Well-posedness of an optical flow based optimal control formulation for image registration

TL;DR

This paper investigates the mathematical well-posedness of an image registration method based on optical flow, formulating it as an optimal control problem governed by hyperbolic transport equations, and introduces relaxations for better analysis.

Contribution

It introduces relaxations using smoothed functions and Orlicz spaces to establish well-posedness for the optical flow-based registration model.

Findings

Existence and uniqueness results for the transport equations.

Well-posedness of the relaxed optimization problem.

Analysis of limit behavior with respect to relaxations and discretizations.

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.