图像关于边缘提取算子的微分不变性

THE DIFFERENTIAL INVARIANCE OF IMAGE UNDER BOUNDARY EXTRACTING OPERATOR

图像边缘是一类重要特征， 该文研究图像关于边缘提取算子的微分不变性， 其目的是考察图像在边缘提取的过程失去了什么信息． 利用Ｌｉｅ变换群在节丛上的延拓公式以及由此导出的相关微分方程的不变性条件， 文中给出了构造Ｌａｐｌａｃｅ边缘算子微分不变量的详细推导过程． 为使该文自封闭， 作者发展了所需的数学工具， 包括一系列延拓Ｌｉｅ导数的计算公式和有关微分方程解的不变性条件． 最后给出了一个包含尺度、灰度、旋转变换的数值实例．

The boundary extracted from images is a sort of important feature. The object of this paper is to study the differential invariance of images under the boundary extracting operator so as to probe what kinds of information are lost during the extraction procedure. By the prolongation formula in jet bundle of the prolongated Lie transformation group and the invariant condition to the related differential equations, this paper constructs the differential invariants of Laplace boundary operator by detailed deduction. In order to have the paper self contained, a series of calculating formula for the prolongation of Lie derivatives and the condition of invariant solutions of the differential equations are developed. Finally, a digital experiment involving scale, gray level and rotation transformation is given. The method presented by this paper can be actually adopted to a large scale of differential operators in image processing area.