point problem is decomposed into a sequence of u subproblem and p subproblem by the alternate direction method. Then the fast Fourier transform (FFT) is applied to solve the u subproblem and the shrinkage operator is applied to solve the p subproblem. The selec-tion of parameters has a great influence on the numerical simulation. This paper has carried out a lot of experiments on the selection of parameters and selected better parameters. The experiments show that the numerical algorithm is convergent, and has a good inpainting effect on the experi-mental images with text occlusion and artificial scratches.