Building Irregular Pyramids by Dual Graph Contraction

Many image analysis tasks lead to or make use of graph structures that are related through the analysis process with the planar layout of a digital image. This paper presents a theory that allows to build diierent types of hierarchies on top of such image graphs. The theory is based on the properties of a pair of dual image graphs that the reduction process should preserve, e.g. the structure of a particular input graph. The reduction process is controlled by decimation parameters, i.e. a selected subset of vertices, called survivors, and a selected subset of the graph's edges, the parent-child connections. It is formally shown that two phases of contractions transform a dual image graph to a dual image graph built by the surviving vertices. Phase one operates on the original (neighborhood) graph and eliminates all non-surviving vertices. Phase two operates on the dual (face) graph and eliminates all degenerated faces that have been created in phase one. The resulting graph preserves the structure of the survivors, it is minimal and unique with respect to the selected decimation parameters. The result is compared with two modiied speciications, the one already in use for building stochastic and adaptive irregular pyramids.