Discontinuity preserving image registration for breathing induced sliding organ motion

Image registration is a powerful tool in medical image analysis and facilitates
the clinical routine in several aspects. It became an indispensable device for
many medical applications including image-guided therapy systems. The
basic goal of image registration is to spatially align two images that show a
similar region of interest. More speci�cally, a displacement �eld respectively
a transformation is estimated, that relates the positions of the pixels or
feature points in one image to the corresponding positions in the other one.
The so gained alignment of the images assists the doctor in comparing and
diagnosing them. There exist di�erent kinds of image registration methods,
those which are capable to estimate a rigid transformation or more generally
an a�ne transformation between the images and those which are able to
capture a more complex motion by estimating a non-rigid transformation.
There are many well established non-rigid registration methods, but those
which are able to preserve discontinuities in the displacement �eld are rather
rare. These discontinuities appear in particular at organ boundaries during
the breathing induced organ motion.
In this thesis, we make use of the idea to combine motion segmentation
with registration to tackle the problem of preserving the discontinuities in
the resulting displacement �eld. We introduce a binary function to represent
the motion segmentation and the proposed discontinuity preserving
non-rigid registration method is then formulated in a variational framework.
Thus, an energy functional is de�ned and its minimisation with respect to
the displacement �eld and the motion segmentation will lead to the desired
result. In theory, one can prove that for the motion segmentation a global
minimiser of the energy functional can be found, if the displacement �eld
is given. The overall minimisation problem, however, is non-convex and a
suitable optimisation strategy has to be considered. Furthermore, depending
on whether we use the pure L1-norm or an approximation of it in the formulation
of the energy functional, we use di�erent numerical methods to solve
the minimisation problem. More speci�cally, when using an approximation
of the L1-norm, the minimisation of the energy functional with respect to the displacement �eld is performed through Brox et al.'s �xed point iteration
scheme, and the minimisation with respect to the motion segmentation
with the dual algorithm of Chambolle. On the other hand, when we make
use of the pure L1-norm in the energy functional, the primal-dual algorithm
of Chambolle and Pock is used for both, the minimisation with respect to
the displacement �eld and the motion segmentation. This approach is clearly
faster compared to the one using the approximation of the L1-norm and also
theoretically more appealing. Finally, to support the registration method
during the minimisation process, we incorporate additionally in a later approach
the information of certain landmark positions into the formulation of
the energy functional, that makes use of the pure L1-norm. Similarly as before,
the primal-dual algorithm of Chambolle and Pock is then used for both,
the minimisation with respect to the displacement �eld and the motion segmentation.
All the proposed non-rigid discontinuity preserving registration
methods delivered promising results for experiments with synthetic images
and real MR images of breathing induced liver motion.