From equivalent weighting functions to equivalent contraction kernels
Walter G. Kropatsch · Proceedings of SPIE, the International Society for Optical Engineering/Proceedings of SPIE · 1998
Burt introduced 1983 'equivalent weighting function': 'Iterative pyramid generation is equivalent to convolving the image g0 with a set of 'equivalent weighting functions' hl' gl equals hl * g0 equals h * gl-1, l > 1. It allowed him to study the effects of iterated reduction using the single parameter hl without giving up the efficient 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 he same time, the topological relations among the 'surviving' components are preserved. Repeated application produces a stack of successively smaller graphs: a pari 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 pay a similar role for graph pyramids than the convolution kernels of Gaussian pyramids. Equivalent contraction kernels combine two or more contraction kernels int 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.