March 24, 2005
Dear morphmetters,
Good morning from Vienna. This is a quick response to
Pablo Jarrin's question of yesterday regarding symmetry in landmarks.
The best and clearest paper on the subject is still,
in my view, my paper with Kanti Mardia in the
2000 volume of Biometrika:
Mardia, K.V., F. L. Bookstein, and I. J. Moreton.
Statistical assessment of bilateral symmetry of shapes.
Biometrika 87:285-300, 2000.
Divide the landmark list into the "paired" and the "unpaired," and
define an operation that consists of reflecting the configuration
(in any mirroring plane whatsoever) while at the same time
swapping the labels (left <--> right) of all pairs of paired
landmarks. (The paper calls this "reflected relabelling.")
Then, whether or not there are ANY unpaired
landmarks, the usual Procrustes procedure provides the symmetrization
of the original and the mirrored forms you want, and you can rotate the
(now exact) axis or plane of symmetry to any position you want
(horizontal and vertical are the two commonest choices, of course).
This is because the symmetrization is the Procrustes average of the
original form and the reflected-relabelled form no matter what plane
is used for the mirroring. To produce these symmetrized forms,
you mirror the entire data set (don't forget to relabel the left-right
labels of the paired landmarks), fit all original and mirrored
forms to a new (and symmetric) grand mean, then average pairs of fitted forms
(original and mirrored) individual by individual.
Our 2000 paper is fairly rigorous and abstract. For a
more accessible presentation, please see
Bookstein, F. L., and K. V. Mardia. The five components of
directional asymmetry. Pp. 35-40 in R. Aykroyd et al., eds.,
Stochastic Geometry, Biological Structure, and
Images. Department of Statistics, University of Leeds, 2003.
I'd be glad to email a copy of this item to anybody who wants one.
As for those discarded degrees of freedom, there
is no need ever to consider them in the statistics.
They have variance exactly zero after the averaging, so a proper
generalized inverse algorithm will simply omit them, or you can
use only the nontrivial relative warps of the configurations.
The formula for the count of those df is given in the papers.
Finally, regarding how one draws the resulting splines,
it is important that variability of the unpaired landmarks is
in fact part of the result: it is to be examined, not discarded.
The variety of splines shown in the Mardia-Bookstein and Bookstein-
Mardia papers cover many versions of this investigation. See also
my chapter in the book Modern Morphometrics in Physical Anthropology,
edited by Dennis Slice, that has just been published by Kluwer.
I know this note is not the equivalent of a full lecture
on the subject, but Mr. Jarrin and other morphmet readers should
be comforted that all of his questions have rigorous answers, and
that no special programming is necessary beyond the usual
Procrustes and TPS software on which most of us rely already.
The Procrustes formalism is indeed unusually helpful in thinking
through these problems of symmetry, since it doesn't require you
to locate a mirroring plane, and the corresponding splines deal
correctly with the origin of the data in symmetrized coordinates
even if drawn using all of the landmarks, paired and unpaired.
Regards to all, Fred Bookstein
[EMAIL PROTECTED]
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