From Equivalent Weighting Functions to Equivalent Contraction Kernels

Burt introduced 1983 'equivalent weighting function': \Iterative pyramid generation is equivalent to convolving the image g 0 with a set of 'equivalent weighting functions' h l :" g l = h l g 0 = h g l?1 ; l > 1. It allowed him to study the eeects of iterated reduction (e.g. the low-pass character of Gaussian pyramids) using the single parameter h l without giving up the eecient iterative computation. A similar concept applies to graph pyramids built by dual graph contraction. This new algorithm 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. These play a similar role for graph pyramids than the convolution kernels of Gaussian pyramids. 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. The basic concepts are elaborated and discussed. The new theory opens a large variety of possibilities to explore the domain of 'all' graph pyramids.