Despite being an ill-posed problem, non-rigid image registration has been the subject of numerous works, which apply the framework on different applications where rigid and affine transformations cannot completely model the variations between image sets. One such application of non-rigid registration is to register pre- and post-contrast breast MR images for estimating contrast uptake, which in turn is an indicator of the tumor malignancy. Due to large variations between the pre- and post-contrast images in terms of patient movement, and breast movement which is both global and local, registration of these images become challenging. Classic methods cannot capture the exact semantics of movements that the images exhibit.

To this end, the authors propose a non-rigid registration method which combines advantages of voxel-based similarity measures like Mutual Information (MI) as well as non-rigid transformation models of the breast. The method is built using a hierarchical transformation model which capture both the global and local movement of the breast across pre- and post-contrast scans. 

The proposed method consists of two interesting contributions which model the motion of the breast across scans using a global and local model. The global motion model conists of a 3D affine transformation parameterized by 12 degrees of freedom. 

The local model is based upon FFD model, which is based upon B-splines which is a powerful tool for modeling 3D deformable objects. The main idea behind this approach is that FFDs can be used to deform an object by manipulating the underlying mesh of the control points, which is estimated in the form of a B-spline. The formulation exposes a trade-off between computational running time and accurate modelling of the object (breast). In order to achieve the best compromise, a hierarchical multi-resolution approach is implemented in which the resolution of the control mesh is increased along with the image resolution  in a coarse to fine fashion. 

In addition to modeling the movement of the breast, a regularization term is also added to the final optimization function which forces the B-spline based FFD transformation to be smooth. The term is zero in case of affine transformation, and only penalizes non-affine transformations. 

The function that the method optimizes is as follows:

$\mathcal{C}(\theta, \phi) = - \mathcal{C}_{similarity}\left( I(t_0), T(I(t))\right) + \lambda \mathcal{C}_{smooth}(T)$

The optimization of the above objective function is performed in multiple stages, and by using gradient descent algorithm which takes steps towards the gradient vector with a certain step size of $\mu$. Local optimum is assumed if $||\nabla \mathcal{C}|| <= \epsilon$.

In order to assess the quality of the proposed method, the method is tested on both clinical and volunteer 3D MR data. The results show that the rigid and affine only transformations based methods perform significantly worse than the proposed method. Moreover, it was shown that the results improve with better control point resolution. However the use of SSD as a quantitative metric is debatable since the contrast enhanced and pre-contrast images will have varying distributions of intensity values.