中文摘要：

图像欧拉数是数字拓扑学的重要特征参数之一，计算图像欧拉数的方法被不断探索更新．为了更好理解三维图像欧拉数的本质和方便计算三维图像的欧拉数，通过对三维图像连通性的深入研究，在定义三维图段和三维相邻数两个基本概念的基础上，提出局部计算三维图像欧拉数的公式和计算三维相邻数的方法，并用归纳法证明该公式与全局计算公式的等价性．不同于以往对像素和连通性的描述，为局部计算三维图像的欧拉数提供新途径．

英文摘要：

The Euler number for digital images is one of the most important features of the digital topology parameter. The method for calculating the Euler number has been constantly explored to understand the nature of the Euler number for three-dimensional images better and to conveniently calculate the 3D image Euler number. Through the in-depth study of the connectivity of three-dimensional images, a new formula to locally calculate the Euler number for 3D images is proposed based on the two basic definitions of a 3D foreground run and a 3D neighbor number. Equivalence between the new formula and the global formula is proved by the induction method. A new way to locally calculate the Euler number for 3D images is provided which is unlike the description of the previous pixels and connectivity.