In this study, we discuss mainly the image denoising and texture retention issues. Usually, the time-fractional derivative has an adjustable fractional order to control the diffusion process, and its memory effect can nicely retain the image texture when it is applied to image denoising. Therefore, we design a new Rudin-Osher-Fatemi model with a time-fractional derivative based on a classical one, where the discretization in space is based on the integer-order difference scheme and the discretization in time is the approximation of the Caputo derivative (i.e., Caputo-like difference is applied to discretize the Caputo derivative). Stability and convergence of such an explicit scheme are analyzed in detail. We prove that the numerical solution to the new model converges to the exact solution with the order of O(τ^2−α+h²), where τ, α, and h are the time step size, fractional order, and space step size, respectively. Finally, various evaluation criteria including the signal-to-noise ratio, feature similarity, and histogram recovery degree are used to evaluate the performance of our new model. Numerical test results show that our improved model has more powerful denoising and texture retention ability than existing ones.

在这项研究中，我们主要讨论图像降噪和纹理保留问题。通常，时间分数导数具有可调节的分数阶以控制扩散过程，当将其应用于图像去噪时，其记忆效果可以很好地保留图像纹理。因此，我们基于经典模型设计了具有时间分数导数的新Rudin-Osher-Fatemi模型，其中空间离散化基于整数阶差分方案，时间离散化是Caputo导数的近似值（即，使用类似Caputo的差异来离散化Caputo导数）。详细分析了这种显式方案的稳定性和收敛性。我们证明新模型的数值解收敛到精确解，其阶次为O（τ ^{2- α} + ħ ²），其中，τ，α，和^ h是时间步长，分数阶，和空间步长大小，分别。最后，各种评估标准（包括信噪比，特征相似度和直方图恢复度）用于评估新模型的性能。数值测试结果表明，改进后的模型比现有模型具有更强大的去噪和纹理保持能力。