Nonrigid image registration is a key tool in medical imaging. Because of high degrees of freedom in nonrigid transforms,
there have been many efforts to regularize the deformation based on some reasonable assumptions. Especially, motion
invertibility and local tissue rigidity have been investigated as reasonable priors in image registration. There have been
several papers on exploiting each constraint separately.
These constraints are reasonable in respiratory motion estimation because breathing motion is invertible and there are
some rigid structures such as bones. Using both constraints seems very attractive in respiratory motion registration since
using invertibility prior alone usually causes bone warping in ribs. Using rigidity prior seems natural and straightforward.
However, the "sliding effect" near the interface between rib cage and diaphragm makes problem harder because it is not
locally invertible. In this area, invertibility and rigidity priors have opposite forces.
Recently, we proposed a simple piecewise quadratic penalty that encourages the local invertibility of motions. In this
work we relax this penalty function by using a Geman-type function that allows the deformation to be piecewise smooth
instead of globally smooth. This allows the deformation to be discontinuous in the area of the interface between rib
cage and diaphragm. With some small sacrifice of regularity, we could achieve more realistic discontinuous motion near
diaphragm, better data fitting error as well as less bone warping. We applied this Geman-type function penalty only to the
x- and y-direction partial derivatives of the z-direction deformation to address the sliding effect. 192 × 128 × 128 3D CT
inhale and exhale images of a real patient were used to show the benefits of this new penalty method.
Motion artifacts are a significant issue in medical image reconstruction. There are many methods for incorporating motion
information into image reconstruction. However, there are fewer studies that focus on deformation regularization in motioncompensated
image reconstruction. The usual choice for deformation regularization has been penalty functions based on
the assumption that tissues are elastic. In the image registration field, there have been some methods proposed that impose
deformation invertibility using constraints or regularization, assuming that organ motions are invertible transformations.
However, most of these methods require very high memory or computation complexity, making them poorly suited for
dealing with multiple images simultaneously in motion-compensated image reconstruction. Recently we proposed an
image registration method that uses a simple penalty function based on a sufficient condition for the local invertibility of
deformations.<sup>1</sup> That approach encourages local invertibility in a fast and memory-efficient way. This paper investigates
the use of that regularization method for the more challenging problem of joint image reconstruction and nonrigid motion
estimation. A 2D PET simulation (based on realistic motion from real patient CT data) demonstrates the benefits of such
motion regularization for joint image reconstruction/registration.