  Diffeomorphic registration is a widely used technique for finding a smooth
and invertible transformation between two coordinate systems, which are
measured using template and reference images. The point-wise volume-preserving
constraint $\det(\nabla\bm{\varphi}(\bm{x})) =1$ is effective in some cases,
but may be too restrictive in others, especially when local deformations are
relatively large. This can result in poor matching when enforcing large local
deformations. In this paper, we propose a new bi-variant diffeomorphic image
registration model that introduces a soft constraint on the Jacobian equation
$\det(\nabla\bm{\varphi}(\bm{x})) = f(\bm{x}) > 0$. This allows local
deformations to shrink and grow within a flexible range
$0<\kappa_{m}<\det(\nabla\bm{\varphi}(\bm{x}))<\kappa_{M}$. The Jacobian
determinant of transformation is explicitly controlled by optimizing the
relaxation function $f(\bm{x})$. To prevent deformation folding and improve the
smoothness of the transformation, a positive constraint is imposed on the
optimization of the relaxation function $f(\bm{x})$, and a regularizer is used
to ensure the smoothness of $f(\bm{x})$. Furthermore, the positivity constraint
ensures that $f(\bm{x})$ is as close to one as possible, which helps to achieve
a volume-preserving transformation on average. We also analyze the existence of
the minimizer for the variational model and propose a penalty-splitting
algorithm with a multilevel strategy to solve this model. Numerical experiments
demonstrate the convergence of the proposed algorithm and show that the
positivity constraint can effectively control the range of relative volume
without compromising the accuracy of the registration. Moreover, the proposed
model generates diffeomorphic maps for large local deformations and outperforms
several existing registration models in terms of performance.
