Well-posedness of an optical flow based optimal control formulation for image registration
Johannes Haubner, Christian Clason
TL;DR
The paper addresses image registration by formulating it as an optimal control problem governed by a linear hyperbolic transport equation. It develops a rigorous framework for well-posedness in a non-reflexive Banach setting by relaxing the $L^{\infty}$-norm regularization through smoothed max/min (via entropy-based Orlicz spaces) and proves existence, uniqueness, and stability results for the relaxed problem and the transport equation. The authors establish a discretization strategy, prove convergence as the discretization is refined and as the relaxation parameter vanishes, and show that the relaxed problem approximates the original $W^{1,\infty}$-regularized formulation. The work provides theoretical guarantees and practical guidance for reliable image registration with Lipschitz regularity, supported by a detailed treatment of Orlicz spaces and the transport dynamics.
Abstract
We consider image registration as an optimal control problem using an optical flow formulation, i.e., we discuss an optimization problem that is governed by a linear hyperbolic transport equation. Requiring Lipschitz continuity of the vector fields that parametrize the transformation leads to an optimization problem in a non-reflexive Banach space. We introduce relaxations of the optimization problem involving smoothed maximum and minimum functions and appropriate Orlicz spaces. To derive well-posedness results for the relaxed optimization problem, we revisit and establish new existence and uniqueness results for the linear hyperbolic transport equations. We further discuss limit considerations with respect to the relaxation parameter and discretizations.