Motion Estimation - Motion Correction in Thoracic Positron Emission Tomography - page 31

a b

4 4

2 2

− 3 − 2 − 1 1 2 3 − 3 − 2 − 1 1 2 3

− 2 − 2

− 4 − 4

Mother spline b Derivative of mother spline ∂ b

Fig. 2.2 The mother spline b defined in Eq. ( 2.27 )isshownin( a ). b is continuously differentiable
and its derivative is plotted in ( b )

Parametric transformations are implicitly regularized by the reduced number of
parameters compared to non-parametric transformations, as discussed in Sect. 2.1.2 .
Hence, regularization of, e.g., rigid or affine transformations, is usually not neces-
sary. However, regularization becomes of importance for spline transformations due
to the non-linearity [ 48 ]. Possible regularization variants for spline transformations
are discussed in the following. For parametric transformations the functional in
Eq. ( 2.22 ) becomes

!
=

J ( p ) : = D ( M ( T , y p ) , R )+ α ·S M ( p ) min , (2.28)

α ∈ R > 0

for a regularization term S M and a scalar weighting factor balancing

between the data and regularization term.

2.1.3.1 Hyperelastic Regularization

Hyperelastic regularization, introduced for non-parametric transformations in
Definition 2.7 , can also be applied to the B-spline transformation model. The
transformation y p (

x )= x + u p ( x ) is written in the discrete setting of [ 95 ]as

y = x + Q · p . (2.29)