[with (hopefully) mild reformatting by the moderator]
December 4, 2004
TO: morphmet readers
FROM: Fred Bookstein
RE: why the thin-plate spline?
I would like to thank Olumide from Germany for the original
question about the thin-plate spline a couple of days ago and also to
thank those who have posted answers so far. I'd like to chime in on the
subject, too, partly agreeing with the previous posts and partly
disagreeing.
As I explained in the Orange Book, the principal justification of
the thin-plate spline is that it is a globally optimal interpolant for a
natural figure of merit, specifically, the global integral of summed
squared second partial derivatives. (The word "natural" is used here in
a mathematical sense, not in an evolutionary or any other
natural-science meaning.) This is a consequence of some deep theorems
proved as recently as the 1970's. The kernel function (r^2 log r, in
two dimensions; |r|, in three dimensions) is chosen not as "a convenient
radial basis function" or "a matter of habit" but because it is the
fundamental solution of the _biharmonic equation_ \Del^4 u = \delta for
a space of the appropriate dimensionality. And the reason for _that_ is
the formal equivalence of the local differential equation to that global
optimization criterion, an equivalence that is not at all obvious. We
call this value "bending energy" because that's the physical
interpretation when the same equation is solved in the theory of plates
and shells; but the "bending" in the biological application is wholly
irreal, being from the dimensions of one picture (or space) into the
dimensions of a wholly different one. It is a further extraordinary
convenience that both this optimum integral and the optimizing
interpolant itself can be computed in closed matrix form from the usual
bordered-matrix equation that pretty much everybody has seen by now.
Still, there is a biological interpretation of that global
bending. We can't do much about the fact that it is an integral from
minus infinity to infinity (i.e. over the whole of the plane or space,
even the parts outside the boundary of the organism), but inside that
boundary the second derivatives are exactly what we ought to be looking
at to decide to what extent a transformation is uniform or localized.
In biological explanations, usually the explanations of uniform
transformations are biomechanical, having to do with forces, while the
explanations of focal transformations are more morphogenetic, having to
do with details of function or of life history. It took us some years
of work, in fact, to get the uniform component back INTO thin-plate
biometrics, via the uniform term from the Procrustes representation.
In addition to the biological interpretation here, there is a
useful statistical interpretation. In another remarkable mathematical
coincidence, that figure of merit (the integral of squared second
derivatives) is a quadratic form in the data (when referred to a fixed
"reference form" such as the ensemble average). That means that it has
statistics that can be related to the appropriate Euclidean gauge metric
(in this case the Procrustes metric, as the interpolation is invariant
against changes of scale, position, or orientation), which is where the
principal warps mentioned by a previous post come from: these are
eigenvalues of the bending energy with respect to normalized Euclidean
shape distance. The principal warps, together with the uniform term,
supply a very useful basis for the space in which we compute "relative
warps," which are just principal components of form after normalization
for the isometric group in the customary way. Relative warps are not
properties OF the spline-- they are principal components drawn VIA the
spline. The tools that derive from the spline are the partial warps,
which are vector multiples of the principal warps, which are
eigenfunctions of the bending energy. I am responsible for all these
names, so I have to take responsibility for any confusion as well.
The thin-plate spline thus contributes to morphometrics in two
ways, not just one: as an interpolator and as a statistical structure.
As an interpolator, it optimizes a global figure of merit that often has
a useful biological interpretation (even for the space of no bending).
As a statistical structure, it is linear in the data, and it embodies
much of what we mean by large-scale versus small-scale biological
variability, in a formalism that is furthermore completely consistent
with all the rest of the modern multivariate biometric toolkit. There
are other strong mathematical properties, too, built into this specific
interpolant when treated as a prediction function for the parts of the
image in-between the observed data (this is a third function, along with
the graphical and the biometrical). But these points are too technical
to type in an email like this one.
There are costs to this elegant approach, of course. One is the
restriction to data in the form of biologically corresponding locations
(landmarks and semilandmarks). The methods don't apply easily to sand
grains, for instance, or to handwritten digits (the familiar post-office
application). They don't apply well to questions about shape that are
not easily answered in terms of deformation, such as questions about
image textures or arrangement-independent substructures. Another
limitation is built into those strong symmetries of the method. Data
that are _known_ to arise by special sorts of biomechanical processes,
such as rigid motions, or that do not have rotational symmetry, like
processes strongly controlled by gravity or some other polarity, do not
suit these interpolants. They don't work well for very large shape
changes---the Procrustes shape coordinates themselves become somewhat
unwieldy, and the splines inform you of this by folding, which is no
longer a biologically reasonable display. For bilaterally symmetric
data, the appropriate modifications of the basic spline were worked out
only a couple of years ago; for textures and the like, we are still
working. I have an essay coming out in late 2004 or early 2005 (in a
book edited by our beloved morphmet moderator Dennis Slice) that shows
how some of these limitations, though not all, can be lifted by modest
extensions of the underlying mathematics.
I hope this answers some of the questions currently floating
around, even if at the cost of raising others.
Fred Bookstein
Vienna/Ann Arbor/Seattle
(at the moment, Ann Arbor)
[EMAIL PROTECTED]
--
Replies will be sent to the list.
For more information visit http://www.morphometrics.org
