A Bi-variant Variational Model for Diffeomorphic Image Registration with Relaxed Jacobian Determinant Constraints

Diffeomorphic registration has become a powerful approach for seeking a smooth and invertible spatial transformation between two coordinate systems which have been measured via the template and reference images. While the pointwise volume-preserving constraint is effective for some problems, it is too stringent for many other problems especially when the local deformations are relatively large, because it may lead to a poor large-deformation for enforcing local matching.In this paper, we propose a novel bi-variant diffeomorphic image registration model with the soft constraint of Jacobian equation, which allows local deformations to shrink and grow in a flexible range.The Jacobian determinant of the transformation is explicitly controlled by optimizing the relaxation function. To prevent deformation folding and enhance the smoothness of deformation, we not only impose a positivity constraint in optimizing the relaxation function, but also employ a regularizer to ensure the smoothness of the relaxation function.Furthermore, the positivity constraint ensures that is as close to one as possible, which helps to obtain a volume-preserving transformation on average.We further analyze the existence of the minimizer for the variational model and propose a penalty splitting method with a multilevel strategy to solve this model. Numerical experiments show that the proposed algorithm is convergent, and the positivity constraint can control the range of relative volume and not compromise registration accuracy. Moreover, the proposed model produces diffeomorphic maps for large deformation, and achieves better performance compared to the several existing registration models.