A unifying measure for neighbourhood preservation in topographic mappings

In this paper, the abstract computational principles underlying topographic maps are discussed. We give a definition of a “perfectly neighbourhood preserving” map, which we call a topographic homeomorphism, and we prove that this has certain desirable properties. It is argued that when a topographic homeomorphism does not exist (the usual case), many equally valid choices are available for quantifying the quality of a map. We introduce a particular measure that encompasses several previous proposals, and discuss its relation to other work. This formulation of the problem sets it within the well-known class of quadratic assignment problems.

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