Interpolation between Images by Constrained Optimal Transport

In this paper, the recently proposed technique of constrained optimal transport is used to interpolate between images under specified constraints. The intensity values in both images are considered as mass distributions, and a flow of minimum kinetic energy is computed to transport the initial distribution to the final one, while satisfying specified constraints on the intermediate mass as well as the the velocity or the momentum field. As an application, the proposed method is used for interpolating between images with a common unchanged part, as well as under constraint on the volume expansion and contraction. The latter is achieved by imposing bounds on the divergence of the velocity field of the flow. This constraint is discretized then integrated into the problem Lagrangian using the augmented Lagrangian method. A variation of the solution is also presented, where the constraint is decoupled into two constraints coordinated by an additional Lagrange multiplier. This allows a considerable speedup, though numerical robustness decreases in certain cases. Constrained optimal transport can potentially be used in image registration. In particular, the proposed method for controlling the volume change has potential application in registration of images under volume change constraints as it is the case for medical images depicting muscle movements or those with contrast enhancing structures.