A simple penalty that encourages local invertibility and considers sliding effects for respiratory motion

摘要

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, motioninvertibility and local tissue rigidity have been investigated as reasonable priors in image registration. There have beenseveral papers on exploiting each constraint separately.These constraints are reasonable in respiratory motion estimation because breathing motion is invertible and there aresome rigid structures such as bones. Using both constraints seems very attractive in respiratory motion registration sinceusing 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 notlocally 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 thiswork we relax this penalty function by using a Geman-type function that allows the deformation to be piecewise smoothinstead of globally smooth. This allows the deformation to be discontinuous in the area of the interface between ribcage and diaphragm. With some small sacrifice of regularity, we could achieve more realistic discontinuous motion neardiaphragm, better data fitting error as well as less bone warping. We applied this Geman-type function penalty only to thex- and y-direction partial derivatives of the z-direction deformation to address the sliding effect. 192 × 128 × 128 3D CTinhale and exhale images of a real patient were used to show the benefits of this new penalty method.