Biomedical Engineering Reference
In-Depth Information
9
basis functions.
In the case of trigonometric basis functions this corresponds
to a spectral representation of the deformation field where each basis func-
tion describes a particular frequency of the deformation. Restricting the
summation in Equation (15.3) to the first
) has the
effect of limiting the frequency spectrum of the transformation to the
N
terms (where 1
N
n
N
low-
est frequencies.
13.2.2
Registration Using Splines
The term spline originally referred to the use of long flexible strips of wood
or metal to model the surfaces of ships and planes. These splines were bent
by attaching different weights along their length. A similar concept can be
used to model spatial transformations. For example, a 2D transformation can
be represented by two separate surfaces whose height above the plane corre-
sponds to the displacement in the horizontal or vertical direction. An exam-
ple of such a transformation is shown in Figure 13.2.
Many registration techniques using splines are based on the assumption
that a set of corresponding points or landmarks can be identified in the
source and target images. This is analagous to the use of point landmarks for
rigid or affine registration using the Procrustes method described in Chapter 3.
These corresponding points are often referred to as
. At these
control points, spline-based transformations either interpolate or approxi-
mate the displacements which are necessary to map the location of the con-
trol point in the target image into its corresponding counterpart in the source
image. Between control points, they provide a smoothly varying displace-
ment field. The interpolation condition can be written as
control points
T
()
i
i
1,…,
n
(13.4)
FIGURE 13.2
An example of a nonrigid transformation required to warp a square into a circle. The corres-
ponding transformation is shown as two separate surfaces defining (a) the displacement in
the horizontal direction and (b) the displacement in the vertical direction.
