Marching cubes method with connectivity

In this paper, we solve the topological problem of isosurfaces generated by the marching cubes method using the approach of combinatorial topology. For each marching cube, we examine the connectivity of polyhedral configuration in the sense of combinatorial topology. For the cubes where the connectivities are not considered, we modify the polyhedral configurations with the connectivity and construct polyhedral isosurfaces with the correct topologies.