Equivalent Contraction Kernels and the Domain of Dual Irregular Pyramids 1

Dual graph contraction reduces the number of vertices and of edges of a pair of dual image graphs while, at the same time, the topological relations among the 'surviving' components are preserved. Repeated application produces a stack of successively smaller graphs: a pair of dual irregular pyramids. The process is controlled by selected decimation parameters which consist of a subset of surviving vertices and associated contraction kernels. Equivalent contraction kernels (ECKs) combine two or more contraction kernels into one single contraction kernel which generates the same result in one single dual contraction. Decima-tion parameters of any individual pyramid level can be reconstructed from the ECK of the pyramid's apex if both vertices and edges of this ECK receive labels indicating their annihilation level in the pyramid. This is a labeled spanning tree (LST) of the base graph which allows eecient design and control of diierent types of dual irregular pyramids. Since the LST determines the pyramid, primitive modiications of the LST transform also pyramids into other pyramids on the same base graph. They open a large variety of possibilities to explore the domain of 'all' pyramids.