将分数阶微分演算引入数字图像处理

为了系统论述图像分数阶微分对纹理细节的增强能力及其侧抑制原理，提出数字图像分数阶微分掩模及其运算规则。论述了分数阶微分的动力学物理意义并推导了分数阶微积分与经典时-频分析之间的关系，分析了在一定条件下二维分数阶微分的可分离性；其次，从信号处理和生物视觉神经模型两个角度提出图像分数阶微分的高斯差感受野模型，并分析其产生的特殊马赫现象；最后，提出并论述了数字图像分数阶微分掩模及其运算规则。计算机数值实验结果表明，对于纹理细节信息丰富的图像信号而言，分数阶微分对灰度变化不大的平滑区域中的纹理细节信息的增强效果明显优于整数阶微分运算。

Application of Fractional Differential Approach to Digital Image Processing

The capabilities of the fractional differential approach for enhancing textural features of two-dimensional digital images,involved Lateral Inhibition Principle,the multiform covering templates and algorithms of digital image's fractional differentiation were discussed.Firstly,the kinetic physical meaning of fractional differential and the relationship between fractional calculus and classical time-frequency analysis were deduced,and used to handle x-y separability of two-dimensional fractional calculus in undemanding conditions.Secondly,the difference of two Gaussians receptive field between fractional and integral differential of digital image was discussed in view of signal processing and biologic vision nerve model respectively.An analysis of its Mach band was also included.Finally,multiform covering templates and a numerical algorithm for computing fractional differential for digital image were discussed.Numerical experiments showed that the textural detail enhancing capabilities of fractional differential-based texture operator are better than that of integral differential-based one for rich-grained digital images.