Spherical Flattening of the Cortex Surface

Abstract

We present a novel technique to ‘unfold’ the curved convoluted outer surface of the brain known as the cortex and map it onto a sphere. The mapping procedure is constructed by first measuring the inter geodesic distances between points on the cortical surface. Next, a multi-dimensional scaling (MDS) technique is applied to map the whole or a section of the surface onto the sphere. The geodesic distances on the cortex are measured by the ‘fast marching on triangulated domains’ algorithm. It calculates the geodesic distances from a vertex on a triangulated surface to the rest of the vertices in O(n) operations, where n is the number of vertices that represent the surface. Using this procedure, a matrix of the geodesic distances between every two vertices on the surface is computed. Next, a constrained MDS procedure finds the coordinates of points on a sphere such that the inter geodesic distances between points on the sphere are as close as possible to the geodesic distances measured between the corresponding points on the cortex. Thereby, our approach maximizes the goodness of fit of distances on the cortex surface to distances on the sphere. We apply our algorithm to sections of the human cortex, which is an extremely complex folded surface.