Image registration has several applications, amongst others in computer vision and medical imaging. The goal is to align different images, e.g., measured at different times or by different sensors. By parameterizing the transformations to align the images via velocity fields, we consider image registration as an optimal control problem using an optical flow formulation. More precisely, we aim to solve an optimization problem that is governed by a linear hyperbolic transport equation. In order to be able to obtain non-smooth features and ensure bi-Lipschitz continuity of the transformation, we aim for Lipschitz regularity of the velocity that parametrizes the transformation. This can be realized via adding a non-differentiable regularization term or including additional inequality constraints. In order to avoid this, we introduce relaxations of the optimization problem involving smoothed maximum and minimum functions. We discuss the choice of the relaxation parameter, challenges for the numerical implementation, and present first numerical results for the proposed approach. Moreover, we address numerical artifacts include zigzagging or unresolved edges when transforming a mesh with the optimized transformation.