Exploiting 2D Topology in Labeling Polyhedral Images

Abftract: A polyhedral image is a segmentation of the image plane into connected regions, called faces, joined by vertices and edges. The segmentation is represented by a planar network of nodes (vertices, edges, faces) linked by adjacency links. The labeling constraints at a node are all local labelings of the node consistent with itself and all its adjacent neighbors. The local labelings are represented by junctions, junction-pairs, and junction-loops respectively for the vertices, edges, and face boundaries of the image. Constraint satisfaction and propagation is done uniformly over all nodes in the image, from each node to its adjacent neighbors. The result is local consistency or inconsistency at all the nodes in the planar network. We show that globally consistent labelings of the image exist, if and only if all the nodes in the network have locally consistent labelings. The planar network of nodes, with labels and local labelings attached to each node, represents all locally/globally consistent labelings of the polyhedral image. 1. A Labeling Example Figure 1 traces the parallel labeling of two blocks, one on top of the other. The parallel labeling starts with the input image and default labels (?) for all the nodes (i.e., vertices, edges, and faces) in the image. Then, it finds all local labeling constraints at all vertices, edges, and face boundaries of the image, described respectively by junctions, junction-pairs, and junction-loops, frames 1 to 3. Attached to each node is the number of local labelings. Constraint satisfaction and propagation (CSP) is done uniformly at all nodes in the image, from every node to its neighbors, through vertex-edge, edge-face, and face-vertex links. Note the decrease in the number of local labelings at each node, frames 4 to 6. The face surrounding the two blocks has 16 local labelings, corresponding to the blocks floating in air, or resting against some imaginary surface at some of its bounding edges. The two interpretations correspond to the face labeled as image background (B) or as polyhedral face (F). Note that the blocks sitting on top of a horizontal surface can be thought as the blocks floating in air, and infinitesimally touching the surface. The surrounding face is a touchable background, and so is labeled by 8, and has only 1 junction-loop. The detection of the background face surrounding the blocks further restricts the labeling of the blocks, Frames 7 and 8 show the final labels at the …