In this paper, we propose a new curvature penalized minimal path model for image segmentation via closed contour detection based on the weighted Euler elastica curves, firstly introduced to the field of computer vision in . Our image segmentation method extracts a collection of curvature penalized minimal geodesics, concatenated to form a closed contour, by connecting a set of user-specified points. Globally optimal minimal paths can be computed by solving an Eikonal equation. This first order PDE is traditionally regarded as unable to penalize curvature, which is related to the path acceleration in active contour models. We introduce here a new approach that enables finding a global minimum of the geodesic energy including a curvature term. We achieve this through the use of a novel Finsler metric adding to the image domain the orientation as an extra space dimension. This metric is non-Riemannian and asymmetric, defined on an orientation lifted space, incorporating the curvature penalty in the geodesic energy. Experiments show that the proposed Finsler minimal path model indeed outperforms state-of-the-art minimal path models in both synthetic and real images.