Motion Estimation - Motion Correction in Thoracic Positron Emission Tomography - page 32

The deformation u is represented by a projection of the spline coefficients, given by
the parameter vector p , into the image space by the projection matrix Q

u = Q · p . (2.30)

For the hyperelastic regularization the transformation y is first determined according
to Eq. ( 2.29 ) and is then regularized as described in Eq. ( 2.17 ). The regularization
functional in Eq. ( 2.28 ) is then defined as

hyper

S M ( p ) : = S ( x + Q · p ) . (2.31)

2.1.3.2 Coefficient-Based Regularization

The regularization term in Eq. ( 2.28 ) is based on the coefficient vector p in

the case of coefficient-based regularization and is a weighted norm of the form

S M ( p )

2
M :

p T Mp S M ( p ) : = p = . (2.32)

The matrix M determines the regularization type. Instead of regularizing the
deformation u p (

3 as in the non-parametric case
(cf. Sect. 2.1.2 ) we now regularize directly on p . We speak of coefficient-based
regularization in image space if the regularization on p considers the projection
matrix Q and thus regularizes Q

x for each spatial position x

) ∈ R

p . If only p is taken into account we speak of
coefficient-based regularization in coefficient space .
With Tikhonov regularization we will discuss one option for coefficient-based
regularization in coefficient space. Tikhonov regularization punishes gradients in
the spline coefficients and is the coefficient-based analog of diffusion regularization
introduced in Definition 2.3 .

·

Definition 2.9 (Tikhonov regularization (coefficient space)). Tikhonov

regularization in coefficient space is defined by the choice of

M tc D T D

= (2.33)

in Eq. ( 2.32 ). The superscript tc denotes ( t ) ikhonov in ( c ) oefficient space. D

is defined by

⎡ ⎤

D 1
D 2
D 3

⎣ ⎦ .

D = I 3 ⊗ (2.34)

(continued)