In this paper, we introduce a novel interactive image segmentation method based on the geodesic voting formulation. In contrast to the classical geodesic voting model which utilizes the geodesics only depending on the path position, the proposed model also takes into account the image edge anisotropic and asymmetric features by adapting the minimal paths relying on the asymmetric quadratic metric for geodesic voting. Furthermore, an adaptive cut-based closed contour computation scheme is invoked to depict the target boundary, by tracing two asymmetric minimal paths from a source point located at the adaptive cut to an end point along two opposite sides of the cut. The proposed asymmetric geodesic voting model is then applied to get the complex structure segmentation, benefiting from the asymmetric features and cut-convexity constraint. Experimental results show that the proposed model indeed outperforms state-of-the-art minimal paths-based image segmentation approaches.