Quasi - Conformally Flat Mapping the HumanCerebellumMonica

We present a novel approach to creating at maps of the brain. It is impossible to atten a curved surface in 3D space without metric and areal distortion; however, the Riemann Mapping Theorem implies that it is theoretically possible to preserve conformal (angular) information under attening. Our approach attempts to preserve the con-formal structure between the original cortical surface in 3-space and the attened surface. We demonstrate this with data from the human cere-bellum and we produce maps in the conventional Euclidean plane, as well as in the hyperbolic plane and on a sphere. Since our at maps exhibit quasi-conformal behavior, they ooer several advantages over existing approaches. In particular, conformal mappings are determined canonically, meaning that they are uniquely determined once certain normalizations have been chosen, and this allows one to impose a coordinate system on the surface when attening in the hyperbolic or spherical setting. Unlike existing methods, our approach does not require cuts to be introduced into the original surface. In addition, hyperbolic and spherical maps allow the map focus to be transformed interactively to correspond to any anatomical landmark and adjust the locations of map distortion.