Quantifying neighbourhood preservation in topographic mappings

Abstract

Mappings that preserve neighbourhood relationships are relevant in both practical and biological contexts. It is important to be clear about precisely what preserving neighbourhoods could mean. We give a definition of a "perfectly neighbourhood preserving" map, which we call a topographic homeomorphism, and prove that this has certain desirable properties. When a topographic homeomorphism does not exist (the usual case), many choices are available for quantifying the quality of a map. We introduce a particular measure, C, which has the form of a quadratic assignment problem. We also discuss other measures that have been proposed, some of which are related to C. A comparison of seven measures applied to the same simple mapping problem reveals interesting similarities and differences between the measures, and challenges common intuitions as to what constitutes a "good" map. 1 Introduction Problems of mapping occur frequently both in understanding biological processes and in formulatin...