目的 传统的边缘检测算法对于具有分形结构等复杂纹理的图像和弱边缘图像检测精度较低。方法 针对该问题，将Grünwald-Letnikov（G-L）分数阶微分引入到Canny算子中，设计了一种新的基于G-L定义的分数阶微分掩模，在分数阶阶次的选取上更灵活（阶次可取正数和负数），分析了分数阶微分掩模中的参数与边缘检测精度之间的关系，并引用了3种评价指标来评定算法的性能。结果 将G-L分数阶梯度代替Canny中传统的梯度算子，不但可以增强图像的细节信息，而且可以增强灰度均匀和弱纹理区域的梯度信息，从而提高了边缘检测的精度和稳定性；设计了一种新的基于G-L定义的分数阶微分掩模，该掩模在分数阶阶次的选取上更灵活，具有差分方向可调性，其应用范围更广；并通过实验给出了边缘检测精度与模板参数之间的关系，从而为最佳模板参数的选取提供了依据。用综合图像和真实图像进行了实验，并与传统的5种边缘检测算子和3种基于分数阶微分的边缘检测算法进行比较，从检测精度，检测效率和抗噪性能3方面验证本文算法性能，大量的实验结果表明，本文算法在检测精度，检测效率和抗躁性能方面都有较大的提升。结论 理论分析和实验结果均表明，该算法可用于检测图像中的纹理细节和弱边缘，且检测精度和稳定性都有明显的提高，本文算法是Canny算法应用的一个重要延伸。

Objective The edge of an image holds important visual information, which plays an important role in the subsequent image understanding and scene perception. Edge detection is used to extract image edge information and eliminate irrelevant information, which significantly reduce the amount of data for the subsequent analysis. Numerous scholars have proposed various edge operators for different requirements. However, the detection accuracy of traditional edge detectors is slightly low when extracting edges. Furthermore, the texture details in an image, which include complex texture with a fractal structure and weak edges, and noise immunity capability, are also weak. A fractional order derivative has the advantage of strengthening and extracting the textural features and weak edges of digital images. A person can choose a different order for the fractional calculus according to various images and interesting features to improve high-frequency signals, nonlinearly enhance intermediate-frequency signals, and retain low-frequency signals. Method To address the aforementioned problem, this study proposes an improved Canny edge detector based on a fractional order derivative. The method calculates image gradient using the classical Grünwald-Letnikov (G-L) fractional order differential definition instead of the derivative of the Gaussian function. Furthermore, a new edge detection mask based on the G-L fractional order calculus is suggested. Then, the quantitative relation curves between edge detection accuracy and the tuning parameters (α and M) are presented, which are helpful when selecting the optimal parameters. In addition, three effective evaluation criterions are introduced to assess the proposed method performance. Result We design a new edge detection mask based on the G-L definition, which is flexible in choosing the degree of fractional order derivative and capable of adjusting the direction of difference, and thus, can have extensive applications. We obtain the quantitative relation curves between edge detection accuracy and the parameters (α and M), which can be a guide in obtaining optimal parameters to extract desirable edges. We take many experiments using synthesize and real images, and compare the proposed method with five traditional edge detecting methods and three kinds of methods based on fraction calculus. The detection accuracy, detection efficiency and the robustness of the new method proposed in this paper are improved. Conclusion In a digital image, a high-frequency component corresponds to edges and noises, an intermediate-frequency component corresponds to the texture detail, and a low-frequency component corresponds to the smoothing area. Using a fractional order differential for edge detection can completely extract weak edges and texture detail. Moreover, using a fractional order differential with suitable parameters can improve noise immunity capability. Edge-detecting efficiency is improved compared with other methods. Thus, the proposed method can be used in many real-time image-processing systems. Considerable experiment results show that the proposed method is a valid edge detector for a textured image, and even exhibit an evident advantage over other techniques. The proposed method is a significant extension of the traditional Canny detector. Overall, the proposed method is a valid and effective edge detection technique.