Abstract: This paper is an attempted application of fractional differential in image processing.Firstly through theoretical analysis gets the conclusion that fractional differential can greatly improve high frequency，reinforce medium frequency and non-linear preserve low frequency of signals.According to the theoretical analysis gets result that fractional differential operator has higher SNR than traditional first and second differential operators to extract edge information.Then using classical fractional differential G-L definition deduces fractional order differential difference function and constructs an approximate fractional order differential Tiansi module.Finally edge extraction experiments show that fractional differential operator not only can effectively extract edge information but also has higher SNR than integer differential operators.This paper attempts to expand application areas of fractional differential and carry outs a significant exploration.

Key words: fractional differential, edge detection, differential order, cover module

摘要： 文章是分数阶微分在图像处理中的尝试性应用。首先通过理论上分析得出分数阶微分可以大幅提升信号高频成分，增强信号的中频成分，非线性保留信号的甚低频。据此分析得出分数阶微分应用于图像边缘信息提取将获得高于传统基于一、二阶微分的方法的信噪比。然后由经典的分数阶微分定义出发，推导出了分数阶差分方程，构建了近似的分数阶Tiansi微分模板。最后通过图像边缘提取的实验表明：基于分数阶微分算子不仅可以有效提取图像边缘，而且比整数阶微分算子具有更高的信噪比。为拓展分数阶微分的应用领域，进行了有意义的探索。

关键词: 分数阶微分, 边缘检测, 微分阶数, 掩模模板