We present SurfCut, an algorithm for extracting a smooth simple surface with unknown boundary from a noisy 3D image and a seed point. In contrast to existing approaches that extract smooth simple surfaces with boundary, our method requires less user input, i.e., a seed point, rather than a 3D boundary curve. Our method is built on the novel observation that certain ridge curves of a front propagated using the Fast Marching algorithm are likely to lie on the surface. Using the framework of cubical complexes, we design a novel algorithm to robustly extract such ridge curves and form the surface of interest. Our algorithm automatically cuts these ridge curves to form the surface boundary, and then extracts the surface. Experiments show the robustness of our method to errors in the data, and that we achieve higher accuracy with lower computational cost than comparable methods. © Springer International Publishing AG 2016.