Motion Estimation - Motion Correction in Thoracic Positron Emission Tomography - page 33

Definition 2.9 (continued)
Here I 3 is the 3 denotes the Kronecker product [ 16 ].

The 1D derivative operators are defined by

× 3 identity matrix and ⊗

D 1 d 1

= ⊗ ⊗ ,

I n 3 I n 2 (2.35)

D 2 d 1 ⊗

= ⊗ ,

I n 3 I n 1 (2.36)

D 3 d 1 ⊗

= ⊗ .

I n 2 I n 1 (2.37)

The I n i denote identity matrices with the size n i , and the n i

defined according to Definition 2.8 . The discrete 1D first derivative operators
for each dimension are defined by

n i

× , i ∈{ 1 , 2 , 3 }

⎡ ⎤

− 11 0

⎣ ⎦ ∈ R

. . . . . .

n i − 1 , n i

d i 1 = , i ∈{ 1 , 2 , 3 } . (2.38)

−

0 11

The analog of the above definition in image space is given with the following
Definition 2.10 . As the real displacements and not the corresponding coefficients are
processed, this version is even more related to the diffusion regularization energy in
Definition 2.3 .

Definition 2.10 (Tikhonov regularization (image space)). Let the number

n 1 n 2 n 3

3

of spline coefficients be n =( , , ) ∈ N and b i ( x )= b ( x − i ) for i ∈

n d

{ , be the translated spline basis function b defined in

Eq. ( 2.27 ). The Tikhonov regularization matrix in image space is defined as

1 ,..., } , d ∈{ 1 , 2 , 3 }

M ti T 3 T 2 T 1 = I 3 ⊗ ⊗ ⊗ . (2.39)

n d , n d

The matrices T d ∈ R are defined by

T i , j =

d ∂

b i ( x ) · ∂ b j ( x ) dx , d ∈{ 1 , 2 , 3 } , (2.40)

Ω

n d n d

d

where i ∈{ 1 ,..., } , j ∈{ 1 ,..., } , and Ω ⊂ R is the 1D subspace of Ω

in dimension d ∈{ 1 , 2 , 3 } .

(continued)