Image Processing Techniques in Emission Tomography - Correction Techniques in Emission Tomography - page 133

set p minimizing the distance between R and the transformed input image T p :

arg mi p D(T p ;R) ; (7.14)

where T p is the template image T transformed with the transformation ac-
cording to the parameter vector p andDis the distance according to some
similarity measure (see Section 7.4.1.2).
The problem of finding the optimal parameter set has to be solved in
an optimization step. Optimization is a very important part of the registra-
tion process. Information about popular non-linear least square methods like
Gauss{Newton or Levenberg{Marquardt and quasi-Newton methods such as
(L-)BFGS or SR1 are given in [47].
Additional regularization is useful for some parametric methods like free-
form deformations. Regularization is commented on further at the end of the
following paragraph.

Non-parametric registration methods
In contrast to parametric registration, the transformation in non-parametric
registration is not represented by a relatively small number of parameters.
Instead, a non-parametric transformation t :R d !R d describes the deforma-
tion for every spatial position independently. The variational formulation of
the registration problem is:

arg mi t D(T t ;R) + S(t) : (7.15)

Here T t is the template image T transformed according to the transformation
t andDmeasures the distance.Sdenotes the regularization of the transfor-
mation and 2R

0 is a weighting factor.
As non-parametric image registration is ill-posed [18], regularization is
essential to find reasonable transformations. Regularization restricts the space
of possible transformations to a smaller set of reasonable functions, e.g., by
penalizing non-smooth transformations. The most important regularizers are
elastic, fluid, diffusion and curvature regularization. Elastic regularization is
a common choice in medical imaging. More information about regularization
and additional constraints can be found in [18, 43, 46].

7.4.1.2 Similarity measure

To take up the questions that came up at the very beginning of Section 7.4,
the definition of similarity is a very important part of image registration. In
Equations (7.14) and (7.15) the similarity measure is represented by the dis-
tance functionalD. Equivalently, it is possible to use dissimilarity for compar-
ison of two images (see Definitions 6 and 7). AsDhas to be minimized, such
dissimilarity measures require some slight modifications.
The similarity measure has to be chosen according to the nature of the
data to be registered. In monomodal studies it is usually sucient to use the
fast and easy to implement sum of squared differences.