Isotropic Total Variation Regularization of Displacements in Parametric Image Registration

Spatial regularization is essential in image registration, which is an ill-posed problem. Regularization can help to avoid both physically implausible displacement fields and local minima during optimization. Tikhonov regularization (squared $\ell _{2}$ -norm) is unable to correctly represent non-smooth displacement fields, that can, for example, occur at sliding interfaces in the thorax and abdomen in image time-series during respiration. In this paper, isotropic Total Variation (TV) regularization is used to enable accurate registration near such interfaces. We further develop the TV-regularization for parametric displacement fields and provide an efficient numerical solution scheme using the Alternating Directions Method of Multipliers (ADMM). The proposed method was successfully applied to four clinical databases which capture breathing motion, including CT lung and MR liver images. It provided accurate registration results for the whole volume. A key strength of our proposed method is that it does not depend on organ masks that are conventionally required by many algorithms to avoid errors at sliding interfaces. Furthermore, our method is robust to parameter selection, allowing the use of the same parameters for all tested databases. The average target registration error (TRE) of our method is superior (10% to 40%) to other techniques in the literature. It provides precise motion quantification and sliding detection with sub-pixel accuracy on the publicly available breathing motion databases (mean TREs of 0.95 mm for DIR 4D CT, 0.96 mm for DIR COPDgene, 0.91 mm for POPI databases).