Elastic image registration with strong mass preserving constraints

We combine the theories of elasticity and optimal mass transportation to construct and numerically solve a constrained optimization problem for matching two different density distributions. The resulting problem is interpreted as an optimal mass transport for continuous media and can be successfully used in tasks of medical image registration, provided the input images describe density of mass (MR or PET images). When using our model, the image densities are aligned in the sense of the pull-back equation and the resulting transformation is guaranteed to be invertible. This work provides an extensive study for the constructed constrained optimization problem. A theoretical analysis concerning the existence of optimal states and the corresponding Lagrange multipliers is conducted and links to related problems in the field of image registration are provided as well. Furthermore, two different numerical methods based on staggered-grid finite difference and linear finite element discretizations are constructed and several experiments are performed.