Digital topology in brain image segmentation and registration

Topological concepts are critically important in the computation of anatomical representations, establishing correspondences between structures, and quantifying shape differences in medical image analysis. Early work in the preservation of an object’s topology to brain imaging involved generating digital reconstructions of the cerebral cortex, which is known to have a fixed topology. Such reconstructions facilitate the analysis and visualization of functional activity and allow group comparisons of cortical geometry in a standardized space. These applications are possible because topology preservation guarantees a mapping equivalence between specific shapes, such as between two cortical surfaces or between the cortex and a sphere. More recent work has shown that enforcing topology-preserving segmentations can maintain relationships between multiple objects and are robust to noise without biasing shape. In this chapter, we present the core principles of digital topology and describe their use in several fundamental tools for topology-preserving image analysis. We further demonstrate how these principles and tools can be applied to derive brain segmentation and registration algorithms that explicitly maintain multi-object topology. The advantage of these algorithms is that structures are reconstructed in a manner consistent with the underlying anatomy, thereby improving accuracy and readily enabling diffeomorphic shape analyses