Regularization in Deformable Registration of Biomedical Images Based on Divergence and Curl Operators

Background: Similarity measures in medical images do not uniquely determine the correspondence between two voxels in deformable image registration. Uncertainties in the final computed deformation exist, questioning the actual physiological consistency of the deformation between the two images. Objectives: We developed a deformable image registration method that regularizes the deformation field in order to model a deformation with physiological properties, relying on vector calculus based operators as a regularization function. Method: We implemented a 3D multi-resolution parametric deformable image registration, containing divergence and curl of the deformation field as regularization terms. Exploiting a BSpline model, we fit the transformation to optimize histogram-based mutual information similarity measure. In order to account for compression/expansion, we extract sink/source/circulation components as irregularities in the warped image and compensate them. The registration performance was evaluated using Jacobian determinant of the deformation field, inverse-consistency, landmark errors and residual image difference along with displacement field errors. Finally, we compare our results to a robust combination of second derivative regularization, as well as to non-regularized methods. Results: The implementation was tested on synthetic phantoms and clinical data, leading to increased image similarity and reduced inverse-consistency errors. The statistical analysis on clinical cases showed that regularized methods are able to achieve better image similarity than non regularized methods. Also, divergence/curl regularization improves anatomical landmark errors compared to second derivative regularization. Conclusion: The implemented divergence/curl regularization was successfully tested, leading to promising results in comparison with competitive regularization methods. Future work is required to establish parameter tuning and reduce the computational cost.