Abstract: It is very ambitious to search the optimal degree of fractional order differential operator manually, lack of order adaptive in image enhancement. In order to tackle this problem, a mathematical model of adaptive degree for fractional order differential operator is presented, which takes the arctangent function as prototype and adopts local image information as independent variables, such as the image gradient information, local information entropy, brightness and contrast. Therefore, the relationship between the optimal degree and local image information can be built up, and the optimal degree at every pixel location can be calculated automatically according to the characteristics of image. And then, the model provided is applied to fractional order differential Tiansi operator in image enhancement. Images from standard library with texture are selected as the object for the experiment to verify the validity of the mode. The experimental results are analyzed qualitatively and quantitatively. In quantitative analysis, image information entropy, image average gradient, image definition and image contrast are selected to evaluate the enhancement effect and comparison results to Laplacian and Tiansi operator are also provided. Theoretical analysis and experimental results have shown that the model presented is effective and the gray image can be enhanced continuously to the optimal level of fractional order differential enhancement effect, satisfying people’s visual perception.

Key words: fractional order differential, Tiansi operator, adaptive degree model, image enhancement

摘要： 为了解决分数阶微分算子在图像增强中需要人工寻找最佳阶次，缺乏阶次自适应的问题，构造了分数阶微分阶次自适应数学模型。该模型以反正切函数为原型，以图像的梯度信息、局部信息熵、亮度和对比度为自变量，建立了微分阶次与图像局部信息之间的关系，从而可以根据图像的局部信息特征自动计算图像中各个像素点的最佳阶次，并将该模型应用在分数阶微分Tiansi算子的图像增强中。为了验证该模型的有效性，选用标准图像库中的多幅纹理图像进行实验，对实验结果进行了定性和定量分析，在定量分析中采用图像信息熵、平均梯度、清晰度和对比度四个评价指标衡量图像增强的效果，并与二阶微分Laplacian算子、Tiansi算子进行比较。理论分析和实验结果均表明建立该模型的有效性，对灰度图像可以得到连续变化的增强效果，接近于最佳分数阶微分增强效果，符合人们的视觉感受。

关键词: 分数阶微分, Tiansi算子, 阶次自适应模型, 图像增强