Elastic Registration for Biomedical Applications - Biomedical Image Analysis - page 348

We agree that landmarks are points in space, as opposed to just coordinate
values. Similarly, the correspondence function g is more than a mathemat-
ical function: it describes correspondence of real points. It is an object
in space, anchored to the landmarks. Consequently, it seems reasonable
to require that the interpolated function g be invariant with respect to
the choice of the coordinate system. In other words, the correspondence
between points in the two images should remain the same, regardless of
how we measure the position of these points.

The interpolation problem should always have a solution, if possible
a unique one.

Another property worth having is the reproduction of identity [81]. In ad-
dition, we might want the reproduction property for other simple transfor-
mations, such as shifts or scalings; more generally, affine transformations.

We want the reconstructed correspondence function to be close to the
(unknown) true underlying correspondence function. We want the recon-
struction error to decrease rapidly with the number of landmarks—the
method should have good approximation properties [82]. This way we
can adapt the landmark density to ensure that the error is below any a
priori given tolerance threshold.

Finally, we want the interpolation procedure to accommodate easily non-
exact fits, useful when the landmark positions are only known approxi-
mately. In this approximation setting, the reconstructed correspondence
function will pass close to the landmarks, making a compromise between
the closeness of the fit and the overall smoothness.

9.3.2 Thin-Plate Splines

The use of thin-plate spline technique for landmark interpolation is attributed to
Bookstein [1]. Here, we present the method from the variational point of view, as
a preparation for the extensions presented in section 9.3.3. Instead of imposing
an empirical interpolation formula, the essence of the variational formula-
tion consists of choosing a variational criterion J ( g ) and then finding among
all possible functions passing through the landmarks the one that minimizes
J [83, 84].