Hi Alex, There are two options for using symmetry to complete your fossil crania:
(1) The best way to do this is to use reflected relabelling (there was a post by Fred Bookstein on March 24th explaining the method in detail -- I can forward it to you if you don't have it.) This approach does not require the definition of a midsagittal plane, rather incomplete specimens are least-squares superimposed with their reflected configurations in Procrustes space and missing data are reconstructed from their homologous counterparts on the other side. (BTW: If you average the original and its relabelled reflection after Procrustes superimposition, you will get a perfectly symmetric cranium). (2) For the alternative you will have to define a midsagittal plane, so you need at least three landmarks on the midsagittal. At least one of these three (or more) points should be on 'the other end' of the cranium (you could use for example: Nasion, Glabella and Lambda). You calculate the centroid for these midsagittal landmarks and subtract it from all landmarks. Then you calculate the first two eigenvectors of the covariance- matrix of these mean-centered midsagittal points. Then you multiply all mean-centered landmarks with the matrix-transpose of these two eigenvectors. Finally you mirror this configuration by multiplying it with a matrix that looks like this: 1 0 0 0 1 0 0 0 -1 ... and add the centroid again. You can then substitute the missing landmarks by the reflection of the opposite side. I recommend to use (1) reflected relabelling, which can be done by hand e.g. in Morpheus if you don't want to program it yourself. All the best from Leipzig Philipp * Dr. Philipp Gunz Max-Planck-Institute for Evolutionary Anthropology Dept. of Human Evolution Deutscher Platz 6, D-04103 Leipzig, Germany -- Replies will be sent to the list. For more information visit http://www.morphometrics.org
