The deformable surface approach when applied to brain images is illustrated in Fig. 2, right. Here we can again see the difficulty of pushing the surface through the many narrow openings into deep sulci, while obtaining a smooth surface at the end of the procedure. In order to resolve this problem, we had the insight that while driving a spherical model to lie at each point of the cortical surface was extremely difficult, the opposite approach, that of taking a cortical surface, regardless of its topology, and driving it outwards towards the surface of a sphere, is relatively easy (Fischl et al., 2001). The fundamental notion of topological equivalence states that if the original surface is not topologically equivalent to a sphere, that is if it contains topological defects, then it is not possible to find a continuous, one-to-one and invertible (i.e. homeomorphic) mapping. More interestingly, we realized that the regions in which the mapping was many-to-one were precisely those that contained topological defects. This enabled us to localize the defects, and limit the topology correction to what is typically only a small fraction of the surface and thus allowed us to focus on geometric accuracy everywhere else.
