3D image
From the above image (several voxels in 3D image) we can observe that in 3D image, two neighboring pixels could have 3 different relationships: face-connected (red voxel VS gray voxel, distance=1), edge-connected (red voxel VS blue voxel, distance=sqrt(2)) and corner-connected (red voxel VS green voxel, distance=sqrt(3)).
Therefore, given red voxel is the object, drawing its border could have 3 different results: face-connected (only gray voxel is the border, distance=1), edge-connected (gray and blue voxels are the border, where the distance is equal to or less than sqrt(2)) and corner-connected (gray, blue and green voxels are all the border, where the distance is equal to or less than sqrt(3)).
2D image
2D image only has face-connected neighbors and edge-connected neighbors.
Note: This post is to illustrate the fully_connected argumennt in seg-metrics