基于时间-空间分数阶偏微分方程的图像去噪模型

为了在去噪的同时更多地保留图像的细节信息，将分数阶微积分理论和梯度下降流有效结合，提出了分数阶梯度下降流的概念，并证明了能量泛函的分数阶梯度下降流在一定微分阶次范围内是收敛的。在此基础上，将时间因素引入到改进的基于空间分数阶偏微分方程的去噪模型中，从而构建了基于时间-空间分数阶偏微分方程的去噪模型，该模型实现了在时间方向上和空间平面内的同时去噪。实验结果表明，提出的基于时间-空间分数阶偏微分方程的图像去噪模型较基于空间分数阶偏微分方程的图像去噪模型不仅可以提高信噪比，而且可以大幅减少图像获得最大信噪比所需要的迭代次数。

Research on image denoising based on time-space fractional partial differential equations

In order to preserve more image details information while image denoising,the concept of fractional-order gradient descent flow is proposed by combining fractional calculus and gradient descent flow,and the fractional-order gradient descent flow of an energy function is convergent within a certain range of differential order.On this base,the denoising model based on time-space fractional partial equations is constructed by adding a time factor to the improved denoising model based on space fractional partial equations.The proposed denoising model can be implemented to remove noise at the time and space direction simultaneously.The experimental results show that,compared with the existing denoising model,the improved image denoising model based on time-space fractional partial differential equations could make the visual effect better and has a faster computing speed.In addition,compared with the image denoising model based on space fractional partial differential equations,the image denoising model based on time-space fractional partial differential equations can appropriately increase the signal-to-noise ratio of images and significantly reduce the iteration number under the condition that the signal-to-noise ratio of the denoising image getting the maximum.