3D Well-Composed Pictures

complement. For example, in Fig. 1a a continuous analog of the eight-point digital set is the union of the eight cubes. By a segmented image, we mean a digital image in which each point is assigned a unique label that indicates the object In the graph interpretation of a digital image, a face in a to which it belongs. By the foreground (objects) of a segmented surface of a continuous analog corresponds to a pair of 6image, we mean the objects whose properties we want to analyze adjacent points (p, q), where p belongs to the object and and by the background, all the other objects of a digital image. q belongs to its complement [9]. A different approach is If one adjacency relation is used for the foreground of a 3D taken in Kong and Roscoe [10], where, for example, a cube segmented image (e.g., 6-adjacency) and a different relation belongs to a continuous analog of a (6, 26) binary digital for the background (e.g., 26-adjacency), then interchanging picture if all of its eight corners belong to the digital object the foreground and the background can change the connected (set of black points), and a face of a cube belongs to the components of the digital picture. Hence, the choice of foresurface of a continuous analog of a digital object if the ground and background is critical for the results of the subsefour corners of the face belong to the boundary of the quent analysis (like object grouping), especially in cases where digital object. For example, in Fig. 1b such a continuous it is not clear at the beginning of the analysis what constitutes analog of the eight-point digital set is the single cube that the foreground and what the background, since this choice has the eight points as its corners. If we treat the corners immediately determines the connected components of the digiof faces as points of a (6, 26) digital picture, then the tal picture. A special class of segmented digital 3D pictures called ‘‘well-composed pictures’’ will be defined. Well-comcorresponding digital surface is composed of picture points. posed pictures have very nice topical and geometrical properSuch surfaces are analyzed in Morgenthaler and Rosenfeld ties; in particular, the boundary of every connected component [14], Kong and Roscoe [10], Francon [6], and Chen and is a Jordan surface and there is only one type of connected Zhang [3, 4]. component in a well-composed picture, since 6-, 14-, 18-, and We will interpret Z3 as the set of points with integer 26-connected components are equal. This implies that for a coordinates in 3D space R3. We will denote the set of the well-composed digital picture, the choice of the foreground and closed unit upright cubes which are centered at points of the background is not critical for the results of the subsequent Z3 by C, and the set of closed faces of cubes in C by F , analysis. Moreover, a very natural definition of a continuous i.e., each f [ F is a unit closed square in R3 parallel to analog for well-composed digital pictures leads to regular propone of the coordinate planes. erties of surfaces. This allows us to give a simple proof of a A three-dimensional digital set (i.e., a finite subset of digital version of the 3D Jordan–Brouwer separation theorem.  1997 Academic Press Z3) can be identified with a union of upright unit cubes which are centered at its points. This gives us an intuitive and simple correspondence between points in Z3 and cubes