3D Well-Composed Pictures

By asegmentedimage, we mean a digital image in which each point is assigned a unique label that indicates the object to which it belongs. By the foreground (objects) of a segmented image, we mean the objects whose properties we want to analyze and by the background, all the other objects of a digital image. If one adjacency relation is used for the foreground of a 3D segmented image (e.g., 6-adjacency) and a different relation for the background (e.g., 26-adjacency), then interchanging the foreground and the background can change the connected components of the digital picture. Hence, the choice of foreground and background is critical for the results of the subsequent analysis (like object grouping), especially in cases where it is not clear at the beginning of the analysis what constitutes the foreground and what the background, since this choice immediately determines the connected components of the digital picture. A special class of segmented digital 3D pictures called “well-composed pictures” will be defined. Well-composed pictures have very nice topical and geometrical properties; in particular, the boundary of every connected component is a Jordan surface and there is only one type of connected component in a well-composed picture, since 6-, 14-, 18-, and 26-connected components are equal. This implies that for a well-composed digital picture, the choice of the foreground and the background is not critical for the results of the subsequent analysis. Moreover, a very natural definition of a continuous analog for well-composed digital pictures leads to regular properties of surfaces. This allows us to give a simple proof of a digital version of the 3D Jordan–Brouwer separation theorem.