How do L2 norm and Jacobian act as a regularisation term to encourage smoothness in a deformation field?

Question

How do L2 norm and The Jacobian act as a regularisation term to encourage smoothness in a deformation field? from the VoxelMorph original paper (here) they used Jacobian as a means to smoothen the deformation field, a similar paper (here) made use of L2 norm to smoothen the deformation field. However, I don't understand how this is achieved using either L2 norm or Jacobian. Thanks in advance.

Answer 1

TL,DR Use the Jacobian to get smoothness and the L2 norm to limit the magnitude + they could be complementary.

I just skimmed at the paper (that you haven't really referenced) and the regularization term using the derivatives makes sense to make $\phi$ smooth. It's basically penalizing large changes between neighboring values of the registration field. Using the L2 norm would penalize the magnitude of the registration field, not the local variations.