Yes, we had co-funded the implementation of TPS in GDAL.

Some background info I have found here:
http://www.ncgia.ucsb.edu/conf/SANTA_FE_CD-ROM/sf_papers/fogel_david/santafe.html#radial
http://www.pcigeomatics.com/cgi-bin/pcihlp/GCPWORKS%7CTheory%7CThin+Plate+Spline

and some comments :)
http://lists.osgeo.org/pipermail/gdal-dev/2007-March/012113.html
http://www.gdal.org/gdal__alg_8h.html#245802b88a8126c138d24febe6c9822a