Motion Estimation - Motion Correction in Thoracic Positron Emission Tomography - page 21

2.1 Image Registration

Image registration is a versatile approach to motion estimation and provides a full
range of application-specific options. We give an introduction to image registration
including suitable options for the data term in Sect. 2.1.1 and the regularization
functional in Sect. 2.1.2 . The alternative to non-parametric image registration in
terms of B-spline transformations is discussed in Sect. 2.1.3 . One of the main mes-
sages of this topic is the mass-preserving transformation model introduced in
Sect. 2.1.4 which is tailored for gated PET. We refer to the comprehensive reviews
of image registration for further reading [ 18 , 49 , 63 , 66 , 88 , 90 , 94 , 95 , 139 ].
The aim of image registration is to spatially align two corresponding images. In
order to formulate this definition of image registration mathematically, we need to
find a way to measure the similarity of two images. The objective is then to find a
meaningful spatial transformation that, applied to one of the images, maximizes the
similarity.
For motion estimation, a so-called template image

T : Ω → R is registered onto

3

a reference image R : Ω → R , where Ω ⊂ R is the image domain . This yields

3 3

a spatial transformation y : R → R representing point-to-point correspondences

between T and R . To find y , the following functional has to be minimized

arg min
y

J ( y ) : = D ( M ( T , y ) , R )+ α S ( y ) . (2.1)

Here denotes the distance functional measuring the dissimilarity between the

transformed template image and the fixed reference image.

D

is the transformation
model specifying how the transformation y should be applied to the template
image

M

is the regularization functional which penalizes non-smooth transfor-
mations and thus enforces meaningful solutions.
The standard transformation model is simply given by an interpolation of

T . S

T at

the transformed grid y

std

M ( T , y ) : = T◦ y = T ( y ) . (2.2)

We will discuss an alternative transformation model for mass-preserving image
registration in Definition 2.11 . A discussion of different options for the data term
D is given in the next Sect. 2.1.1 and for the regularization term S in Sect. 2.1.2 .

2.1.1 Data Terms

The data term measures the dissimilarity, i.e., reversal of the similarity, of two
input images. In Eq. ( 2.1 ) the transformed template image is compared to the
reference image. For simplicity we will use the original template image (without
transformation) for the following definitions without loss of generality.