Quasi-Conformally Flat Mapping the Human Cerebellum

Abstract

Abstract. We present a novel approach to creating flat maps of the brain. It is impossible to flatten 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 flattening. Our approach attempts to preserve the conformal structure between the original cortical surface in 3-space and the flattened surface. We demonstrate this with data from the human cerebellum and we produce maps in the conventional Euclidean plane, as well as in the hyperbolic plane and on a sphere. Conformal mappings are uniquely determined once certain normalizations have been chosen, and this allows one to impose a coordinate system on the surface when flattening in the hyperbolic or spherical setting. Unlike existing methods, our approach does not require that cuts be introduced in the original surface. In addition, hyperbolic and spherical maps allow the map focus to be transformed interactively to correspond to any anatomical landmark.

Mesh Parameterization Methods and Their Applications

“…Hurdal et al [56] and Bowers and Hurdal [15] compute anglepreserving parameterizations using a variation of the circle-packing formulation described later by Collins and Stephenson [23]. They observe that, in the circle-packing setting, angle preservation corresponds to preservation of specified distances between circles.…”

A Tool for Topographic Analysis of Electrode Contacts in Human Cortical Stimulation

“…Because of the homeomorphic property, we applied the Circle Packing algorithm [6,10] The circle packing algorithm is an iterative method that minimizes angular distortions and produces a bijective map. A circle is first associated to each vertex of the original mesh, with a radius value arbitrarily fixed.…”

“…Haker et al [9] used differential geometry to produce a quasi-conformal bijective mapping of surfaces homeomorphic to a ball. However in our application, cortical surface is not homeomorphic to a ball, first because we are only working on a small part of the brain, second because data we are using are MRI: in this context, gaps between two parts of the cortex may be less than the voxel size, producing a connection on the Favreau [10] computed a bijective flattening of a cerebellum using 27 MRI acquisitions from a single subject. In clinical routine it is not realistic to obtain so many images and the method should be automatic.…”

Quasi-conformal Flat Representation of Triangulated Surfaces for Computerized Tomography

“…This means that there is a point of the brain (actually a neighborhood of a point), which will not map conformally to the plane, and in this neighborhood the dilatation will be infinitely large. Hurdal et al solve this problem by removing the corpus callosum, thus obtaining a surface homeomorphic to a 1-punctured sphere, and thus conformally equivalent to a disk ( [10], [11]). An additional problem arises due to the necessary assumption that the surface triangulation is homogeneous in the sense that all triangles are equilateral.…”