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. The author presents a theory that allows the building of different 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 (neighbourhood) graph, and eliminates all nonsurviving 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 modified specifications already in use for building stochastic and adaptive irregular pyramids