-------- Original Message --------
Subject: RE: TPS 3D negative bending energy
Date: Fri, 11 Feb 2011 10:49:30 -0500
From: John Kent <[email protected]>
To: [email protected]
I have noticed the recent discussion on morphmet about whether the
function U(r) = |r| or -|r| is appropriate in thin-plate spline
calculations. Here is an explanation that may offer some insight
about why the minus sign is appropriate. I have slightly changed
notation.
1. Recall a function f(h), where h \in R^d represents a vector "lag",
is called positive definite if for all n and all sequences of distinct
sites t_1, ..., t_n in R^d, the nxn matrix
F = (f_{ij}), f_{ij} = f(t_i - t_j)
is positive definite. That is
a^T F a = \sum_{ij} a_i a_j f_{ij} >0
for all nonzero vectors of coefficients a = (a_1, ..., a_n)^T.
Here "^T" means the transpose of a vector or matrix.
2. A related concept is that of a conditionally positive definite
function (with respect to the constant functions, say, for the
purposes of this note). Then a function f is cpd if for all n and all
sequences of distinct sites t_1, ..., t_n, the nxn matrix
F = (f_{ij}), f_{ij} = f(t_i - t_j)
is cpd. That is
a^T F a = \sum_{ij} a_i a_j f_{ij} >0
for all nonzero vectors of coefficients a such that \sum a_i = 0 (that
is, a must be orthogonal to the constant vector). Sometimes it is
said that the coefficient vector a is an "increment" (with respect to
the constant functions).
3. Perhaps the simplest example of a cpd function is the function f(h)
= -|h|, which is cpd with respect to the constant functions in all
dimensions, d>=1. To get some intution about why the minus sign is
needed, take the simplest possible case, n=2, with two distinct sites
t_1 and t_2, so that
| 0 -|t_1 - t_2| |
F = | |.
| -|t_1 - t_2| 0 |
An increment vector a must take the form a = (alpha, -alpha)^T for
some number alpha. Hence
a^T F a = 2 alpha (-alpha)(-|t_1 - t_2|) = 2 alpha^2 |t_1 - t_2| >0
provided t_1 and t_2 are distinct and alpha is not 0. Note
that F is not positive definite; in particular its determinant is
negative.
4. Of course more work is needed for n>1 to confirm that f(h) = -|h|
is cpd. One version of the mathematical details can be found in
Kent, J T and Mardia, K V (1994). The link between kriging and
thin-plate splines. In Probability, Statistics and Optimization: A
Tribute to Peter Whittle, ed. F P Kelly. Wiley, pp 325--339.
5. The function f(h) = -|h| is of most interest in shape analysis in
d=3 dimensions when it is a building block in the thin-plate spline. The
fact that f is cpd ensures that the bending energy matrix, which is a
sort of inverse of F, is positive semi-definite.
6. In d=2 dimensions the function f(h) = +|h|^2 log |h| is of particular
interest as a building block in the thin-plate spline. Note the "+"
sign for this function. In this case f is cpd with respect to the
larger class of linear functions (not just the constant functions),
though there is not space here to describe what this means.
7. For many purposes the scaling of f does not matter; in the formula
for the thin-plate spline, the scaling factor cancels out of the
equations. In particular the scaling factor can be taken to be
negative without having any algebraic ill-effects. This may be the
reason some authors do not take great care over the choice of sign.
Professor John T. Kent, Head of Department
Department of Statistics, University of Leeds, Leeds LS2 9JT, UK
tel (direct) (+44) 113-343-5103 (fax) (+44) 113-343-5090
e-mail: [email protected]
web: http://www.maths.leeds.ac.uk/~john
