Motion Estimation - Motion Correction in Thoracic Positron Emission Tomography - page 27

E

μ =

+ ν ) ,

(2.15)

2 ( 1

1
2

λ > 0 and μ > 0 ⇐⇒ 0 < ν < and E > 0 . (2.16)

The linear dependency of Young's modulus E and the Lamé constants can be
seen in Eqs. ( 2.14 ) and ( 2.15 ). Together with Eq. ( 2.11 ) it gets obvious that E
simply scales the regularization functional. For practical reasons it can thus be set
to E

= 1[ 73 ].

2.1.2.3 Hyperelastic Regularization

In many medical applications of image registration the user has a priori knowledge
about the processed data concerning the allowed deformations. For example the
invertibility of the transformation is a general requirement. More specific knowl-
edge, like, e.g., the degree of compressibility of tissue can also be controlled. We
have seen in the linear elastic case that the Poisson ratio (Definition 2.6 )givesa
measure for compressibility of small deformations. In the non-linear case, the deter-
minant of the Jacobian of the transformation measures the volume change and can
thus be used to control the compression behavior - even for large deformations. This
is done with polyconvex hyperelastic regularization [ 22 , 30 , 40 ]. The regularization
functional

hyper controls changes in length, area of the surface, and volume of y and
guarantees thereby in particular the invertibility of the estimated transformation.

S

hyper

α l , α a , α v ∈ R > 0

Definition 2.7 ( S - Hyperelastic regularization). Let

be constants and p , q ≥ 2. Further, let Γ a , Γ v : R → R be positive and strictly

convex functions, with Γ v satisfying lim z → 0 + Γ v ( z )= lim z → ∞ Γ v ( z )= ∞ .The

3 3

hyperelastic regularization energy caused by the transformation y : R → R

3 3

with y ( x )= x + u ( x ) and u : R → R is defined as

hyper length area vol

S ( y )= α l ·S ( y )+ α a ·S ( y )+ α v ·S ( y ) . (2.17)

The three summands individually control changes in length, area of the
surface, and volume

p
2 dx

p
2 dx

length

y y x x u x

S ( ) : =

Ω ∇ (

( ) − ) =

Ω ∇

( ) (2.18)

q
2

area

S ( y ) : =

Ω Γ a (

Cof ( ∇ y ( x )) ) dx (2.19)

(continued)