%0 Journal Article
%T 基于加权的分数阶变指数全变差模型研究
A Weighted Variable Exponential Fractional Total Variational Model
%A 张望哲
%J Pure Mathematics
%P 1-13
%@ 2160-7605
%D 2025
%I Hans Publishing
%R 10.12677/pm.2025.154103
%X 全变差(Total Variation, TV)模型作为一类重要的图像正则化技术，因其约束图像梯度结构的独特能力，在图像处理与信号分析等领域受到广泛关注。为解决传统TV模型中图像细节丢失与阶梯效应显现等问题。进一步有效保持图像边缘信息并实现重要区域的适度平滑，本文提出一种融合分数阶变指数的改进加权全变差模型。首先，基于log-exp函数特性构建新型加权变指数分数阶全变差模型，通过引入加权函数对图像边缘区域赋予较小权值，而对平滑区域赋予较大权值；其次，运用变分方法推导模型的Euler-Lagrange方程，将优化问题转化为梯度下降流方程进行求解；最后进行了对比实验，结果表明该方法在相关性能上有显著提升，与现有方法相比具有竞争力。
As a prominent image regularization technique, the Total Variation (TV) model has garnered extensive attention due to its unique capability to constrain gradient structures in images. To address the inherent limitations of conventional TV models—such as loss of fine details and emergence of staircase artifacts—this paper proposes an enhanced weighted total variation model that integrates fractional-order variable exponents. The proposed framework aims to preserve edge information effectively while achieving adaptive smoothing in homogeneous regions. First, a novel weighted fractional-order variable-exponent TV model is constructed based on the log-exp function, where edge regions are assigned to smaller regularization weights and smoother areas receive larger weights to balance structural fidelity and noise suppression. Second, variational principles are employed to derive the Euler-Lagrange equation, transforming the optimization problem into a gradient descent flow for numerical implementation. Finally, comparative experiments demonstrate that the proposed method achieves significant improvements in both edge preservation and artifact reduction, exhibiting competitive performance against state-of-the-art techniques in terms of quantitative metrics and visual quality.
%K 全变差模型，
%K 分数阶变指数，
%K Log-Exp函数，
%K 梯度下降流
Total Variation Model
%K Fractional Variable Exponent
%K Log-Exp Function
%K Gradient Descent Flow
%U http://www.hanspub.org/journal/PaperInformation.aspx?PaperID=110773