Topographic Mappings - Tech Report available

[Geoff Goodhill]

The following paper is available via

      ftp://salk.edu/pub/geoff/goodhill_finch_sejnowski_tech95.ps.Z
or
      http://cnl.salk.edu/~geoff


   QUANTIFYING NEIGHBOURHOOD PRESERVATION IN TOPOGRAPHIC MAPPINGS

  Geoffrey J. Goodhill(1), Steven Finch(2) & Terrence J. Sejnowski(3)

         (1) The Salk Institute for Biological Studies
   10010 North Torrey Pines Road, La Jolla, CA 92037, USA

           (2) Human Communication Research Centre
         University of Edinburgh, 2 Buccleuch Place
              Edinburgh EH8 9LW, GREAT BRITAIN

           (3) The Howard Hughes Medical Institute 
          The Salk Institute for Biological Studies
     10010 North Torrey Pines Road, La Jolla, CA 92037, USA
                                &
                     Department of Biology
    University of California San Diego, La Jolla, CA 92037, USA,

     Institute for Neural Computation Technical Report Series
                     INC-9505, November 1995

                            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.

17 pages, uncompressed postscript = 154K