分数阶原始对偶去噪模型及其数值算法

目的：本文结合分数阶微积分理论和对偶理论，提出了一种与分数阶ROF去噪模型等价的分数阶原始对偶模型。从理论上分析了该模型与具有鞍点结构的优化模型在结构上的相似性，从而可使用求解鞍点问题的数值算法求解该模型。方法：使用求解鞍点问题的基于预解式的原始对偶算法对提出模型进行求解，并采用自适应变步长迭代优化策略提高寻优效率，弥补了传统数值算法对步长要求过高的缺陷。同时论证了确保算法收敛性的参数取值范围。结果：实验结果表明，提出的分数阶原始对偶模型能够有效地抑制“阶梯效应”，保护纹理和细节信息，同时采用的数值算法具有较快的收敛速度。结论：本文提出了一种分数阶原始对偶去噪模型，该模型可采用一种基于预解式的原始对偶算法进行求解。实验结果表明，提出的模型能有效改善图像的视觉效果，采用的数值算法能有效快速收敛。

Fractional-order primal-dual model and numerical algorithm for denoising

Objective: By combining fractional calculus and duality theory, a novel fractional-order primal-dual model which is equivalent with the fractional ROF model is proposed. We theoretically analyze its structural similarity with the saddle-point optimization model. So the algorithms for solving the saddle-point problem can be used for solving the model. Method: The primal-dual algorithm based on resolvent for solving the saddle-point problem is used for solving the proposed model. The adaptive variable step size iterative optimization strategy is used, which can improve the optimizing efficiency, and remedy the step size limitation of the traditional numerical algorithms. In order to guarantee the convergence of the algorithm, the range of the parameter is given. Result: The experiment results show that the proposed fractional-order primal-dual model is effective in avoiding the staircase effect and preserving texture and detail information, and the adoptive numerical algorithm has faster convergence speed. Conclusion: This paper proposes a fractional-order primal-dual denoising model, which can be solved by a primal-dual algorithm based on resolvent. The experiment results show that the proposed model can improve the image visual effect effectively, and the adoptive numerical algorithm has faster convergence speed.