Locally constrained cubic B-spline deformations to control volume variations

It is implicitly assumed in most non-rigid registration methods that either corresponding structures can be found in both images, or that the regularization associated with the class of transformations chosen produce pertinent deformations elsewhere. However, when the images to register have a different contrast and resolution (e.g. in vivo / post mortem), or to minimize the deformation of some specific tissues (e.g. tumor, bones), it is necessary to have local control on the displacement field in these regions. We propose in this work an original registration method based on cubic B-spline transformations which allows to bound the variation of volume induced by the estimated transformation (variation of the determinant of the Jacobian of the transformation relatively to 1) in a predefined region. Lagrange multipliers are used to perform the constrained minimization of mutual information. We tested our method on artificial images with different tolerances on the volume variation and on one real 3D image: it performed efficiently while respecting the constraints.