Motion Estimation - Motion Correction in Thoracic Positron Emission Tomography - page 42

1. Blurring with a Gaussian kernel (simulation of the PVE),
2. Forward projection,
3. Simulation of Poisson noise, and
4. Reconstruction.

The Gaussian kernel is chosen with a FWHM of 3 85 mm according to the simulated

scanner [ 51 ]. The blurred images are forward projected into measurement space,
where Poisson noise is simulated. The amount of noise is adjusted approximately
equal to real patient data. In a final step, the sinograms are reconstructed with the
EM RECON software [ 78 ]. It should be mentioned that the image acquisition process
could alternatively be simulated using the simulation tool GATE [ 69 ].
We applied the proposed V AMPIRE approach, once with the SSD and once with
the SAD distance measure, to the XCAT data. The results of the SSD and SAD
V AMPIRE approach are visualized in Figs. 2.7 respectively 2.8 for one representative
slice. The gate in maximum inspiration and systole, shown in each case in Fig. 2.7 a,
is chosen as the template image

.

T , as it is the most different gate compared to

the reference image in maximum expiration and diastole, shown in Fig. 2.7 b.

Additionally, the transformed image according to the estimated transformation, i.e.,
M

R

MP MP

, and a vector plot are given in Fig. 2.7 c, d. Some
quantitative numbers based on the ground-truth transformation are further given at
the end of Sect. 2.3 for the V AMPIRE results and the Mass-Preserving Optical Flow
(MPOF) approach described in the next section.

( T , y SSD ) and M ( T , y SAD )

2.2 Optical Flow

The motion between two images can not only be estimated by means of image
registration. A common alternative is the so-called optical flow . The objective is
indeed the same, but the problem formulation and implementation are different.
Optical flow algorithms are used to estimate a dense flow field between two
images/volumes.
In this section we will first derive the basic equation of optical flow. It will
directly lead us to the early works on this topic: the local approach of Lucas and
Kanade in Sect. 2.2.1 and the global approach of Horn and Schunck in Sect. 2.2.2 .
Advanced optical flow methods will be discussed in Sect. 2.2.3 . A mass-preserving
optical flow algorithm will be presented in Sect. 2.2.4 . After a brief discussion
of a multi-level implementation in Sect. 2.2.5 we finally present some results in
Sect. 2.2.6 .
The basic idea behind optical flow estimation is that the intensity of a voxel

x T

3 remains constant over time. While changing its spatial position,
the intensity of a moving point does not change. This constraint is called brightness
constancy constraint and can be formulated for 3D volumes as

=( x , y , z ) ∈ R

I ( x , y , z , t )= I ( x + u , y + v , z + w , t + 1 ) , (2.53)