Motion Estimation - Motion Correction in Thoracic Positron Emission Tomography - page 41

2.1.5.2 BFGS and l-BFGS

The BFGS (named after Broyden, Fletcher, Goldfarb, and Shanno) method is a
quasi-Newton method, which means that the Hessian matrix is estimated at runtime.
The Hessian matrix is approximated from function and gradient values. So only first
derivatives are necessary for optimization [ 100 ].
The l in l-BFGS stands for limited memory [ 85 , 95 ]. In l-BFGS the Hessian
matrix is never built or stored explicitly. Vectors representing the Hessian matrix
implicitly are used instead. These implementations are particularly appropriate for
large problems.

2.1.5.3 Trust Region

In trust region optimization the objective function is treated only in a small (trusted)
region where it is approximated by a quadratic model. This is a somehow reverse
approach to line search methods. In trust region methods the step size is chosen
first (in terms of the region size) and the search direction is computed subsequently.
The proceeding in line search methods is vice versa: given a search direction, an
appropriate step size is searched.

2.1.6 Results

Software phantoms like the XCAT phantom [ 123 ] are helpful to validate the
proposed motion estimation algorithms as they provide morphological (anatomy)
and functional (physiology) ground-truth information. The XCAT phantom is
popular in emission tomography for human anatomy as it provides a model of the
subject's anatomy and physiology. It is possible to include respiratory and cardiac
motion resulting in 4D/5D images. In addition to the generation of PET images and
attenuation maps, the XCAT phantom provides information about the ground-truth
motion being used to generate the different motion states. This motion information
can thus be used to quantitatively evaluate the estimated motion fields of the motion
estimation approaches. Some results based on the ground-truth motion are given in
Sect. 2.3 .
For the evaluation in the following, a 5-by-5 dual gating data set is generated with
the XCAT software phantom. The maximum diaphragm motion is set to 20 mm
and the dual gates cover the whole range of the breathing and cardiac cycle. The
total size of each volume is 175

375 mm in
each dimension, simulating the Siemens Biograph TM Sensation 16 PET/CT scanner
(Siemens Medical Solution).
The XCAT phantom data is trimmed to be comparable to real patient data in
terms of tracer distribution, noise, and the scanning system, cf. [ 51 ]. To simulate
nearly realistic PET images the following steps are performed:

× 175 × 47 with a voxel size of 3 .