A proof of the spherical homeomorphism conjecture for surfaces

Abstract—The human cerebral cortex is topologically equivalent to a sphere when it is viewed as closed at the brain stem. Due to noise and/or res-olution issues, magnetic resonance imaging may see “handles ” that need to be eliminated to reflect the true spherical topology. Shattuck and Leahy [2] present an algorithm to correct such an image. The basis for their correc-tion strategy is a conjecture, which they call the spherical homeomorphism conjecture, stating that the boundary between the foreground region and the background region is topologically spherical if certain associated fore-ground and background multigraphs are both graph-theoretic trees. In this paper, we prove the conjecture, and its converse, under the assumption that the foreground/background boundary is a surface. Index Terms—Magnetic resonance imaging, segmentation, topology, topological correction. I