The enumeration of shortest paths in cubic grid is presented herein, which could have importance in image processing and also in the network sciences. The cubic grid considers three neighborhoods - namely, 6-, 18- and 26-neighborhood related to face connectivity, edge connectivity and vertex connectivity, respectively. The formulation for distance metrics is given. L1, D18, and L_$\infty$ are the three metrics for 6-neighborhood, 18-neighborhood and 26-neighborhood. The task is to count the number of minimal paths, based on given neighborhood relations, from any given point to any other, in the three-dimensional cubic grid. Based on the coordinate triplets describing the grid, the formulations for the three neighborhoods are presented in this work. The problem both of theoretical importance and has several practical aspects.

翻译：本文提出了立方网格中最短路径的枚举方法，该方法在图像处理和网络科学中可能具有重要意义。立方网格考虑了三种邻域关系——即分别与面连通性、边连通性和顶点连通性相关的6-邻域、18-邻域和26-邻域。文中给出了距离度量的公式：L1、D18和L_$\infty$分别是6-邻域、18-邻域和26-邻域的三种度量。本研究的任务是基于给定的邻域关系，在三维立方网格中计算从任意给定点到其他任意点的最短路径数量。基于描述网格的坐标三元组，本文给出了三种邻域关系的计算公式。该问题既具有理论重要性，又具有多方面的实际应用价值。